Research ArticleThe Strict AKNS Hierarchy Its Structure and Solutions

G F Helminck

Korteweg-de Vries Institute University of Amsterdam PO Box 94248 1090 GE Amsterdam Netherlands

Correspondence should be addressed to G F Helminck gfhelminckuvanl

Received 4 April 2016 Accepted 2 August 2016

Academic Editor Boris G Konopelchenko

Copyright copy 2016 G F Helminck This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We discuss an integrable hierarchy of compatible Lax equations that is obtained by a wider deformation of a commutative algebrain the loop space of sl

2than that in the AKNS case and whose Lax equations are based on a different decomposition of this loop

space We show the compatibility of these Lax equations and that they are equivalent to a set of zero curvature relations We presenta linearization of the system and conclude by giving a wide construction of solutions of this hierarchy

1 Introduction

Integrable hierarchies often occur as the evolution equationsof the generators of a deformation of a commutative subal-gebra inside some Lie algebra g The deformation and theevolution equations are determined by a splitting of g inthe direct sum of two Lie subalgebras like in the Adler-Kostant-Symes Theorem (see [1]) This gives then rise to acompatible set of Lax equations a so-called hierarchy andthe simplest nontrivial equation often determines the nameof the hierarchy The hierarchy we discuss here correspondsto a somewhat different splitting of an algebra of sl

2loops

that with respect to the usual decomposition yields theAKNS hierarchy Following the terminology used in similarsituations (see [2]) we use the name strictAKNShierarchy forthe system of Lax equations corresponding to this differentdecomposition

The letters AKNS refer to the following fact Ablowitzet al showed (see [3]) that the initial value problem of thefollowing system of equations for two complex functions 119902and 119903 depending on the variables 119909 and 119905

119894120597

120597119905119902 (119909 119905) fl 119894119902

119905= minus

1

2119902119909119909+ 1199022119903

119894120597

120597119905119903 (119909 119905) fl 119894119903

119905=1

2119903119909119909minus 1199021199032

(1)

could be solved with the Inverse Scattering Transform andthat is why this system (1) is called the AKNS equations It

can be rewritten in a zero curvature form for 2 times 2-matricesas follows consider the matrices

1198751= (

0 119902

119903 0)

1198752= (

minus119894

2119902119903

119894

2119902119909

minus119894

2119903119909

119894

2119902119903

)

(2)

A direct computation shows that the following identities holdfor them

120597

120597119909(1198752) = (

119894

2(119902119909119903 + 119902119903

119909)

119894

2119902119909119909

minus119894

2119903119909119909

minus119894

2(119902119909119903 + 119902119903

119909)

)

[1198752 1198751] = (

119894

2(119902119909119903 + 119902119903

119909) 119894119902

2119903

minus1198941199021199032

minus119894

2(119902119909119903 + 119902119903

119909)

)

(3)

By combining them one sees that the AKNS equations areequivalent to

120597

120597119905(1198751) minus

120597

120597119909(1198752) minus [119875

2 1198751] = 0 (4)

In the next section we will show that (4) is the simplestcompatibility condition of a system of Lax equations that

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 3649205 10 pageshttpdxdoiorg10115520163649205

2 Advances in Mathematical Physics

consequently carries the name AKNS hierarchy (see [4])Besides that we introduce here also another system ofLax equations for a more general deformation of the basicgenerator of the commutative algebra It relates to a differentdecomposition of the relevant Lie algebra Since its analoguefor theKP hierarchy is called the strict KP hierarchy we use inthe present context the term strict AKNS hierarchy We showfirst of all that this new system is compatible and equivalentwith a set of zero curvature relations Further we describesuitable linearizations that can be used to construct solutionsof both hierarchiesWe conclude by constructing solutions ofboth hierarchies

2 The AKNS Hierarchy and Its Strict Version

We present here an algebraic description of the AKNS hier-archy and its strict version that underlines the deformationcharacter of these hierarchies as pointed at in IntroductionAt(4) one has to deal with 2times2-matriceswhose coefficientswerepolynomial expressions of a number of complex functionsand their derivatives with respect to some parameters Thesame is true for the other equations in both hierarchies Weformalize this as follows our starting point is a commutativecomplex algebra 119877 that should be seen as the source fromwhich the coefficients of the 2 times 2-matrices are taken Wewill work in the Lie algebra sl

2(119877)[119911 119911

minus1) consisting of all

elements

119883 =

119873

sum

119894=minusinfin

119883119894119911119894 with all 119883

119894isin sl2(119877) (5)

and the bracket

[119883 119884] = [

[

119873

sum

119894=minusinfin

119883119894119911119894

119872

sum

119895=minusinfin

119884119895119911119895]

]

fl119873

sum

119894=minusinfin

119872

sum

119895=minusinfin

[119883119894 119884119895] 119911119894+119895

(6)

We will also make use of the slightly more general Lie algebragl2(119877)[119911 119911

minus1) where the coefficients in the 119911-series from (5)

are taken fromgl2(119877) instead of sl

2(119877) and the bracket is given

by the same formula In the Lie algebra gl2(119877)[119911 119911

minus1) we

decompose elements in two ways The first is as follows

119883 =

119873

sum

119894=minusinfin

119883119894119911119894=

119873

sum

119894=0

119883119894119911119894+

minus1

sum

119894=minusinfin

119883119894119911119894 š (119883)

⩾0+ (119883)

lt0(7)

and this induces the splitting

gl2(119877) [119911 119911

minus1) = gl

2(119877) [119911 119911

minus1)⩾0

oplus gl2(119877) [119911 119911

minus1)lt0

(8)

where the two Lie subalgebras gl2(119877)[119911 119911

minus1)⩾0

and gl2(119877)[119911

119911minus1)lt0

are given by

gl2(119877) [119911 119911

minus1)⩾0

= 119883 isin gl2(119877) [119911 119911

minus1) | 119883 = (119883)

⩾0=

119873

sum

119894=0

119883119894119911119894

gl2(119877) [119911 119911

minus1)lt0

= 119883 isin gl2(119877) [119911 119911

minus1) | 119883 = (119883)

lt0= sum

119894lt0

119883119894119911119894

(9)

By restriction it leads to a similar decomposition forsl2(119877)[119911 119911

minus1) which is relevant for the AKNS hierarchy The

second way to decompose elements of gl2(119877)[119911 119911

minus1) is

119883 =

119873

sum

119894=minusinfin

119883119894119911119894=

119873

sum

119894=1

119883119894119911119894+

0

sum

119894=minusinfin

119883119894119911119894

š (119883)gt0+ (119883)

⩽0

(10)

This yields the splitting

gl2(119877) [119911 119911

minus1) = gl

2(119877) [119911 119911

minus1)gt0

oplus gl2(119877) [119911 119911

minus1)⩽0

(11)

By restricting it to sl2(119877)[119911 119911

minus1) we get a similar decom-

position for this Lie algebra which relates as we will seefurther on to the strict version of the AKNS hierarchy TheLie subalgebras gl

2(119877)[119911 119911

minus1)gt0

and gl2(119877)[119911 119911

minus1)⩽0

in (11)are defined in a similar way as the first two Lie subalgebras

gl2(119877) [119911 119911

minus1)gt0

= 119883 isin gl2(119877) [119911 119911

minus1) | 119883 = (119883)

gt0=

119873

sum

119894=1

119883119894119911119894

gl2(119877) [119911 119911

minus1)⩽0

= 119883 isin gl2(119877) [119911 119911

minus1) | 119883 = (119883)

⩽0= sum

119894⩽0

119883119894119911119894

(12)

Both hierarchies correspond to evolution equations fordeformations of the generators of a commutative Lie algebrain the first component Denote thematrix ( minus119894 0

0 119894) by119876

0 Inside

sl2(119877)[119911 119911

minus1)⩾0 this commutative complex Lie subalgebra119862

0

is chosen to be the Lie subalgebra with the basis 1198760119911119898| 119898 ⩾

0 and in sl2(119877)[119911 119911

minus1)gt0our choice will be the Lie subalgebra

1198621with the basis 119876

0119911119898| 119898 ⩾ 1 In the first case that we

work with 1198620 we assume that the algebra 119877 possesses a set

120597119898| 119898 ⩾ 0 of commutingC-linear derivations 120597

119898 119877 rarr 119877

where each 120597119898should be seen as the derivation corresponding

to the flow generated by 1198760119911119898 The data (119877 120597

119898| 119898 ⩾ 0) is

also called a setting for the AKNS hierarchy In the case thatwe work with the decomposition (11) we merely need a set ofcommutingC-linear derivations 120597

119898 119877 rarr 119877119898 ⩾ 1 with the

same interpretation Staying in the same line of terminologywe call the data (119877 120597

119898| 119898 ⩾ 1) a setting for the strict AKNS

hierarchy

Example 1 Examples of settings for the respective hierarchiesare the algebras of complex polynomials C[119905

119898] in the vari-

ables 119905119898| 119898 ⩾ 0 and 119905

119898| 119898 ⩾ 1 respectively or the

Advances in Mathematical Physics 3

formal power series C[[119905119898]] in the same variables with both

algebras equipped with the derivations 120597119898= 120597120597119905

119898 119898 ⩾ 0

in the AKNS setup or 120597119898= 120597120597119905

119898119898 ⩾ 1 in the strict AKNS

case

We let each derivation 120597119898occurring in some setting act

coefficient-wise on the matrices from gl2(119877) and that defines

then a derivation of this algebra The same holds for theextension to gl

2(119877)[119911 119911

minus1) defined by

120597119898(119883) fl

119873

sum

119895=minusinfin

120597119898(119883119895) 119911119895 (13)

We have now sufficient ingredients to discuss the AKNShierarchy and its strict version In the AKNS case our interestis in certain deformations of the Lie algebra 119862

0obtained

by conjugating with elements of the group corresponding togl2(119877)[119911 119911

minus1)lt0 At the strict version we are interested in

certain deformations of the Lie algebra1198621obtained by conju-

gating with elements from a group linked to gl2(119877)[119911 119911

minus1)⩽0

Note that for each 119883 isin gl2(119877)[119911 119911

minus1)lt0

the exponentialmap yields a well-defined element of the form

exp (119883) =infin

sum

119896=0

1

119896119883119896= Id + 119884 119884 isin gl

2(119877) [119911 119911

minus1)lt0

(14)

and with the formula for the logarithm one retrieves 119883 backfrom 119884 One verifies directly that the elements of the form(14) form a group with respect to multiplication and this wesee as the group 119866

lt0corresponding to gl

2(119877)[119911 119911

minus1)lt0

In the case of the Lie subalgebra gl2(119877)[119911 119911

minus1)⩽0 one

cannot move back and forth between the Lie algebra and itsgroup Nevertheless one can assign a proper group to thisLie algebra A priori the exponential exp(119884) of an element119884 isin gl

2(119877)[119911 119911

minus1)⩽0

does not have to define an element ingl2(119877)[119911 119911

minus1)⩽0 That requires convergence conditions How-

ever if it does then it belongs to

119866⩽0=

119870 =

infin

sum

119895=0

119870119895119911minus119895| all 119870

119895isin gl2(119877) 119870

0

isin gl2(119877)lowast

(15)

where gl2(119877)lowast denotes the elements in gl

2(119877) that have a

multiplicative inverse in gl2(119877) It is a direct verification that

119866⩽0

is a group and we see it as a proper group correspondingto the Lie algebra gl

2(119877)[119911 119911

minus1)⩽0 In fact 119866

⩽0is isomorphic

to the semidirect product of 119866lt0

and gl2(119877)lowast

Now there holds the following

Lemma 2 The group 119866⩽0

acts by conjugation on sl2(119877)[119911

119911minus1)

Proof Take first any 119892 isin 119866lt0 Then there is an119883 isin gl

2(119877)[119911

119911minus1)lt0

such that 119892 = exp(119883) Now there holds for every 119884 isinsl2(119877)[119911 119911

minus1)

119892119884119892minus1= exp (119883) 119884 exp (minus119883) = 119890ad(119883) (119884)

= 119884 +

infin

sum

119896=1

1

119896ad (119883)119896 (119884)

(16)

and this shows that the coefficients for the different powersof 119911 in this expression are commutators of elements of gl

2(119877)

and sl2(119877) and that proves the claim for elements from 119866

lt0

Since conjugation with an element from gl2(119877)lowast maps sl

2(119877)

to itself the same holds for sl2(119877)[119911 119911

minus1) This proves the full

claim

Next we have a look at the different deformations Thedeformations of119862

0by elements of119866

lt0are determined by that

of1198760Therefore we focus on that element andwe consider for

some 119892 = exp(119883) = exp(suminfin119895=1119883119895119911minus119895) isin 119866lt0

the deformation

119876 = 1198921198760119892minus1 fl

infin

sum

119894=0

119876119894119911minus119894

= (minus119894 0

0 119894) + [119883

1 1198760] 119911minus1

+ [1198832 1198760] +

1

2[1198831 [1198831 1198760]] 119911minus2+ sdot sdot sdot

(17)

of 1198760 From this formula we see directly that if each 119883

119894=

(minus120572119894 120573119894120574119894 120572119894

) 119894 = 1 2 then

1198761= (

0 21198941205731

minus2119894120574119894

0) = (

0 119902

119903 0)

1198762= (

1199021111990212

1199022111990222

)

= (minus211989412057311205741

2119894 (1205732minus 12057211205731)

minus2119894 (1205742+ 12057211205741) 2119894120573

11205741

)

(18)

In particular we get in this way that 11990211= minus119894(1199021199032) and

11990222= 119894(1199021199032) Since 119876 is the deformation of 119876

0and Id119911 is

central the deformation of each 1198760119911119898 119898 isin Z is given by

119876119911119898The deformation of the Lie algebra119862

1by elements from

119866⩽0

is basically determined by that of the element 1198760119911 So

we focus on the deformation of this element Using the samenotations as at the deformation of 119876

0by 119866lt0 we get that the

deformation of 1198760119911 by a 119870119892 isin 119866

⩽0 with 119870 isin gl

2(119877)lowast and

119892 isin 119866lt0 looks like

119885 = 1198701198921198760119911119892minus1119870minus1 fl

infin

sum

119894=0

119870119876119894119870minus11199111minus119894=

infin

sum

119894=0

1198851198941199111minus119894

= 1198850119911 + [119870119883

1119870minus1 1198850] + sdot sdot sdot

(19)

Consequently the corresponding deformation of each 1198760119911119898

119898 ⩾ 1 is 119885119911119898minus1

4 Advances in Mathematical Physics

Now we are looking for deformations 119876 of the form (17)such that the evolution with respect to the 120597

119898 satisfies the

following for all119898 ⩾ 0

120597119898(119876) = [(119876119911

119898)⩾0 119876] = minus [(119876119911

119898)lt0 119876] (20)

where the second identity follows from the fact that all 119876119911119898commute Equations (20) are called the Lax equations ofthe AKNS hierarchy and the deformation 119876 satisfying theseequations is called a solution of the hierarchy Note that119876 = 119876

0is a solution of the AKNS hierarchy and it is

called the trivial one AKNS equation (4) occurs among thecompatibility equations of this system as we will see furtheron Note that (20) for 119898 = 0 is simply 120597

0(119876) = [119876

0 119876]

Therefore if 1205970= 120597120597119905

0and both 119890minus1198941199050 and 1198901198941199050 belong to the

algebra 119877 of matrix coefficients then we can introduce theloop isin sl

2(119877)[119911 119911

minus1) given by

fl exp (minus11990501198760) 119876 exp (119905

01198760)

= sum

119896⩾0

exp (minus11990501198760) 119876119896exp (119905

01198760) 119911minus119896

(21)

which is easily seen to satisfy 1205970() = 0 This handles then

the dependence of 119876 of 1199050

Among the deformations 119885 of the form (19) we look for119885 such that their evolution with respect to 120597

119898 satisfies the

following for all119898 ⩾ 1

120597119898(119885) = [(119885119911

119898minus1)gt0 119885] = minus [(119885119911

119898minus1)⩽0 119885] (22)

where the second identity follows from the fact that all119885119911119898minus1 commute Since (22) correspond to the strict cut-

off (10) they are called the Lax equations of the strict AKNShierarchy and the deformation 119885 is called a solution of thishierarchy Again there is at least one solution 119885 = 119876

0119911 It is

called the trivial solution of the hierarchyFor both systems (20) and (22) one can speak of compat-

ibility Namely there holds the following result

Proposition 3 Both sets of Lax equations (20) and (22) arethe so-called compatible systems that is the projections 119861

119898fl

(119876119911119898)⩾0| 119898 ⩾ 0 of a solution 119876 to (20) satisfy the zero

curvature relations

1205971198981(1198611198982) minus 1205971198982(1198611198981) minus [119861

1198981 1198611198982] = 0 (23)

and the projections 119862119898fl (119885119911

119898minus1)gt0| 119898 ⩾ 1 corresponding

to a solution 119885 to (22) satisfy the zero curvature relations

1205971198981(1198621198982) minus 1205971198982(1198621198981) minus [119862

1198981 1198621198982] = 0 (24)

Proof The idea is to show that the left-hand side of (23) and(24) respectively belongs to

sl2(119877) [119911 119911

minus1)⩾0cap sl2(119877) [119911 119911

minus1)lt0

respectively sl2(119877) [119911 119911

minus1)gt0cap sl2(119877) [119911 119911

minus1)⩽0

(25)

and thus has to be zero We give the proof for 119862119898 that for

119861119898 is similar and is left to the reader The inclusion in the

first factor is clear as both 119862119898and 120597119899(119862119898) belong to the Lie

subalgebra sl2(119877)[119911 119911

minus1)gt0To show the other one we use Lax

equations (22) Note that the same Lax equations hold for allthe 119911119873119885 | 119873 ⩾ 0

120597119898(119911119873119885) = [(119885119911

119898minus1)gt0 119911119873119885] (26)

By substituting 119862119898119894

= 119911119898119894minus1119885 minus (119911

119898119894minus1119885)lt0 we get the

identities for

1205971198981(1198621198982) minus 1205971198982(1198621198981)

= 1205971198981(1199111198982minus1119885) minus 120597

1198981((1199111198982minus1119885)

⩽0)

minus 1205971198982(1199111198981minus1119885) + 120597

1198982((1199111198981minus1119885)

⩽0)

= [1198621198981 1199111198982minus1119885] minus [119862

1198982 1199111198981minus1119885]

minus 1205971198981((1199111198982minus1119885)

⩽0) + 1205971198982((1199111198981minus1119885)

⩽0)

(27)

and

[1198621198981 1198621198982]

= [1199111198981minus1119885 minus (119911

1198981minus1119885)⩽0 1199111198982minus1119885 minus (119911

1198982minus1119885)⩽0]

= minus [(1199111198981minus1119885)

⩽0 1199111198982minus1119885]

+ [(1199111198982minus1119885)

⩽0 1199111198981minus1119885]

+ [(1199111198981minus1119885)

⩽0 (1199111198982minus1119885)

⩽0]

(28)

Taking into account the second identity in (22) we see thatthe left-hand side of (24) is equal to

minus 1205971198981((1199111198982minus1119885)

⩽0) + 1205971198982((1199111198981minus1119885)

⩽0)

minus [(1199111198981minus1119885)

⩽0 (1199111198982minus1119885)

⩽0]

(29)

This element belongs to the Lie subalgebra sl2(119877)[119911 119911

minus1)⩽0

and that proves the claim

Reversely we have the following

Proposition 4 Suppose we have a deformation 119876 of the type(17) or a deformation 119885 of the form (19) Then there holds thefollowing

(1) Assume that the projections 119861119898fl (119876119911

119898)⩾0| 119898 ⩾ 0

satisfy the zero curvature relations (23) Then 119876 is asolution of the AKNS hierarchy

(2) Similarly if the projections 119862119898

fl (119885119911119898minus1)⩾0| 119898 ⩾

1 satisfy the zero curvature relations (24) then 119885 is asolution of the strict AKNS hierarchy

Advances in Mathematical Physics 5

Proof Againwe prove the statement for119885 that for119876 is shownin a similarway So assume that there is one Lax equation (22)that does not hold Then there is a119898 ⩾ 1 such that

120597119898(119885) minus [119862

119898 119885] = sum

119895⩽119896(119898)

119883119895119911119895 with 119883

119896(119898)= 0 (30)

Since both 120597119898(119885) and [119862

119898 119885] are of order smaller than or

equal to one in 119911 we know that 119896(119898) ⩽ 1 Further we can saythat for all119873 ⩾ 0

120597119898(119911119873119885) minus [119862

119898 119911119873119885] = sum

119895⩽119896(119898)

119883119895119911119895+119873

with 119883119896(119898)

= 0

(31)

and we see by letting 119873 go to infinity that the right-handside can obtain any sufficiently large order in 119911 By the zerocurvature relation for119873 and119898 we get for the left-hand side

120597119898(119911119873119885) minus [119862

119898 119911119873119885]

= 120597119898(119862119873) minus [119862

119898 119862119873] + 120597119898((119911119873119885)⩽0)

minus [119862119898 (119911119873119885)⩽0]

= 120597119873(119862119898) + 120597119898((119911119873119885)⩽0) minus [119862

119898 (119911119873119885)⩽0]

(32)

and this last expression is of order smaller or equal to 119898 in119911 This contradicts the unlimited growth in orders of 119911 of theright-hand side Hence all Lax equations (22) have to holdfor 119885

Because of the equivalence between Lax equations (20)for 119876 and the zero curvature relations (23) for 119861

119898 we call

this last set of equations also the zero curvature form of theAKNS hierarchy Similarly the zero curvature relations (24)for 119862

119898 are called the zero curvature form of the strict AKNS

hierarchyBesides the zero curvature relations for the cut-off rsquos

119861119898 and 119862

119898 respectively corresponding to respectively

a solution 119876 of the AKNS hierarchy and a solution 119885 of thestrict AKNS hierarchy also other parts satisfy such relationsDefine

119860119898fl 119861119898minus 119876119911119898 119898 ⩾ 0

119863119898fl 119862119898minus 119885119911119898minus1 119898 ⩾ 1

(33)

Then we have the following result

Corollary 5 The following relations hold

(i) The parts 119860119898| 119898 ⩾ 0 of a solution 119876 of the AKNS

hierarchy satisfy

1205971198981(1198601198982) minus 1205971198982(1198601198981) minus [119860

1198981 1198601198982] = 0 (34)

(ii) The parts 119863119898| 119898 ⩾ 0 of a solution 119885 of the strict

AKNS hierarchy satisfy

1205971198981(1198631198982) minus 1205971198982(1198631198981) minus [119863

1198981 1198631198982] = 0 (35)

Proof Again we show the result only in the strict case Notethat 119885119911119898minus1 satisfy Lax equations similar to 119885

120597119894(119885119911119898minus1) = [119863

119894 119885119911119898minus1] 119894 ⩾ 1 (36)

Now we substitute in the zero curvature relations for 119862119898

everywhere the relation 119862119898= 119863119898+ 119885119911119898minus1 and use the Lax

equations above and the fact that all 119885119911119898minus1 commute Thisgives the desired result

To clarify the link with the AKNS equation considerrelation (23) for119898

1= 2 and119898

2= 1

1205972(1198760119911 + 119876

1) = 1205971(11987601199112+ 1198761119911 + 119876

2)

+ [1198762 1198760119911 + 119876

1]

(37)

Then this identity reduces in sl2(119877)[119911 119911

minus1)⩾0 since 119876

0is

constant to the following two equalities

1205971(1198761) = [119876

0 1198762]

1205972(1198761) = 1205971(1198762) + [119876

2 1198761]

(38)

The first gives an expression of the off-diagonal terms of 1198762

in the coefficients 119902 and 119903 of 1198761 that is

11990212=119894

21205971(119902)

11990221= minus

119894

21205971(119903)

(39)

and the second equation becomes AKNS equations (4) if onehas 1205971= 120597120597119909 and 120597

2= 120597120597119905

3 The Linearization of Both Hierarchies

The zero curvature form of both hierarchies points at thepossible existence of a linear system of which the zerocurvature equations form the compatibility conditions Wepresent here such a system for each hierarchy For theAKNS hierarchy this system the linearization of the AKNShierarchy is as follows take a potential solution119876 of the form(17) and consider the system

119876120595 = 1205951198760

120597119898(120595) = 119861

119898120595 forall119898 ⩾ 0 119861

119898= (119876119911

119898)⩾0

(40)

Likewise for a potential solution 119885 of the strict AKNShierarchy of the form (19) the linearization of the strict AKNShierarchy is given by

119885120593 = 1205931198760119911

120597119898(120593) = 119862

119898120593 forall119898 ⩾ 1 119862

119898= (119885119911

119898minus1)gt0

(41)

Before specifying120595 and120593 we show themanipulations neededto derive the Lax equations of the hierarchy from theirlinearizationWedo this for the strictAKNShierarchy for the

6 Advances in Mathematical Physics

AKNS hierarchy one proceeds similarly Apply 120597119898to the first

equation in (41) and use both equalities in (41) in the sequel

120597119898(119885120593 minus 120593119876

0119911) = 120597

119898(119885) 120593 + 119885120597

119898(120593) minus 120597

119898(120593)1198760119911

= 0 = 120597119898(119885) 120593 + 119885119862

119898120593 minus 119862

1198981205931198760119911

= 120597119898(119885) minus [119862

119898 119885] 120593 = 0

(42)

Hence if we can scratch 120593 from (42) then we obtain the Laxequations of the strict AKNS hierarchy To get the propersetup where all these manipulations make sense we first havea look at the linearization for the trivial solutions119876 = 119876

0and

119885 = 1198760119911 In the AKNS case we have then

11987601205950= 12059501198760

120597119898(1205950) = 119876

01199111198981205950 forall119898 ⩾ 0

(43)

and for its strict version

11987601199111205930= 12059301198760119911

120597119898(1205930) = 119876

01199111198981205930 forall119898 ⩾ 1

(44)

Assuming that each derivation 120597119898equals 120597120597119905

119898 one arrives

for (43) at the solution

1205950= 1205950(119905 119911) = exp(

infin

sum

119898=0

1199051198981198760119911119898) (45)

where 119905 is the short hand notation for 119905119898| 119898 ⩾ 0 and for

(44) it leads to

1205930= 1205930(119905 119911) = exp(

infin

sum

119898=1

1199051198981198760119911119898)

with 119905 = 119905119898| 119898 ⩾ 1

(46)

In general one needs in the linearizations (120595) and (120593)respectively a left action with elements like 119876 119861

119898and 119885

119862119898from gl

2(119877)[119911 119911

minus1) respectively an action of all the 120597

119898

and a right action of 1198760and 119876

0119911 respectively This can

all be realized in suitable gl2(119877)[119911 119911

minus1) modules that are

perturbations of the trivial solutions 1205950and 120593

0 Consider in

a AKNS setting

M0= 119892 (119911) 120595

0=

119873

sum

119894=minusinfin

1198921198941199111198941205950| 119892 (119911)

=

119873

sum

119894=minusinfin

119892119894119911119894isin gl2(119877) [119911 119911

minus1)

(47)

and in a strict AKNS setting

M1= 119892 (119911) 120593

0=

119873

sum

119894=minusinfin

1198921198941199111198941205930| 119892 (119911)

=

119873

sum

119894=minusinfin

119892119894119911119894isin gl2(119877) [119911 119911

minus1)

(48)

where the products 119892(119911)1205950and 119892(119911)120593

0should be seen as

formal and both factors should be kept separate On bothM0

andM1 one can define the required actions for each ℎ(119911) isin

gl2(119877)[119911 119911

minus1) define

ℎ (119911) sdot 119892 (119911) 1205950fl ℎ (119911) 119892 (119911) 120595

0

respectively ℎ (119911) sdot 119892 (119911) 1205930fl ℎ (119911) 119892 (119911) 120593

0

(49)

We define the right-hand action of 1198760and 119876

0119911 respectively

by

119892 (119911) 12059501198760fl 119892 (119911)119876

0 1205950

respectively 119892 (119911) 12059301198760119911 fl 119892 (119911)119876

0119911 1205930

(50)

and the action of each 120597119898by

120597119898(119892 (119911) 120595

0)

=

119873

sum

119894=minusinfin

120597119898(119892119894) 119911119894+

119873

sum

119894=minusinfin

1198921198941198760119911119894+1198981205950

120597119898(119892 (119911) 120593

0)

=

119873

sum

119894=minusinfin

120597119898(119892119894) 119911119894+

119873

sum

119894=minusinfin

1198921198941198760119911119894+1198981205930

(51)

Analogous to the terminology in the scalar case (see [5]) wecall the elements of M

0and M

1oscillating matrices Note

that both M0and M

1are free gl

2(119877)[119911 119911

minus1) modules with

generators 1205950and 120593

0 respectively because for each ℎ(119911) isin

gl2(119877)[119911 119911

minus1) we have

ℎ (119911) sdot 1205950= ℎ (119911) sdot 1 120595

0= ℎ (119911) 120595

0

respectively ℎ (119911) sdot 1205930= ℎ (119911) sdot 1 120595

0= ℎ (119911) 120593

0

(52)

Hence in order to be able to perform legally the scratching of120595 = ℎ(119911)120595

0or 120593 = ℎ(119911)120593

0 it is enough to find oscillating

matrices such that ℎ(119911) is invertible in gl2(119877)[119911 119911

minus1) We will

now introduce a collection of such elements that will occur atthe construction of solutions of both hierarchies

For119898 = (1198981 1198982) isin Z2 let 120575(119898) isin gl

2(119877)[119911 119911

minus1) be given

by

120575 (119898) = (1199111198981 0

0 1199111198982) (53)

Then 120575(119898) has 120575(minus119898) as its inverse in gl2(119877)[119911 119911

minus1) and the

collection Δ = 120575(119898) | 119898 isin Z2 forms a group An element120595 isinM

0is called an oscillating matrix of type 120575(119898) if it has the

form

120595 = ℎ (119911) 120575 (119898) 1205950 with ℎ (119911) isin 119866

lt0 (54)

These oscillatingmatrices are examples of elements ofM0for

which the scratching procedure is valid Let 119876 be a potentialsolution of the AKNS hierarchy of the form (17) If there isan oscillating function 120595 of type 120575(119898) such that for 119876 and 120595

Advances in Mathematical Physics 7

(40) hold then we call120595 awave matrix of the AKNS hierarchyof type 120575(119898) In particular 119876 is then a solution of the AKNShierarchy and the first equation in (40) implies

119876ℎ (119911) 120575 (119898) = ℎ (119911)1198760120575 (119898) 997904rArr

119876 = ℎ (119911)1198760ℎ (119911)minus1

(55)

in other words the solution 119876 is totally determined by thewave matrix Similarly we call an element 120593 isin M

1an

oscillating matrix of type 120575(119898) if it has the form

120593 = ℎ (119911) 120575 (119898) 1205930 with ℎ (119911) isin 119866

⩽0 (56)

This type of oscillating matrices are examples of elements ofM1for which the scratching procedure is valid Let 119885 be a

potential solution of the strict AKNS hierarchy of the form(19) If there is an oscillating function 120593 of type 120575(119898) in M

1

such that for119885 and120593 (41) hold thenwe call120593 awavematrix ofthe strict AKNS hierarchy of type 120575(119898) In particular119885 is thena solution of the strict AKNS hierarchy and the first equationin (41) implies

119885ℎ (119911) 120575 (119898) = ℎ (119911)1198760119911120575 (119898) 997904rArr

119885 = ℎ (119911)1198760119911ℎ (119911)

minus1

(57)

so that also here the wave matrix fully determines thesolution

For both hierarchies there is a milder condition that is tobe satisfied by oscillating matrices of a certain type in orderthat they become a wave matrix of that hierarchy

Proposition 6 Let120595 = ℎ(119911)120575(119898)1205950be an oscillatingmatrix

of type 120575(119898) in M0and 119876 = ℎ(119911)119876

0ℎ(119911)minus1 the corresponding

potential solution of the AKNS hierarchy Similarly let 120593 =

ℎ(119911)120575(119898)1205930be such a matrix in M

1with potential solution

119885 = ℎ(119911)1198760119911ℎ(119911)minus1

(a) If there exists for each 119898 ⩾ 0 an element 119872119898isin

gl2(119877)[119911 119911

minus1)⩾0

such that

120597119898(120595) = 119872

119898120595 (58)

then each119872119898= (119876119911

119898)⩾0

and 120595 is a wave matrix oftype 120575(119898) for the AKNS hierarchy

(b) If there exists for each 119898 ⩾ 1 an element 119873119898isin

gl2(119877)[119911 119911

minus1)gt0

such that

120597119898(120593) = 119873

119898120593 (59)

then each 119873119898= (119885119911

119898minus1)gt0

and 120593 is a wave matrix oftype 120575(119898) for the strict AKNS hierarchy

Proof We give the proof again for the strict AKNS casethe other one is similar By using the fact that M

1is a free

gl2(119877)[119911 119911

minus1)module with generator 120593

0we can translate the

relations 120597119898(120593) = 119873

119898120593 into equations in gl

2(119877)[119911 119911

minus1) This

yields

120597119898(ℎ (119911)) + ℎ (119911)119876

0119911119898= 119873119898ℎ (119911) 997904rArr

120597119898(ℎ (119911)) ℎ (119911)

minus1+ 119885119911119898minus1

= 119873119898

(60)

Projecting the right-hand side on gl2(119877)[119911 119911

minus1)gt0

gives us theidentity

(119885119911119898minus1)gt0= 119873119898 (61)

we are looking for

In the next section we produce a context where we canconstruct wave matrices for both hierarchies in which theproduct is real

4 The Construction of the Solutions

In this section we will show how to construct a wide class ofsolutions of both hierarchies This is done in the style of [6]We first describe the group of loops we will work with Foreach 0 lt 119903 lt 1 let 119860

119903be the annulus

119911 | 119911 isin C 119903 ⩽ |119911| ⩽1

119903 (62)

Following [7] we use the notation 119871anGL2(C) for the col-lection of holomorphic maps from some annulus 119860

119903into

GL2(C) It is a group with respect to pointwise multiplication

and contains in a natural way GL2(C) as a subgroup as the

collection of constant maps into GL2(C) Other examples

of elements in 119871anGL2(C) are the elements of Δ However119871anGL2(C) is more than just a group it is an infinitedimensional Lie group Its manifold structure can be read offfrom its Lie algebra 119871angl2(C) consisting of all holomorphicmaps 120574 119880 rarr gl

2(C) where 119880 is an open neighborhood of

some annulus 119860119903 0 lt 119903 lt 1 Since gl

2(C) is a Lie algebra

the space 119871angl2(C) becomes a Lie algebra with respect to thepointwise commutator Topologically the space 119871angl2(C) isthe direct limit of all the spaces 119871an119903gl2(C) where this lastspace consists of all 120574 corresponding to the fixed annulus119860119903 One gives each 119871an119903gl2(C) the topology of uniform

convergence and with that topology it becomes a Banachspace In this way 119871angl2(C) becomes a Frechet space Thepointwise exponential map defines a local diffeomorphismaround zero in 119871angl2(C) (see eg [8])

Now each 120574 isin 119871angl2(C) possesses an expansion in aFourier series

120574 =

infin

sum

119896=minusinfin

120574119896119911119896 with each 120574

119896isin gl2(C) (63)

that converges absolutely on the annulus 119860119903

infin

sum

119896=minusinfin

10038171003817100381710038171205741198961003817100381710038171003817 119903minus|119896|

lt infin (64)

This Fourier expansion is used to make two different decom-positions of the Lie algebra119871angl2(C)Thefirst is the analoguefor the present Lie algebra 119871angl2(C) of decomposition (8) ofgl2(119877)[119911 119911

minus1) that lies at the basis of the Lax equations of the

AKNS hierarchy Namely consider the subspaces

119871angl2 (C)⩾0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=0

120574119896119911119896

8 Advances in Mathematical Physics

119871angl2 (C)lt0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =minus1

sum

119896=minusinfin

120574119896119911119896

(65)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra The first Lie algebra consists ofthe elements in 119871angl2(C) that extend to holomorphic mapsdefined on some disk 119911 isin C | |119911| ⩽ 1119903 0 lt 119903 lt 1 andthe second Lie algebra corresponds to the maps in 119871angl2(C)that have a holomorphic extension towards some disk aroundinfinity 119911 isin P1(C) | |119911| ⩾ 119903 0 lt 119903 lt 1 and that arezero at infinity To each of the two Lie subalgebras belongsa subgroup of 119871anGL2(C) The pointwise exponential mapapplied to elements of 119871angl2(C)lt0 yields elements of

119880minus= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

minus1

sum

119896=minusinfin

120574119896119911119896 (66)

and the exponential map applied to elements of 119871angl2(C)⩾0maps them into

119875+= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740 +

infin

sum

119896=1

120574119896119911119896 with 120574

0

isin GL2(C)

(67)

Both119880minusand 119875+are easily seen to be subgroups of 119871anGL2(C)

and since the direct sum of their Lie algebras is 119871angl2(C)their product

Ω = 119880minus119875+ (68)

is open in 119871anGL2(C) and is called like in the finite dimen-sional case the big cell with respect to 119880

minusand 119875

+

The second decomposition is the analogue of the splitting(11) of gl

2(119877)[119911 119911

minus1) that led to the Lax equations of the strict

AKNS hierarchy Now we consider the subspaces

119871angl2 (C)gt0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=1

120574119896119911119896

119871angl2 (C)⩽0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =0

sum

119896=minusinfin

120574119896119911119896

(69)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra 119871angl2(C)gt0 consists of mapswhose holomorphic extension to 119911 = 0 equals zero inthat point and 119871angl2(C)⩽0 is the set of maps that extendhomomorphically to a neighborhood of infinity To eachof them belongs a subgroup of 119871anGL2(C) The pointwiseexponential map applied to elements of 119871angl2(C)gt0 yieldselements of

119880+= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

infin

sum

119896=1

120574119896119911119896 (70)

and the exponential map applied to elements of 119871angl2(C)⩽0maps them into

119875minus= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740

+

minus1

sum

119896=minusinfin

120574119896119911119896 with 120574

0isin GL2(C)

(71)

Both119875minusand119880

+are easily seen to be subgroups of 119871anGL2(C)

Since the direct sum of their Lie algebras is 119871angl2(C) theirproduct 119875

minus119880+is open and because 119875

+= GL

2(C)119880+=

119880+GL2(C) and 119875

minus= GL2(C)119880minus= 119880minusGL2(C) this gives the

equality

119875minus119880+= Ω = 119880

minus119875+ (72)

the big cell in 119871anGL2(C)The next two subgroups of 119871anGL2(C) correspond to

the exponential factors in both linearizations The group ofcommuting flows relevant for the AKNS hierarchy is thegroup

Γ0= 1205740(119905) = exp(

infin

sum

119894=0

1199051198941198760119911119894) | 1205740isin 119875+ (73)

and similarly for the strict AKNS hierarchy we use thecommuting flows from

Γ1= 1205741(119905) = exp(

infin

sum

119894=1

1199051198941198760119911119894) | 1205741isin 119880+ (74)

Besides the groups Δ Γ0 and Γ

1 there are more subgroups in

119871anGL2(C) that commute with those three groups and theyare in a sense complimentary to Δ Γ

0and Δ Γ

1 respectively

That is why we use the following notations for them in 119880minus

there is

Γ119888

0= 120574119888

0(119905) = exp(

infin

sum

119896=1

1199041198961198760119911minus119896) | 120574119888

0isin 119880minus (75)

and in 119875minuswe have

Γ119888

1= 120574119888

1(119905) = exp(

infin

sum

119896=0

1199041198961198760119911minus119896) | 120574119888

1isin 119875minus (76)

We have now all ingredients to describe the constructionof the solutions to each hierarchy and we start with theAKNS hierarchy Take inside the product 119871anGL2(C) times Δthe collection 119878

0of pairs (119892 120575(119898)) such that there exists a

1205740(119905) isin Γ

0satisfying

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898) isin Ω = 119880

minus119875+ (77)

For such a pair (119892 120575(119898)) we take the collection Γ0(119892 120575(119898))

of all 1205740(119905) satisfying the condition (77) This is an open

nonempty subset of Γ0 Let 119877

0(119892 120575(119898)) be the algebra of

analytic functions Γ0(119892 120575(119898)) rarr C This is the algebra of

Advances in Mathematical Physics 9

functions 119877 that we associate with the point (119892 120575(119898)) isin 1198780

and for the commuting derivations of 1198770(119892 120575(119898)) we choose

120597119894fl 120597120597119905

119894 119894 ⩾ 0 By property (77) we have for all 120574

0(119905) isin

Γ0(119892 120575(119898))

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898)

= 119906minus(119892 120575 (119898))

minus1

119901+(119892 120575 (119898))

(78)

with 119906minus(119892 120575(119898)) isin 119880

minus119901+(119892 120575(119898)) isin 119875

+Then all thematrix

coefficients in the Fourier expansions of the elements 119906minus(119892

120575(119898)) and 119901+(119892 120575(119898)) belong to the algebra 119877

0(119892 120575(119898)) It is

convenient to write relation (78) in the form

Ψ119892120575(119898)

fl 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905)

= 119901+(119892 120575 (119898)) 120575 (119898) 120574

0(119905) 119892minus1

= 119901+(119892 120575 (119898)) 120574

0(119905) 120575 (119898) 119892

minus1

= 119902+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(79)

with 119902+(119892 120575(119898)) isin 119875

+ Clearly Ψ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M0 where all the products in the

decomposition 119906minus(119892 120575(119898))120575(119898)120574

0(119905) are no longer formal

but real This is our potential wave matrix for the AKNShierarchy

To get the potential wave function for the strict AKNShierarchy we proceed similarly only this time we use thedecomposition Ω = 119875

minus119880+ Consider again the product

119871anGL2(C) times Δ and the subset 1198781of pairs (119892 120575(119898)) such that

there exists a 1205741(119905) isin Γ

1satisfying

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898) isin Ω = 119875

minus119880+ (80)

For such a pair (119892 120575(119898)) we define Γ1(119892 120575(119898)) as the set of

all 1205741(119905) satisfying condition (77) This is an open nonempty

subset of Γ1 Let 119877

1(119892 120575(119898)) be the algebra of analytic

functions Γ1(119892 120575(119898)) rarr C This is the algebra of functions

119877 that we associate with the point (119892 120575(119898)) isin 1198781and for the

commuting derivations of 1198771(119892 120575(119898)) we choose 120597

119894fl 120597120597119905

119894

119894 gt 0 By property (80) we have for all 1205741(119905) isin Γ

1(119892 120575(119898))

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898)

= 119901minus(119892 120575 (119898))

minus1

119906+(119892 120575 (119898))

(81)

with 119901minus(119892 120575(119898)) isin 119875

minus 119906+(119892 120575(119898)) isin 119880

+ Then all

the matrix coefficients in the Fourier expansions of theelements 119901

minus(119892 120575(119898)) and 119906

+(119892 120575(119898)) belong to the algebra

1198771(119892 120575(119898)) It is convenient to write relation (81) in the form

Φ119892120575(119898)

fl 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905)

= 119906+(119892 120575 (119898)) 120575 (119898) 120574

1(119905) 119892minus1

= 119906+(119892 120575 (119898)) 120574

1(119905) 120575 (119898) 119892

minus1

= 119908+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(82)

with 119908+(119892 120575(119898)) isin 119880

+ Clearly Φ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M1where all the products in the

decomposition 119901minus(119892 120575(119898))120575(119898)120574

1(119905) are no longer formal

but realTo show that Ψ

119892120575(119898)is a wave matrix for the AKNS

hierarchy of type 120575(119898) and Φ119892120575(119898)

one for the strict AKNShierarchy we use the fact that it suffices to prove the relationsin Proposition 6 We treat first Ψ

119892120575(119898) In (79) we presented

two different decompositions of Ψ119892120575(119898)

that we will use tocompute 120597

119894(Ψ119892120575(119898)

) With respect to 119906minus(119892 120575(119898))120575(119898)120574

0(119905)

there holds120597119894(Ψ119892120575(119898)

)

= 120597119894(119906minus(119892 120575 (119898))) 119906

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905) = 119872

119894Ψ119892120575(119898)

(83)

with119872119894isin gl2(119877(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911 If we take

the decomposition 119902+(119892 120575(119898))120575(119898)119892

minus1 then we get

120597119894(Ψ119892120575(119898)

)

= 120597119894(119902+(119892 120575 (119898))) 119902

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(84)

with 120597119894(119902+(119892 120575(119898)))119902

+(119892 120575(119898))

minus1 of the formsum119895⩾0119902119895119911119895 with

all 119902119895isin gl2(119877(119892 120575(119898))) As Ψ

119892120575(119898)is invertible we obtain

119872119894= sum119895⩾0119902119895119911119895 and thus 119872

119894= sum119894

119895=0119902119895119911119895 and that is the

desired resultWe handle the case of Φ

119892120575(119898)in a similar way Also in

(79) you can find two different decompositions of Φ119892120575(119898)

that we will use to compute 120597119894(Φ119892120575(119898)

) With respect to theexpression 119901

minus(119892 120575(119898))120575(119898)120574

1(119905) we get for all 119894 ⩾ 1

120597119894(Φ119892120575(119898)

)

= 120597119894(119901minus(119892 120575 (119898))) 119901

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905) = 119873

119894Φ119892120575(119898)

(85)

again with 119873119894isin gl2(1198771(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911

Next we take the decomposition 119908+(119892 120575(119898))120575(119898)119892

minus1 andthat yields

120597119894(Φ119892120575(119898)

)

= 120597119894(119908+(119892 120575 (119898)))119908

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(86)

Since the zeroth order term of 119908+(119892 120575(119898)) is Id we note

that the element 120597119894(119908+(119892 120575(119898)))119908

+(119892 120575(119898))

minus1 has the formsum119895⩾1119908119895119911119895 with all 119908

119895isin gl2(1198771(119892 120575(119898))) As Φ

119892120575(119898)is also

invertible we obtain119873119894= sum119895⩾1119908119895119911119895 and thus119873

119894= sum119894

119895=1119908119895119911119895

and that is the result we were looking for in the strict caseThe abovementioned constructions are not affected seri-

ously by a number of transformations For example if wewrite 120575 = 120575((1 1)) then all 120575119896 119896 isin Z are central elementsand we have for all (119892 120575(119898)) isin 119878

0and 119878

1respectively and all

119896 isin Z

Ψ119892120575(119898)120575

119896 = Ψ119892120575(119898)

120575119896

respectively Φ119892120575(119898)120575

119896 = Φ119892120575(119898)

120575119896

(87)

10 Advances in Mathematical Physics

Also for any 1199011isin 119875+ each 120574119888

0(119904) isin Γ

119888

0 and all (119892 120575(119898)) isin

1198780 we see that the element (120574119888

0(119904)119892120575(minus119898)119901

1120575(119898) 120575(119898)) also

belongs to 1198780 for

120575 (119898) 1205740(119905) 120574119888

0(119904) 119892120575 (minus119898) 119901

1120575 (119898) 120574

0(119905)minus1120575 (minus119898)

= 120574119888

0(119904) 119906minus1

minus119901+1205740(119905) 11990111205740(119905)minus1

(88)

In particular we see that

Ψ120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575(119898)

= 119906minus120574119888

0(119904)minus1120575 (119898) 120574

0(119905) (89)

Its analogue in the strict case is as follows for any 1199061isin

119880+ each 120574119888

1(119904) isin Γ

119888

1 and all (119892 120575(119898)) isin 119878

1 the element

(120574119888

1(119904)119892120575(minus119898)119906

1120575(119898) 120575(119898)) also belongs to 119878

1and there

holds

Φ120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575(119898)

= 119901minus120574119888

1(119904)minus1120575 (119898) 120574

1(119905) (90)

We resume the foregoing results in the following theorem

Theorem 7 Consider the product spaceΠ fl 119871anGL2(C) times ΔThen the following holds

(a) For each point (119892 120575(119898)) isin Π that satisfies condition(77) we have a wave matrix of type 120575(119898) of the AKNShierarchyΨ

119892120575(119898) defined by (79) and this determines a

solution 119876119892120575(119898)

= 119906minus(119892 120575(119898))119876

0119906minus(119892 120575(119898))

minus1 of theAKNS hierarchy For each 119901

1isin 119875+ every 120574119888

0(119904) isin Γ

119888

0

and all powers of 120575 one has

119876120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575

119896120575(119898)

= 119876119892120575(119898)

(91)

(b) For each point (119892 120575(119898)) isin Π that satisfies condition(80) we have a wave matrix of type 120575(119898) of thestrict AKNS hierarchy Φ

119892120575(119898) defined by (82) and it

determines a solution 119885119892120575(119898)

= 119901minus(119892 120575(119898))119876

0119911119901minus(119892

120575(119898))minus1 of the strict AKNS hierarchy For each 119906

1isin 119880+

every 1205741198881(119904) isin Γ

119888

1 and all powers of 120575 one has

119885120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575

119896120575(119898)

= 119885119892120575(119898)

(92)

Competing Interests

The author declares that they have no competing interests

References

[1] M Adler P vanMoerbeke and P VanhaeckeAlgebraic Integra-bility Painleve Geometry and Lie Algebras vol 47 of Ergebnisseder Mathematik und ihrer Grenzgebiete Springer 2004

[2] G F Helminck A G Helminck and E A PanasenkoldquoIntegrable deformations in the algebra of pseudodifferentialoperators from a Lie algebraic perspectiverdquo Theoretical andMathematical Physics vol 174 no 1 pp 134ndash153 2013

[3] M J Ablowitz D J Kaup A C Newell and H Segur ldquoTheinverse scattering transform-Fourier analysis for nonlinearproblemsrdquo Studies in Applied Mathematics vol 53 no 4 pp249ndash315 1974

[4] H Flashka A CNewell and T Ratiu ldquoKac-Moody Lie algebrasand soliton equations II Lax equations associated with 119860

1

(1)rdquoPhysicaDNonlinear Phenomena vol 9 no 3 pp 300ndash323 1983

[5] EDateM JimboMKashiwara andTMiwa ldquoTransformationgroups for soliton equationsrdquo in Proceedings of the RIMSSymposium on Nonlinear Integrable SystemsmdashClassical Theoryand Quantum Theory T Jimbo and T Miwa Eds WorldScience Publisher Kyoto Japan 1981

[6] G Segal and G Wilson ldquoLoop groups and equations of KdVtyperdquo Publications Mathematiques de lrsquoIHES vol 61 pp 5ndash651985

[7] A Pressley and G Segal Loop Groups Oxford MathematicalMonographs Clarendon Press 1986

[8] R S Hamilton ldquoThe inverse function theorem of Nash andMoserrdquoBulletin of the AmericanMathematical Society vol 7 no1 pp 65ndash222 1982

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Stochastic AnalysisInternational Journal of

2 Advances in Mathematical Physics

consequently carries the name AKNS hierarchy (see [4])Besides that we introduce here also another system ofLax equations for a more general deformation of the basicgenerator of the commutative algebra It relates to a differentdecomposition of the relevant Lie algebra Since its analoguefor theKP hierarchy is called the strict KP hierarchy we use inthe present context the term strict AKNS hierarchy We showfirst of all that this new system is compatible and equivalentwith a set of zero curvature relations Further we describesuitable linearizations that can be used to construct solutionsof both hierarchiesWe conclude by constructing solutions ofboth hierarchies

2 The AKNS Hierarchy and Its Strict Version

We present here an algebraic description of the AKNS hier-archy and its strict version that underlines the deformationcharacter of these hierarchies as pointed at in IntroductionAt(4) one has to deal with 2times2-matriceswhose coefficientswerepolynomial expressions of a number of complex functionsand their derivatives with respect to some parameters Thesame is true for the other equations in both hierarchies Weformalize this as follows our starting point is a commutativecomplex algebra 119877 that should be seen as the source fromwhich the coefficients of the 2 times 2-matrices are taken Wewill work in the Lie algebra sl

2(119877)[119911 119911

minus1) consisting of all

elements

119883 =

119873

sum

119894=minusinfin

119883119894119911119894 with all 119883

119894isin sl2(119877) (5)

and the bracket

[119883 119884] = [

[

119873

sum

119894=minusinfin

119883119894119911119894

119872

sum

119895=minusinfin

119884119895119911119895]

]

fl119873

sum

119894=minusinfin

119872

sum

119895=minusinfin

[119883119894 119884119895] 119911119894+119895

(6)

We will also make use of the slightly more general Lie algebragl2(119877)[119911 119911

minus1) where the coefficients in the 119911-series from (5)

are taken fromgl2(119877) instead of sl

2(119877) and the bracket is given

by the same formula In the Lie algebra gl2(119877)[119911 119911

minus1) we

decompose elements in two ways The first is as follows

119883 =

119873

sum

119894=minusinfin

119883119894119911119894=

119873

sum

119894=0

119883119894119911119894+

minus1

sum

119894=minusinfin

119883119894119911119894 š (119883)

⩾0+ (119883)

lt0(7)

and this induces the splitting

gl2(119877) [119911 119911

minus1) = gl

2(119877) [119911 119911

minus1)⩾0

oplus gl2(119877) [119911 119911

minus1)lt0

(8)

where the two Lie subalgebras gl2(119877)[119911 119911

minus1)⩾0

and gl2(119877)[119911

119911minus1)lt0

are given by

gl2(119877) [119911 119911

minus1)⩾0

= 119883 isin gl2(119877) [119911 119911

minus1) | 119883 = (119883)

⩾0=

119873

sum

119894=0

119883119894119911119894

gl2(119877) [119911 119911

minus1)lt0

= 119883 isin gl2(119877) [119911 119911

minus1) | 119883 = (119883)

lt0= sum

119894lt0

119883119894119911119894

(9)

By restriction it leads to a similar decomposition forsl2(119877)[119911 119911

minus1) which is relevant for the AKNS hierarchy The

second way to decompose elements of gl2(119877)[119911 119911

minus1) is

119883 =

119873

sum

119894=minusinfin

119883119894119911119894=

119873

sum

119894=1

119883119894119911119894+

0

sum

119894=minusinfin

119883119894119911119894

š (119883)gt0+ (119883)

⩽0

(10)

This yields the splitting

gl2(119877) [119911 119911

minus1) = gl

2(119877) [119911 119911

minus1)gt0

oplus gl2(119877) [119911 119911

minus1)⩽0

(11)

By restricting it to sl2(119877)[119911 119911

minus1) we get a similar decom-

position for this Lie algebra which relates as we will seefurther on to the strict version of the AKNS hierarchy TheLie subalgebras gl

2(119877)[119911 119911

minus1)gt0

and gl2(119877)[119911 119911

minus1)⩽0

in (11)are defined in a similar way as the first two Lie subalgebras

gl2(119877) [119911 119911

minus1)gt0

= 119883 isin gl2(119877) [119911 119911

minus1) | 119883 = (119883)

gt0=

119873

sum

119894=1

119883119894119911119894

gl2(119877) [119911 119911

minus1)⩽0

= 119883 isin gl2(119877) [119911 119911

minus1) | 119883 = (119883)

⩽0= sum

119894⩽0

119883119894119911119894

(12)

Both hierarchies correspond to evolution equations fordeformations of the generators of a commutative Lie algebrain the first component Denote thematrix ( minus119894 0

0 119894) by119876

0 Inside

sl2(119877)[119911 119911

minus1)⩾0 this commutative complex Lie subalgebra119862

0

is chosen to be the Lie subalgebra with the basis 1198760119911119898| 119898 ⩾

0 and in sl2(119877)[119911 119911

minus1)gt0our choice will be the Lie subalgebra

1198621with the basis 119876

0119911119898| 119898 ⩾ 1 In the first case that we

work with 1198620 we assume that the algebra 119877 possesses a set

120597119898| 119898 ⩾ 0 of commutingC-linear derivations 120597

119898 119877 rarr 119877

where each 120597119898should be seen as the derivation corresponding

to the flow generated by 1198760119911119898 The data (119877 120597

119898| 119898 ⩾ 0) is

also called a setting for the AKNS hierarchy In the case thatwe work with the decomposition (11) we merely need a set ofcommutingC-linear derivations 120597

119898 119877 rarr 119877119898 ⩾ 1 with the

same interpretation Staying in the same line of terminologywe call the data (119877 120597

119898| 119898 ⩾ 1) a setting for the strict AKNS

hierarchy

Example 1 Examples of settings for the respective hierarchiesare the algebras of complex polynomials C[119905

119898] in the vari-

ables 119905119898| 119898 ⩾ 0 and 119905

119898| 119898 ⩾ 1 respectively or the

Advances in Mathematical Physics 3

formal power series C[[119905119898]] in the same variables with both

algebras equipped with the derivations 120597119898= 120597120597119905

119898 119898 ⩾ 0

in the AKNS setup or 120597119898= 120597120597119905

119898119898 ⩾ 1 in the strict AKNS

case

We let each derivation 120597119898occurring in some setting act

coefficient-wise on the matrices from gl2(119877) and that defines

then a derivation of this algebra The same holds for theextension to gl

2(119877)[119911 119911

minus1) defined by

120597119898(119883) fl

119873

sum

119895=minusinfin

120597119898(119883119895) 119911119895 (13)

We have now sufficient ingredients to discuss the AKNShierarchy and its strict version In the AKNS case our interestis in certain deformations of the Lie algebra 119862

0obtained

by conjugating with elements of the group corresponding togl2(119877)[119911 119911

minus1)lt0 At the strict version we are interested in

certain deformations of the Lie algebra1198621obtained by conju-

gating with elements from a group linked to gl2(119877)[119911 119911

minus1)⩽0

Note that for each 119883 isin gl2(119877)[119911 119911

minus1)lt0

the exponentialmap yields a well-defined element of the form

exp (119883) =infin

sum

119896=0

1

119896119883119896= Id + 119884 119884 isin gl

2(119877) [119911 119911

minus1)lt0

(14)

and with the formula for the logarithm one retrieves 119883 backfrom 119884 One verifies directly that the elements of the form(14) form a group with respect to multiplication and this wesee as the group 119866

lt0corresponding to gl

2(119877)[119911 119911

minus1)lt0

In the case of the Lie subalgebra gl2(119877)[119911 119911

minus1)⩽0 one

cannot move back and forth between the Lie algebra and itsgroup Nevertheless one can assign a proper group to thisLie algebra A priori the exponential exp(119884) of an element119884 isin gl

2(119877)[119911 119911

minus1)⩽0

does not have to define an element ingl2(119877)[119911 119911

minus1)⩽0 That requires convergence conditions How-

ever if it does then it belongs to

119866⩽0=

119870 =

infin

sum

119895=0

119870119895119911minus119895| all 119870

119895isin gl2(119877) 119870

0

isin gl2(119877)lowast

(15)

where gl2(119877)lowast denotes the elements in gl

2(119877) that have a

multiplicative inverse in gl2(119877) It is a direct verification that

119866⩽0

is a group and we see it as a proper group correspondingto the Lie algebra gl

2(119877)[119911 119911

minus1)⩽0 In fact 119866

⩽0is isomorphic

to the semidirect product of 119866lt0

and gl2(119877)lowast

Now there holds the following

Lemma 2 The group 119866⩽0

acts by conjugation on sl2(119877)[119911

119911minus1)

Proof Take first any 119892 isin 119866lt0 Then there is an119883 isin gl

2(119877)[119911

119911minus1)lt0

such that 119892 = exp(119883) Now there holds for every 119884 isinsl2(119877)[119911 119911

minus1)

119892119884119892minus1= exp (119883) 119884 exp (minus119883) = 119890ad(119883) (119884)

= 119884 +

infin

sum

119896=1

1

119896ad (119883)119896 (119884)

(16)

and this shows that the coefficients for the different powersof 119911 in this expression are commutators of elements of gl

2(119877)

and sl2(119877) and that proves the claim for elements from 119866

lt0

Since conjugation with an element from gl2(119877)lowast maps sl

2(119877)

to itself the same holds for sl2(119877)[119911 119911

minus1) This proves the full

claim

Next we have a look at the different deformations Thedeformations of119862

0by elements of119866

lt0are determined by that

of1198760Therefore we focus on that element andwe consider for

some 119892 = exp(119883) = exp(suminfin119895=1119883119895119911minus119895) isin 119866lt0

the deformation

119876 = 1198921198760119892minus1 fl

infin

sum

119894=0

119876119894119911minus119894

= (minus119894 0

0 119894) + [119883

1 1198760] 119911minus1

+ [1198832 1198760] +

1

2[1198831 [1198831 1198760]] 119911minus2+ sdot sdot sdot

(17)

of 1198760 From this formula we see directly that if each 119883

119894=

(minus120572119894 120573119894120574119894 120572119894

) 119894 = 1 2 then

1198761= (

0 21198941205731

minus2119894120574119894

0) = (

0 119902

119903 0)

1198762= (

1199021111990212

1199022111990222

)

= (minus211989412057311205741

2119894 (1205732minus 12057211205731)

minus2119894 (1205742+ 12057211205741) 2119894120573

11205741

)

(18)

In particular we get in this way that 11990211= minus119894(1199021199032) and

11990222= 119894(1199021199032) Since 119876 is the deformation of 119876

0and Id119911 is

central the deformation of each 1198760119911119898 119898 isin Z is given by

119876119911119898The deformation of the Lie algebra119862

1by elements from

119866⩽0

is basically determined by that of the element 1198760119911 So

we focus on the deformation of this element Using the samenotations as at the deformation of 119876

0by 119866lt0 we get that the

deformation of 1198760119911 by a 119870119892 isin 119866

⩽0 with 119870 isin gl

2(119877)lowast and

119892 isin 119866lt0 looks like

119885 = 1198701198921198760119911119892minus1119870minus1 fl

infin

sum

119894=0

119870119876119894119870minus11199111minus119894=

infin

sum

119894=0

1198851198941199111minus119894

= 1198850119911 + [119870119883

1119870minus1 1198850] + sdot sdot sdot

(19)

Consequently the corresponding deformation of each 1198760119911119898

119898 ⩾ 1 is 119885119911119898minus1

4 Advances in Mathematical Physics

Now we are looking for deformations 119876 of the form (17)such that the evolution with respect to the 120597

119898 satisfies the

following for all119898 ⩾ 0

120597119898(119876) = [(119876119911

119898)⩾0 119876] = minus [(119876119911

119898)lt0 119876] (20)

where the second identity follows from the fact that all 119876119911119898commute Equations (20) are called the Lax equations ofthe AKNS hierarchy and the deformation 119876 satisfying theseequations is called a solution of the hierarchy Note that119876 = 119876

0is a solution of the AKNS hierarchy and it is

called the trivial one AKNS equation (4) occurs among thecompatibility equations of this system as we will see furtheron Note that (20) for 119898 = 0 is simply 120597

0(119876) = [119876

0 119876]

Therefore if 1205970= 120597120597119905

0and both 119890minus1198941199050 and 1198901198941199050 belong to the

algebra 119877 of matrix coefficients then we can introduce theloop isin sl

2(119877)[119911 119911

minus1) given by

fl exp (minus11990501198760) 119876 exp (119905

01198760)

= sum

119896⩾0

exp (minus11990501198760) 119876119896exp (119905

01198760) 119911minus119896

(21)

which is easily seen to satisfy 1205970() = 0 This handles then

the dependence of 119876 of 1199050

Among the deformations 119885 of the form (19) we look for119885 such that their evolution with respect to 120597

119898 satisfies the

following for all119898 ⩾ 1

120597119898(119885) = [(119885119911

119898minus1)gt0 119885] = minus [(119885119911

119898minus1)⩽0 119885] (22)

where the second identity follows from the fact that all119885119911119898minus1 commute Since (22) correspond to the strict cut-

off (10) they are called the Lax equations of the strict AKNShierarchy and the deformation 119885 is called a solution of thishierarchy Again there is at least one solution 119885 = 119876

0119911 It is

called the trivial solution of the hierarchyFor both systems (20) and (22) one can speak of compat-

ibility Namely there holds the following result

Proposition 3 Both sets of Lax equations (20) and (22) arethe so-called compatible systems that is the projections 119861

119898fl

(119876119911119898)⩾0| 119898 ⩾ 0 of a solution 119876 to (20) satisfy the zero

curvature relations

1205971198981(1198611198982) minus 1205971198982(1198611198981) minus [119861

1198981 1198611198982] = 0 (23)

and the projections 119862119898fl (119885119911

119898minus1)gt0| 119898 ⩾ 1 corresponding

to a solution 119885 to (22) satisfy the zero curvature relations

1205971198981(1198621198982) minus 1205971198982(1198621198981) minus [119862

1198981 1198621198982] = 0 (24)

Proof The idea is to show that the left-hand side of (23) and(24) respectively belongs to

sl2(119877) [119911 119911

minus1)⩾0cap sl2(119877) [119911 119911

minus1)lt0

respectively sl2(119877) [119911 119911

minus1)gt0cap sl2(119877) [119911 119911

minus1)⩽0

(25)

and thus has to be zero We give the proof for 119862119898 that for

119861119898 is similar and is left to the reader The inclusion in the

first factor is clear as both 119862119898and 120597119899(119862119898) belong to the Lie

subalgebra sl2(119877)[119911 119911

minus1)gt0To show the other one we use Lax

equations (22) Note that the same Lax equations hold for allthe 119911119873119885 | 119873 ⩾ 0

120597119898(119911119873119885) = [(119885119911

119898minus1)gt0 119911119873119885] (26)

By substituting 119862119898119894

= 119911119898119894minus1119885 minus (119911

119898119894minus1119885)lt0 we get the

identities for

1205971198981(1198621198982) minus 1205971198982(1198621198981)

= 1205971198981(1199111198982minus1119885) minus 120597

1198981((1199111198982minus1119885)

⩽0)

minus 1205971198982(1199111198981minus1119885) + 120597

1198982((1199111198981minus1119885)

⩽0)

= [1198621198981 1199111198982minus1119885] minus [119862

1198982 1199111198981minus1119885]

minus 1205971198981((1199111198982minus1119885)

⩽0) + 1205971198982((1199111198981minus1119885)

⩽0)

(27)

and

[1198621198981 1198621198982]

= [1199111198981minus1119885 minus (119911

1198981minus1119885)⩽0 1199111198982minus1119885 minus (119911

1198982minus1119885)⩽0]

= minus [(1199111198981minus1119885)

⩽0 1199111198982minus1119885]

+ [(1199111198982minus1119885)

⩽0 1199111198981minus1119885]

+ [(1199111198981minus1119885)

⩽0 (1199111198982minus1119885)

⩽0]

(28)

Taking into account the second identity in (22) we see thatthe left-hand side of (24) is equal to

minus 1205971198981((1199111198982minus1119885)

⩽0) + 1205971198982((1199111198981minus1119885)

⩽0)

minus [(1199111198981minus1119885)

⩽0 (1199111198982minus1119885)

⩽0]

(29)

This element belongs to the Lie subalgebra sl2(119877)[119911 119911

minus1)⩽0

and that proves the claim

Reversely we have the following

Proposition 4 Suppose we have a deformation 119876 of the type(17) or a deformation 119885 of the form (19) Then there holds thefollowing

(1) Assume that the projections 119861119898fl (119876119911

119898)⩾0| 119898 ⩾ 0

satisfy the zero curvature relations (23) Then 119876 is asolution of the AKNS hierarchy

(2) Similarly if the projections 119862119898

fl (119885119911119898minus1)⩾0| 119898 ⩾

1 satisfy the zero curvature relations (24) then 119885 is asolution of the strict AKNS hierarchy

Advances in Mathematical Physics 5

Proof Againwe prove the statement for119885 that for119876 is shownin a similarway So assume that there is one Lax equation (22)that does not hold Then there is a119898 ⩾ 1 such that

120597119898(119885) minus [119862

119898 119885] = sum

119895⩽119896(119898)

119883119895119911119895 with 119883

119896(119898)= 0 (30)

Since both 120597119898(119885) and [119862

119898 119885] are of order smaller than or

equal to one in 119911 we know that 119896(119898) ⩽ 1 Further we can saythat for all119873 ⩾ 0

120597119898(119911119873119885) minus [119862

119898 119911119873119885] = sum

119895⩽119896(119898)

119883119895119911119895+119873

with 119883119896(119898)

= 0

(31)

and we see by letting 119873 go to infinity that the right-handside can obtain any sufficiently large order in 119911 By the zerocurvature relation for119873 and119898 we get for the left-hand side

120597119898(119911119873119885) minus [119862

119898 119911119873119885]

= 120597119898(119862119873) minus [119862

119898 119862119873] + 120597119898((119911119873119885)⩽0)

minus [119862119898 (119911119873119885)⩽0]

= 120597119873(119862119898) + 120597119898((119911119873119885)⩽0) minus [119862

119898 (119911119873119885)⩽0]

(32)

and this last expression is of order smaller or equal to 119898 in119911 This contradicts the unlimited growth in orders of 119911 of theright-hand side Hence all Lax equations (22) have to holdfor 119885

Because of the equivalence between Lax equations (20)for 119876 and the zero curvature relations (23) for 119861

119898 we call

this last set of equations also the zero curvature form of theAKNS hierarchy Similarly the zero curvature relations (24)for 119862

119898 are called the zero curvature form of the strict AKNS

hierarchyBesides the zero curvature relations for the cut-off rsquos

119861119898 and 119862

119898 respectively corresponding to respectively

a solution 119876 of the AKNS hierarchy and a solution 119885 of thestrict AKNS hierarchy also other parts satisfy such relationsDefine

119860119898fl 119861119898minus 119876119911119898 119898 ⩾ 0

119863119898fl 119862119898minus 119885119911119898minus1 119898 ⩾ 1

(33)

Then we have the following result

Corollary 5 The following relations hold

(i) The parts 119860119898| 119898 ⩾ 0 of a solution 119876 of the AKNS

hierarchy satisfy

1205971198981(1198601198982) minus 1205971198982(1198601198981) minus [119860

1198981 1198601198982] = 0 (34)

(ii) The parts 119863119898| 119898 ⩾ 0 of a solution 119885 of the strict

AKNS hierarchy satisfy

1205971198981(1198631198982) minus 1205971198982(1198631198981) minus [119863

1198981 1198631198982] = 0 (35)

Proof Again we show the result only in the strict case Notethat 119885119911119898minus1 satisfy Lax equations similar to 119885

120597119894(119885119911119898minus1) = [119863

119894 119885119911119898minus1] 119894 ⩾ 1 (36)

Now we substitute in the zero curvature relations for 119862119898

everywhere the relation 119862119898= 119863119898+ 119885119911119898minus1 and use the Lax

equations above and the fact that all 119885119911119898minus1 commute Thisgives the desired result

To clarify the link with the AKNS equation considerrelation (23) for119898

1= 2 and119898

2= 1

1205972(1198760119911 + 119876

1) = 1205971(11987601199112+ 1198761119911 + 119876

2)

+ [1198762 1198760119911 + 119876

1]

(37)

Then this identity reduces in sl2(119877)[119911 119911

minus1)⩾0 since 119876

0is

constant to the following two equalities

1205971(1198761) = [119876

0 1198762]

1205972(1198761) = 1205971(1198762) + [119876

2 1198761]

(38)

The first gives an expression of the off-diagonal terms of 1198762

in the coefficients 119902 and 119903 of 1198761 that is

11990212=119894

21205971(119902)

11990221= minus

119894

21205971(119903)

(39)

and the second equation becomes AKNS equations (4) if onehas 1205971= 120597120597119909 and 120597

2= 120597120597119905

3 The Linearization of Both Hierarchies

The zero curvature form of both hierarchies points at thepossible existence of a linear system of which the zerocurvature equations form the compatibility conditions Wepresent here such a system for each hierarchy For theAKNS hierarchy this system the linearization of the AKNShierarchy is as follows take a potential solution119876 of the form(17) and consider the system

119876120595 = 1205951198760

120597119898(120595) = 119861

119898120595 forall119898 ⩾ 0 119861

119898= (119876119911

119898)⩾0

(40)

Likewise for a potential solution 119885 of the strict AKNShierarchy of the form (19) the linearization of the strict AKNShierarchy is given by

119885120593 = 1205931198760119911

120597119898(120593) = 119862

119898120593 forall119898 ⩾ 1 119862

119898= (119885119911

119898minus1)gt0

(41)

Before specifying120595 and120593 we show themanipulations neededto derive the Lax equations of the hierarchy from theirlinearizationWedo this for the strictAKNShierarchy for the

6 Advances in Mathematical Physics

AKNS hierarchy one proceeds similarly Apply 120597119898to the first

equation in (41) and use both equalities in (41) in the sequel

120597119898(119885120593 minus 120593119876

0119911) = 120597

119898(119885) 120593 + 119885120597

119898(120593) minus 120597

119898(120593)1198760119911

= 0 = 120597119898(119885) 120593 + 119885119862

119898120593 minus 119862

1198981205931198760119911

= 120597119898(119885) minus [119862

119898 119885] 120593 = 0

(42)

Hence if we can scratch 120593 from (42) then we obtain the Laxequations of the strict AKNS hierarchy To get the propersetup where all these manipulations make sense we first havea look at the linearization for the trivial solutions119876 = 119876

0and

119885 = 1198760119911 In the AKNS case we have then

11987601205950= 12059501198760

120597119898(1205950) = 119876

01199111198981205950 forall119898 ⩾ 0

(43)

and for its strict version

11987601199111205930= 12059301198760119911

120597119898(1205930) = 119876

01199111198981205930 forall119898 ⩾ 1

(44)

Assuming that each derivation 120597119898equals 120597120597119905

119898 one arrives

for (43) at the solution

1205950= 1205950(119905 119911) = exp(

infin

sum

119898=0

1199051198981198760119911119898) (45)

where 119905 is the short hand notation for 119905119898| 119898 ⩾ 0 and for

(44) it leads to

1205930= 1205930(119905 119911) = exp(

infin

sum

119898=1

1199051198981198760119911119898)

with 119905 = 119905119898| 119898 ⩾ 1

(46)

In general one needs in the linearizations (120595) and (120593)respectively a left action with elements like 119876 119861

119898and 119885

119862119898from gl

2(119877)[119911 119911

minus1) respectively an action of all the 120597

119898

and a right action of 1198760and 119876

0119911 respectively This can

all be realized in suitable gl2(119877)[119911 119911

minus1) modules that are

perturbations of the trivial solutions 1205950and 120593

0 Consider in

a AKNS setting

M0= 119892 (119911) 120595

0=

119873

sum

119894=minusinfin

1198921198941199111198941205950| 119892 (119911)

=

119873

sum

119894=minusinfin

119892119894119911119894isin gl2(119877) [119911 119911

minus1)

(47)

and in a strict AKNS setting

M1= 119892 (119911) 120593

0=

119873

sum

119894=minusinfin

1198921198941199111198941205930| 119892 (119911)

=

119873

sum

119894=minusinfin

119892119894119911119894isin gl2(119877) [119911 119911

minus1)

(48)

where the products 119892(119911)1205950and 119892(119911)120593

0should be seen as

formal and both factors should be kept separate On bothM0

andM1 one can define the required actions for each ℎ(119911) isin

gl2(119877)[119911 119911

minus1) define

ℎ (119911) sdot 119892 (119911) 1205950fl ℎ (119911) 119892 (119911) 120595

0

respectively ℎ (119911) sdot 119892 (119911) 1205930fl ℎ (119911) 119892 (119911) 120593

0

(49)

We define the right-hand action of 1198760and 119876

0119911 respectively

by

119892 (119911) 12059501198760fl 119892 (119911)119876

0 1205950

respectively 119892 (119911) 12059301198760119911 fl 119892 (119911)119876

0119911 1205930

(50)

and the action of each 120597119898by

120597119898(119892 (119911) 120595

0)

=

119873

sum

119894=minusinfin

120597119898(119892119894) 119911119894+

119873

sum

119894=minusinfin

1198921198941198760119911119894+1198981205950

120597119898(119892 (119911) 120593

0)

=

119873

sum

119894=minusinfin

120597119898(119892119894) 119911119894+

119873

sum

119894=minusinfin

1198921198941198760119911119894+1198981205930

(51)

Analogous to the terminology in the scalar case (see [5]) wecall the elements of M

0and M

1oscillating matrices Note

that both M0and M

1are free gl

2(119877)[119911 119911

minus1) modules with

generators 1205950and 120593

0 respectively because for each ℎ(119911) isin

gl2(119877)[119911 119911

minus1) we have

ℎ (119911) sdot 1205950= ℎ (119911) sdot 1 120595

0= ℎ (119911) 120595

0

respectively ℎ (119911) sdot 1205930= ℎ (119911) sdot 1 120595

0= ℎ (119911) 120593

0

(52)

Hence in order to be able to perform legally the scratching of120595 = ℎ(119911)120595

0or 120593 = ℎ(119911)120593

0 it is enough to find oscillating

matrices such that ℎ(119911) is invertible in gl2(119877)[119911 119911

minus1) We will

now introduce a collection of such elements that will occur atthe construction of solutions of both hierarchies

For119898 = (1198981 1198982) isin Z2 let 120575(119898) isin gl

2(119877)[119911 119911

minus1) be given

by

120575 (119898) = (1199111198981 0

0 1199111198982) (53)

Then 120575(119898) has 120575(minus119898) as its inverse in gl2(119877)[119911 119911

minus1) and the

collection Δ = 120575(119898) | 119898 isin Z2 forms a group An element120595 isinM

0is called an oscillating matrix of type 120575(119898) if it has the

form

120595 = ℎ (119911) 120575 (119898) 1205950 with ℎ (119911) isin 119866

lt0 (54)

These oscillatingmatrices are examples of elements ofM0for

which the scratching procedure is valid Let 119876 be a potentialsolution of the AKNS hierarchy of the form (17) If there isan oscillating function 120595 of type 120575(119898) such that for 119876 and 120595

Advances in Mathematical Physics 7

(40) hold then we call120595 awave matrix of the AKNS hierarchyof type 120575(119898) In particular 119876 is then a solution of the AKNShierarchy and the first equation in (40) implies

119876ℎ (119911) 120575 (119898) = ℎ (119911)1198760120575 (119898) 997904rArr

119876 = ℎ (119911)1198760ℎ (119911)minus1

(55)

in other words the solution 119876 is totally determined by thewave matrix Similarly we call an element 120593 isin M

1an

oscillating matrix of type 120575(119898) if it has the form

120593 = ℎ (119911) 120575 (119898) 1205930 with ℎ (119911) isin 119866

⩽0 (56)

This type of oscillating matrices are examples of elements ofM1for which the scratching procedure is valid Let 119885 be a

potential solution of the strict AKNS hierarchy of the form(19) If there is an oscillating function 120593 of type 120575(119898) in M

1

such that for119885 and120593 (41) hold thenwe call120593 awavematrix ofthe strict AKNS hierarchy of type 120575(119898) In particular119885 is thena solution of the strict AKNS hierarchy and the first equationin (41) implies

119885ℎ (119911) 120575 (119898) = ℎ (119911)1198760119911120575 (119898) 997904rArr

119885 = ℎ (119911)1198760119911ℎ (119911)

minus1

(57)

so that also here the wave matrix fully determines thesolution

For both hierarchies there is a milder condition that is tobe satisfied by oscillating matrices of a certain type in orderthat they become a wave matrix of that hierarchy

Proposition 6 Let120595 = ℎ(119911)120575(119898)1205950be an oscillatingmatrix

of type 120575(119898) in M0and 119876 = ℎ(119911)119876

0ℎ(119911)minus1 the corresponding

potential solution of the AKNS hierarchy Similarly let 120593 =

ℎ(119911)120575(119898)1205930be such a matrix in M

1with potential solution

119885 = ℎ(119911)1198760119911ℎ(119911)minus1

(a) If there exists for each 119898 ⩾ 0 an element 119872119898isin

gl2(119877)[119911 119911

minus1)⩾0

such that

120597119898(120595) = 119872

119898120595 (58)

then each119872119898= (119876119911

119898)⩾0

and 120595 is a wave matrix oftype 120575(119898) for the AKNS hierarchy

(b) If there exists for each 119898 ⩾ 1 an element 119873119898isin

gl2(119877)[119911 119911

minus1)gt0

such that

120597119898(120593) = 119873

119898120593 (59)

then each 119873119898= (119885119911

119898minus1)gt0

and 120593 is a wave matrix oftype 120575(119898) for the strict AKNS hierarchy

Proof We give the proof again for the strict AKNS casethe other one is similar By using the fact that M

1is a free

gl2(119877)[119911 119911

minus1)module with generator 120593

0we can translate the

relations 120597119898(120593) = 119873

119898120593 into equations in gl

2(119877)[119911 119911

minus1) This

yields

120597119898(ℎ (119911)) + ℎ (119911)119876

0119911119898= 119873119898ℎ (119911) 997904rArr

120597119898(ℎ (119911)) ℎ (119911)

minus1+ 119885119911119898minus1

= 119873119898

(60)

Projecting the right-hand side on gl2(119877)[119911 119911

minus1)gt0

gives us theidentity

(119885119911119898minus1)gt0= 119873119898 (61)

we are looking for

In the next section we produce a context where we canconstruct wave matrices for both hierarchies in which theproduct is real

4 The Construction of the Solutions

In this section we will show how to construct a wide class ofsolutions of both hierarchies This is done in the style of [6]We first describe the group of loops we will work with Foreach 0 lt 119903 lt 1 let 119860

119903be the annulus

119911 | 119911 isin C 119903 ⩽ |119911| ⩽1

119903 (62)

Following [7] we use the notation 119871anGL2(C) for the col-lection of holomorphic maps from some annulus 119860

119903into

GL2(C) It is a group with respect to pointwise multiplication

and contains in a natural way GL2(C) as a subgroup as the

collection of constant maps into GL2(C) Other examples

of elements in 119871anGL2(C) are the elements of Δ However119871anGL2(C) is more than just a group it is an infinitedimensional Lie group Its manifold structure can be read offfrom its Lie algebra 119871angl2(C) consisting of all holomorphicmaps 120574 119880 rarr gl

2(C) where 119880 is an open neighborhood of

some annulus 119860119903 0 lt 119903 lt 1 Since gl

2(C) is a Lie algebra

the space 119871angl2(C) becomes a Lie algebra with respect to thepointwise commutator Topologically the space 119871angl2(C) isthe direct limit of all the spaces 119871an119903gl2(C) where this lastspace consists of all 120574 corresponding to the fixed annulus119860119903 One gives each 119871an119903gl2(C) the topology of uniform

convergence and with that topology it becomes a Banachspace In this way 119871angl2(C) becomes a Frechet space Thepointwise exponential map defines a local diffeomorphismaround zero in 119871angl2(C) (see eg [8])

Now each 120574 isin 119871angl2(C) possesses an expansion in aFourier series

120574 =

infin

sum

119896=minusinfin

120574119896119911119896 with each 120574

119896isin gl2(C) (63)

that converges absolutely on the annulus 119860119903

infin

sum

119896=minusinfin

10038171003817100381710038171205741198961003817100381710038171003817 119903minus|119896|

lt infin (64)

This Fourier expansion is used to make two different decom-positions of the Lie algebra119871angl2(C)Thefirst is the analoguefor the present Lie algebra 119871angl2(C) of decomposition (8) ofgl2(119877)[119911 119911

minus1) that lies at the basis of the Lax equations of the

AKNS hierarchy Namely consider the subspaces

119871angl2 (C)⩾0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=0

120574119896119911119896

8 Advances in Mathematical Physics

119871angl2 (C)lt0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =minus1

sum

119896=minusinfin

120574119896119911119896

(65)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra The first Lie algebra consists ofthe elements in 119871angl2(C) that extend to holomorphic mapsdefined on some disk 119911 isin C | |119911| ⩽ 1119903 0 lt 119903 lt 1 andthe second Lie algebra corresponds to the maps in 119871angl2(C)that have a holomorphic extension towards some disk aroundinfinity 119911 isin P1(C) | |119911| ⩾ 119903 0 lt 119903 lt 1 and that arezero at infinity To each of the two Lie subalgebras belongsa subgroup of 119871anGL2(C) The pointwise exponential mapapplied to elements of 119871angl2(C)lt0 yields elements of

119880minus= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

minus1

sum

119896=minusinfin

120574119896119911119896 (66)

and the exponential map applied to elements of 119871angl2(C)⩾0maps them into

119875+= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740 +

infin

sum

119896=1

120574119896119911119896 with 120574

0

isin GL2(C)

(67)

Both119880minusand 119875+are easily seen to be subgroups of 119871anGL2(C)

and since the direct sum of their Lie algebras is 119871angl2(C)their product

Ω = 119880minus119875+ (68)

is open in 119871anGL2(C) and is called like in the finite dimen-sional case the big cell with respect to 119880

minusand 119875

+

The second decomposition is the analogue of the splitting(11) of gl

2(119877)[119911 119911

minus1) that led to the Lax equations of the strict

AKNS hierarchy Now we consider the subspaces

119871angl2 (C)gt0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=1

120574119896119911119896

119871angl2 (C)⩽0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =0

sum

119896=minusinfin

120574119896119911119896

(69)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra 119871angl2(C)gt0 consists of mapswhose holomorphic extension to 119911 = 0 equals zero inthat point and 119871angl2(C)⩽0 is the set of maps that extendhomomorphically to a neighborhood of infinity To eachof them belongs a subgroup of 119871anGL2(C) The pointwiseexponential map applied to elements of 119871angl2(C)gt0 yieldselements of

119880+= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

infin

sum

119896=1

120574119896119911119896 (70)

and the exponential map applied to elements of 119871angl2(C)⩽0maps them into

119875minus= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740

+

minus1

sum

119896=minusinfin

120574119896119911119896 with 120574

0isin GL2(C)

(71)

Both119875minusand119880

+are easily seen to be subgroups of 119871anGL2(C)

Since the direct sum of their Lie algebras is 119871angl2(C) theirproduct 119875

minus119880+is open and because 119875

+= GL

2(C)119880+=

119880+GL2(C) and 119875

minus= GL2(C)119880minus= 119880minusGL2(C) this gives the

equality

119875minus119880+= Ω = 119880

minus119875+ (72)

the big cell in 119871anGL2(C)The next two subgroups of 119871anGL2(C) correspond to

the exponential factors in both linearizations The group ofcommuting flows relevant for the AKNS hierarchy is thegroup

Γ0= 1205740(119905) = exp(

infin

sum

119894=0

1199051198941198760119911119894) | 1205740isin 119875+ (73)

and similarly for the strict AKNS hierarchy we use thecommuting flows from

Γ1= 1205741(119905) = exp(

infin

sum

119894=1

1199051198941198760119911119894) | 1205741isin 119880+ (74)

Besides the groups Δ Γ0 and Γ

1 there are more subgroups in

119871anGL2(C) that commute with those three groups and theyare in a sense complimentary to Δ Γ

0and Δ Γ

1 respectively

That is why we use the following notations for them in 119880minus

there is

Γ119888

0= 120574119888

0(119905) = exp(

infin

sum

119896=1

1199041198961198760119911minus119896) | 120574119888

0isin 119880minus (75)

and in 119875minuswe have

Γ119888

1= 120574119888

1(119905) = exp(

infin

sum

119896=0

1199041198961198760119911minus119896) | 120574119888

1isin 119875minus (76)

We have now all ingredients to describe the constructionof the solutions to each hierarchy and we start with theAKNS hierarchy Take inside the product 119871anGL2(C) times Δthe collection 119878

0of pairs (119892 120575(119898)) such that there exists a

1205740(119905) isin Γ

0satisfying

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898) isin Ω = 119880

minus119875+ (77)

For such a pair (119892 120575(119898)) we take the collection Γ0(119892 120575(119898))

of all 1205740(119905) satisfying the condition (77) This is an open

nonempty subset of Γ0 Let 119877

0(119892 120575(119898)) be the algebra of

analytic functions Γ0(119892 120575(119898)) rarr C This is the algebra of

Advances in Mathematical Physics 9

functions 119877 that we associate with the point (119892 120575(119898)) isin 1198780

and for the commuting derivations of 1198770(119892 120575(119898)) we choose

120597119894fl 120597120597119905

119894 119894 ⩾ 0 By property (77) we have for all 120574

0(119905) isin

Γ0(119892 120575(119898))

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898)

= 119906minus(119892 120575 (119898))

minus1

119901+(119892 120575 (119898))

(78)

with 119906minus(119892 120575(119898)) isin 119880

minus119901+(119892 120575(119898)) isin 119875

+Then all thematrix

coefficients in the Fourier expansions of the elements 119906minus(119892

120575(119898)) and 119901+(119892 120575(119898)) belong to the algebra 119877

0(119892 120575(119898)) It is

convenient to write relation (78) in the form

Ψ119892120575(119898)

fl 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905)

= 119901+(119892 120575 (119898)) 120575 (119898) 120574

0(119905) 119892minus1

= 119901+(119892 120575 (119898)) 120574

0(119905) 120575 (119898) 119892

minus1

= 119902+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(79)

with 119902+(119892 120575(119898)) isin 119875

+ Clearly Ψ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M0 where all the products in the

decomposition 119906minus(119892 120575(119898))120575(119898)120574

0(119905) are no longer formal

but real This is our potential wave matrix for the AKNShierarchy

To get the potential wave function for the strict AKNShierarchy we proceed similarly only this time we use thedecomposition Ω = 119875

minus119880+ Consider again the product

119871anGL2(C) times Δ and the subset 1198781of pairs (119892 120575(119898)) such that

there exists a 1205741(119905) isin Γ

1satisfying

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898) isin Ω = 119875

minus119880+ (80)

For such a pair (119892 120575(119898)) we define Γ1(119892 120575(119898)) as the set of

all 1205741(119905) satisfying condition (77) This is an open nonempty

subset of Γ1 Let 119877

1(119892 120575(119898)) be the algebra of analytic

functions Γ1(119892 120575(119898)) rarr C This is the algebra of functions

119877 that we associate with the point (119892 120575(119898)) isin 1198781and for the

commuting derivations of 1198771(119892 120575(119898)) we choose 120597

119894fl 120597120597119905

119894

119894 gt 0 By property (80) we have for all 1205741(119905) isin Γ

1(119892 120575(119898))

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898)

= 119901minus(119892 120575 (119898))

minus1

119906+(119892 120575 (119898))

(81)

with 119901minus(119892 120575(119898)) isin 119875

minus 119906+(119892 120575(119898)) isin 119880

+ Then all

the matrix coefficients in the Fourier expansions of theelements 119901

minus(119892 120575(119898)) and 119906

+(119892 120575(119898)) belong to the algebra

1198771(119892 120575(119898)) It is convenient to write relation (81) in the form

Φ119892120575(119898)

fl 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905)

= 119906+(119892 120575 (119898)) 120575 (119898) 120574

1(119905) 119892minus1

= 119906+(119892 120575 (119898)) 120574

1(119905) 120575 (119898) 119892

minus1

= 119908+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(82)

with 119908+(119892 120575(119898)) isin 119880

+ Clearly Φ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M1where all the products in the

decomposition 119901minus(119892 120575(119898))120575(119898)120574

1(119905) are no longer formal

but realTo show that Ψ

119892120575(119898)is a wave matrix for the AKNS

hierarchy of type 120575(119898) and Φ119892120575(119898)

one for the strict AKNShierarchy we use the fact that it suffices to prove the relationsin Proposition 6 We treat first Ψ

119892120575(119898) In (79) we presented

two different decompositions of Ψ119892120575(119898)

that we will use tocompute 120597

119894(Ψ119892120575(119898)

) With respect to 119906minus(119892 120575(119898))120575(119898)120574

0(119905)

there holds120597119894(Ψ119892120575(119898)

)

= 120597119894(119906minus(119892 120575 (119898))) 119906

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905) = 119872

119894Ψ119892120575(119898)

(83)

with119872119894isin gl2(119877(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911 If we take

the decomposition 119902+(119892 120575(119898))120575(119898)119892

minus1 then we get

120597119894(Ψ119892120575(119898)

)

= 120597119894(119902+(119892 120575 (119898))) 119902

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(84)

with 120597119894(119902+(119892 120575(119898)))119902

+(119892 120575(119898))

minus1 of the formsum119895⩾0119902119895119911119895 with

all 119902119895isin gl2(119877(119892 120575(119898))) As Ψ

119892120575(119898)is invertible we obtain

119872119894= sum119895⩾0119902119895119911119895 and thus 119872

119894= sum119894

119895=0119902119895119911119895 and that is the

desired resultWe handle the case of Φ

119892120575(119898)in a similar way Also in

(79) you can find two different decompositions of Φ119892120575(119898)

that we will use to compute 120597119894(Φ119892120575(119898)

) With respect to theexpression 119901

minus(119892 120575(119898))120575(119898)120574

1(119905) we get for all 119894 ⩾ 1

120597119894(Φ119892120575(119898)

)

= 120597119894(119901minus(119892 120575 (119898))) 119901

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905) = 119873

119894Φ119892120575(119898)

(85)

again with 119873119894isin gl2(1198771(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911

Next we take the decomposition 119908+(119892 120575(119898))120575(119898)119892

minus1 andthat yields

120597119894(Φ119892120575(119898)

)

= 120597119894(119908+(119892 120575 (119898)))119908

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(86)

Since the zeroth order term of 119908+(119892 120575(119898)) is Id we note

that the element 120597119894(119908+(119892 120575(119898)))119908

+(119892 120575(119898))

minus1 has the formsum119895⩾1119908119895119911119895 with all 119908

119895isin gl2(1198771(119892 120575(119898))) As Φ

119892120575(119898)is also

invertible we obtain119873119894= sum119895⩾1119908119895119911119895 and thus119873

119894= sum119894

119895=1119908119895119911119895

and that is the result we were looking for in the strict caseThe abovementioned constructions are not affected seri-

ously by a number of transformations For example if wewrite 120575 = 120575((1 1)) then all 120575119896 119896 isin Z are central elementsand we have for all (119892 120575(119898)) isin 119878

0and 119878

1respectively and all

119896 isin Z

Ψ119892120575(119898)120575

119896 = Ψ119892120575(119898)

120575119896

respectively Φ119892120575(119898)120575

119896 = Φ119892120575(119898)

120575119896

(87)

10 Advances in Mathematical Physics

Also for any 1199011isin 119875+ each 120574119888

0(119904) isin Γ

119888

0 and all (119892 120575(119898)) isin

1198780 we see that the element (120574119888

0(119904)119892120575(minus119898)119901

1120575(119898) 120575(119898)) also

belongs to 1198780 for

120575 (119898) 1205740(119905) 120574119888

0(119904) 119892120575 (minus119898) 119901

1120575 (119898) 120574

0(119905)minus1120575 (minus119898)

= 120574119888

0(119904) 119906minus1

minus119901+1205740(119905) 11990111205740(119905)minus1

(88)

In particular we see that

Ψ120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575(119898)

= 119906minus120574119888

0(119904)minus1120575 (119898) 120574

0(119905) (89)

Its analogue in the strict case is as follows for any 1199061isin

119880+ each 120574119888

1(119904) isin Γ

119888

1 and all (119892 120575(119898)) isin 119878

1 the element

(120574119888

1(119904)119892120575(minus119898)119906

1120575(119898) 120575(119898)) also belongs to 119878

1and there

holds

Φ120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575(119898)

= 119901minus120574119888

1(119904)minus1120575 (119898) 120574

1(119905) (90)

We resume the foregoing results in the following theorem

Theorem 7 Consider the product spaceΠ fl 119871anGL2(C) times ΔThen the following holds

(a) For each point (119892 120575(119898)) isin Π that satisfies condition(77) we have a wave matrix of type 120575(119898) of the AKNShierarchyΨ

119892120575(119898) defined by (79) and this determines a

solution 119876119892120575(119898)

= 119906minus(119892 120575(119898))119876

0119906minus(119892 120575(119898))

minus1 of theAKNS hierarchy For each 119901

1isin 119875+ every 120574119888

0(119904) isin Γ

119888

0

and all powers of 120575 one has

119876120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575

119896120575(119898)

= 119876119892120575(119898)

(91)

(b) For each point (119892 120575(119898)) isin Π that satisfies condition(80) we have a wave matrix of type 120575(119898) of thestrict AKNS hierarchy Φ

119892120575(119898) defined by (82) and it

determines a solution 119885119892120575(119898)

= 119901minus(119892 120575(119898))119876

0119911119901minus(119892

120575(119898))minus1 of the strict AKNS hierarchy For each 119906

1isin 119880+

every 1205741198881(119904) isin Γ

119888

1 and all powers of 120575 one has

119885120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575

119896120575(119898)

= 119885119892120575(119898)

(92)

Competing Interests

The author declares that they have no competing interests

References

[1] M Adler P vanMoerbeke and P VanhaeckeAlgebraic Integra-bility Painleve Geometry and Lie Algebras vol 47 of Ergebnisseder Mathematik und ihrer Grenzgebiete Springer 2004

[2] G F Helminck A G Helminck and E A PanasenkoldquoIntegrable deformations in the algebra of pseudodifferentialoperators from a Lie algebraic perspectiverdquo Theoretical andMathematical Physics vol 174 no 1 pp 134ndash153 2013

[3] M J Ablowitz D J Kaup A C Newell and H Segur ldquoTheinverse scattering transform-Fourier analysis for nonlinearproblemsrdquo Studies in Applied Mathematics vol 53 no 4 pp249ndash315 1974

[4] H Flashka A CNewell and T Ratiu ldquoKac-Moody Lie algebrasand soliton equations II Lax equations associated with 119860

1

(1)rdquoPhysicaDNonlinear Phenomena vol 9 no 3 pp 300ndash323 1983

[5] EDateM JimboMKashiwara andTMiwa ldquoTransformationgroups for soliton equationsrdquo in Proceedings of the RIMSSymposium on Nonlinear Integrable SystemsmdashClassical Theoryand Quantum Theory T Jimbo and T Miwa Eds WorldScience Publisher Kyoto Japan 1981

[6] G Segal and G Wilson ldquoLoop groups and equations of KdVtyperdquo Publications Mathematiques de lrsquoIHES vol 61 pp 5ndash651985

[7] A Pressley and G Segal Loop Groups Oxford MathematicalMonographs Clarendon Press 1986

[8] R S Hamilton ldquoThe inverse function theorem of Nash andMoserrdquoBulletin of the AmericanMathematical Society vol 7 no1 pp 65ndash222 1982

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 3

formal power series C[[119905119898]] in the same variables with both

algebras equipped with the derivations 120597119898= 120597120597119905

119898 119898 ⩾ 0

in the AKNS setup or 120597119898= 120597120597119905

119898119898 ⩾ 1 in the strict AKNS

case

We let each derivation 120597119898occurring in some setting act

coefficient-wise on the matrices from gl2(119877) and that defines

then a derivation of this algebra The same holds for theextension to gl

2(119877)[119911 119911

minus1) defined by

120597119898(119883) fl

119873

sum

119895=minusinfin

120597119898(119883119895) 119911119895 (13)

We have now sufficient ingredients to discuss the AKNShierarchy and its strict version In the AKNS case our interestis in certain deformations of the Lie algebra 119862

0obtained

by conjugating with elements of the group corresponding togl2(119877)[119911 119911

minus1)lt0 At the strict version we are interested in

certain deformations of the Lie algebra1198621obtained by conju-

gating with elements from a group linked to gl2(119877)[119911 119911

minus1)⩽0

Note that for each 119883 isin gl2(119877)[119911 119911

minus1)lt0

the exponentialmap yields a well-defined element of the form

exp (119883) =infin

sum

119896=0

1

119896119883119896= Id + 119884 119884 isin gl

2(119877) [119911 119911

minus1)lt0

(14)

and with the formula for the logarithm one retrieves 119883 backfrom 119884 One verifies directly that the elements of the form(14) form a group with respect to multiplication and this wesee as the group 119866

lt0corresponding to gl

2(119877)[119911 119911

minus1)lt0

In the case of the Lie subalgebra gl2(119877)[119911 119911

minus1)⩽0 one

cannot move back and forth between the Lie algebra and itsgroup Nevertheless one can assign a proper group to thisLie algebra A priori the exponential exp(119884) of an element119884 isin gl

2(119877)[119911 119911

minus1)⩽0

does not have to define an element ingl2(119877)[119911 119911

minus1)⩽0 That requires convergence conditions How-

ever if it does then it belongs to

119866⩽0=

119870 =

infin

sum

119895=0

119870119895119911minus119895| all 119870

119895isin gl2(119877) 119870

0

isin gl2(119877)lowast

(15)

where gl2(119877)lowast denotes the elements in gl

2(119877) that have a

multiplicative inverse in gl2(119877) It is a direct verification that

119866⩽0

is a group and we see it as a proper group correspondingto the Lie algebra gl

2(119877)[119911 119911

minus1)⩽0 In fact 119866

⩽0is isomorphic

to the semidirect product of 119866lt0

and gl2(119877)lowast

Now there holds the following

Lemma 2 The group 119866⩽0

acts by conjugation on sl2(119877)[119911

119911minus1)

Proof Take first any 119892 isin 119866lt0 Then there is an119883 isin gl

2(119877)[119911

119911minus1)lt0

such that 119892 = exp(119883) Now there holds for every 119884 isinsl2(119877)[119911 119911

minus1)

119892119884119892minus1= exp (119883) 119884 exp (minus119883) = 119890ad(119883) (119884)

= 119884 +

infin

sum

119896=1

1

119896ad (119883)119896 (119884)

(16)

and this shows that the coefficients for the different powersof 119911 in this expression are commutators of elements of gl

2(119877)

and sl2(119877) and that proves the claim for elements from 119866

lt0

Since conjugation with an element from gl2(119877)lowast maps sl

2(119877)

to itself the same holds for sl2(119877)[119911 119911

minus1) This proves the full

claim

Next we have a look at the different deformations Thedeformations of119862

0by elements of119866

lt0are determined by that

of1198760Therefore we focus on that element andwe consider for

some 119892 = exp(119883) = exp(suminfin119895=1119883119895119911minus119895) isin 119866lt0

the deformation

119876 = 1198921198760119892minus1 fl

infin

sum

119894=0

119876119894119911minus119894

= (minus119894 0

0 119894) + [119883

1 1198760] 119911minus1

+ [1198832 1198760] +

1

2[1198831 [1198831 1198760]] 119911minus2+ sdot sdot sdot

(17)

of 1198760 From this formula we see directly that if each 119883

119894=

(minus120572119894 120573119894120574119894 120572119894

) 119894 = 1 2 then

1198761= (

0 21198941205731

minus2119894120574119894

0) = (

0 119902

119903 0)

1198762= (

1199021111990212

1199022111990222

)

= (minus211989412057311205741

2119894 (1205732minus 12057211205731)

minus2119894 (1205742+ 12057211205741) 2119894120573

11205741

)

(18)

In particular we get in this way that 11990211= minus119894(1199021199032) and

11990222= 119894(1199021199032) Since 119876 is the deformation of 119876

0and Id119911 is

central the deformation of each 1198760119911119898 119898 isin Z is given by

119876119911119898The deformation of the Lie algebra119862

1by elements from

119866⩽0

is basically determined by that of the element 1198760119911 So

we focus on the deformation of this element Using the samenotations as at the deformation of 119876

0by 119866lt0 we get that the

deformation of 1198760119911 by a 119870119892 isin 119866

⩽0 with 119870 isin gl

2(119877)lowast and

119892 isin 119866lt0 looks like

119885 = 1198701198921198760119911119892minus1119870minus1 fl

infin

sum

119894=0

119870119876119894119870minus11199111minus119894=

infin

sum

119894=0

1198851198941199111minus119894

= 1198850119911 + [119870119883

1119870minus1 1198850] + sdot sdot sdot

(19)

Consequently the corresponding deformation of each 1198760119911119898

119898 ⩾ 1 is 119885119911119898minus1

4 Advances in Mathematical Physics

Now we are looking for deformations 119876 of the form (17)such that the evolution with respect to the 120597

119898 satisfies the

following for all119898 ⩾ 0

120597119898(119876) = [(119876119911

119898)⩾0 119876] = minus [(119876119911

119898)lt0 119876] (20)

where the second identity follows from the fact that all 119876119911119898commute Equations (20) are called the Lax equations ofthe AKNS hierarchy and the deformation 119876 satisfying theseequations is called a solution of the hierarchy Note that119876 = 119876

0is a solution of the AKNS hierarchy and it is

called the trivial one AKNS equation (4) occurs among thecompatibility equations of this system as we will see furtheron Note that (20) for 119898 = 0 is simply 120597

0(119876) = [119876

0 119876]

Therefore if 1205970= 120597120597119905

0and both 119890minus1198941199050 and 1198901198941199050 belong to the

algebra 119877 of matrix coefficients then we can introduce theloop isin sl

2(119877)[119911 119911

minus1) given by

fl exp (minus11990501198760) 119876 exp (119905

01198760)

= sum

119896⩾0

exp (minus11990501198760) 119876119896exp (119905

01198760) 119911minus119896

(21)

which is easily seen to satisfy 1205970() = 0 This handles then

the dependence of 119876 of 1199050

Among the deformations 119885 of the form (19) we look for119885 such that their evolution with respect to 120597

119898 satisfies the

following for all119898 ⩾ 1

120597119898(119885) = [(119885119911

119898minus1)gt0 119885] = minus [(119885119911

119898minus1)⩽0 119885] (22)

where the second identity follows from the fact that all119885119911119898minus1 commute Since (22) correspond to the strict cut-

off (10) they are called the Lax equations of the strict AKNShierarchy and the deformation 119885 is called a solution of thishierarchy Again there is at least one solution 119885 = 119876

0119911 It is

called the trivial solution of the hierarchyFor both systems (20) and (22) one can speak of compat-

ibility Namely there holds the following result

Proposition 3 Both sets of Lax equations (20) and (22) arethe so-called compatible systems that is the projections 119861

119898fl

(119876119911119898)⩾0| 119898 ⩾ 0 of a solution 119876 to (20) satisfy the zero

curvature relations

1205971198981(1198611198982) minus 1205971198982(1198611198981) minus [119861

1198981 1198611198982] = 0 (23)

and the projections 119862119898fl (119885119911

119898minus1)gt0| 119898 ⩾ 1 corresponding

to a solution 119885 to (22) satisfy the zero curvature relations

1205971198981(1198621198982) minus 1205971198982(1198621198981) minus [119862

1198981 1198621198982] = 0 (24)

Proof The idea is to show that the left-hand side of (23) and(24) respectively belongs to

sl2(119877) [119911 119911

minus1)⩾0cap sl2(119877) [119911 119911

minus1)lt0

respectively sl2(119877) [119911 119911

minus1)gt0cap sl2(119877) [119911 119911

minus1)⩽0

(25)

and thus has to be zero We give the proof for 119862119898 that for

119861119898 is similar and is left to the reader The inclusion in the

first factor is clear as both 119862119898and 120597119899(119862119898) belong to the Lie

subalgebra sl2(119877)[119911 119911

minus1)gt0To show the other one we use Lax

equations (22) Note that the same Lax equations hold for allthe 119911119873119885 | 119873 ⩾ 0

120597119898(119911119873119885) = [(119885119911

119898minus1)gt0 119911119873119885] (26)

By substituting 119862119898119894

= 119911119898119894minus1119885 minus (119911

119898119894minus1119885)lt0 we get the

identities for

1205971198981(1198621198982) minus 1205971198982(1198621198981)

= 1205971198981(1199111198982minus1119885) minus 120597

1198981((1199111198982minus1119885)

⩽0)

minus 1205971198982(1199111198981minus1119885) + 120597

1198982((1199111198981minus1119885)

⩽0)

= [1198621198981 1199111198982minus1119885] minus [119862

1198982 1199111198981minus1119885]

minus 1205971198981((1199111198982minus1119885)

⩽0) + 1205971198982((1199111198981minus1119885)

⩽0)

(27)

and

[1198621198981 1198621198982]

= [1199111198981minus1119885 minus (119911

1198981minus1119885)⩽0 1199111198982minus1119885 minus (119911

1198982minus1119885)⩽0]

= minus [(1199111198981minus1119885)

⩽0 1199111198982minus1119885]

+ [(1199111198982minus1119885)

⩽0 1199111198981minus1119885]

+ [(1199111198981minus1119885)

⩽0 (1199111198982minus1119885)

⩽0]

(28)

Taking into account the second identity in (22) we see thatthe left-hand side of (24) is equal to

minus 1205971198981((1199111198982minus1119885)

⩽0) + 1205971198982((1199111198981minus1119885)

⩽0)

minus [(1199111198981minus1119885)

⩽0 (1199111198982minus1119885)

⩽0]

(29)

This element belongs to the Lie subalgebra sl2(119877)[119911 119911

minus1)⩽0

and that proves the claim

Reversely we have the following

Proposition 4 Suppose we have a deformation 119876 of the type(17) or a deformation 119885 of the form (19) Then there holds thefollowing

(1) Assume that the projections 119861119898fl (119876119911

119898)⩾0| 119898 ⩾ 0

satisfy the zero curvature relations (23) Then 119876 is asolution of the AKNS hierarchy

(2) Similarly if the projections 119862119898

fl (119885119911119898minus1)⩾0| 119898 ⩾

1 satisfy the zero curvature relations (24) then 119885 is asolution of the strict AKNS hierarchy

Advances in Mathematical Physics 5

Proof Againwe prove the statement for119885 that for119876 is shownin a similarway So assume that there is one Lax equation (22)that does not hold Then there is a119898 ⩾ 1 such that

120597119898(119885) minus [119862

119898 119885] = sum

119895⩽119896(119898)

119883119895119911119895 with 119883

119896(119898)= 0 (30)

Since both 120597119898(119885) and [119862

119898 119885] are of order smaller than or

equal to one in 119911 we know that 119896(119898) ⩽ 1 Further we can saythat for all119873 ⩾ 0

120597119898(119911119873119885) minus [119862

119898 119911119873119885] = sum

119895⩽119896(119898)

119883119895119911119895+119873

with 119883119896(119898)

= 0

(31)

and we see by letting 119873 go to infinity that the right-handside can obtain any sufficiently large order in 119911 By the zerocurvature relation for119873 and119898 we get for the left-hand side

120597119898(119911119873119885) minus [119862

119898 119911119873119885]

= 120597119898(119862119873) minus [119862

119898 119862119873] + 120597119898((119911119873119885)⩽0)

minus [119862119898 (119911119873119885)⩽0]

= 120597119873(119862119898) + 120597119898((119911119873119885)⩽0) minus [119862

119898 (119911119873119885)⩽0]

(32)

and this last expression is of order smaller or equal to 119898 in119911 This contradicts the unlimited growth in orders of 119911 of theright-hand side Hence all Lax equations (22) have to holdfor 119885

Because of the equivalence between Lax equations (20)for 119876 and the zero curvature relations (23) for 119861

119898 we call

this last set of equations also the zero curvature form of theAKNS hierarchy Similarly the zero curvature relations (24)for 119862

119898 are called the zero curvature form of the strict AKNS

hierarchyBesides the zero curvature relations for the cut-off rsquos

119861119898 and 119862

119898 respectively corresponding to respectively

a solution 119876 of the AKNS hierarchy and a solution 119885 of thestrict AKNS hierarchy also other parts satisfy such relationsDefine

119860119898fl 119861119898minus 119876119911119898 119898 ⩾ 0

119863119898fl 119862119898minus 119885119911119898minus1 119898 ⩾ 1

(33)

Then we have the following result

Corollary 5 The following relations hold

(i) The parts 119860119898| 119898 ⩾ 0 of a solution 119876 of the AKNS

hierarchy satisfy

1205971198981(1198601198982) minus 1205971198982(1198601198981) minus [119860

1198981 1198601198982] = 0 (34)

(ii) The parts 119863119898| 119898 ⩾ 0 of a solution 119885 of the strict

AKNS hierarchy satisfy

1205971198981(1198631198982) minus 1205971198982(1198631198981) minus [119863

1198981 1198631198982] = 0 (35)

Proof Again we show the result only in the strict case Notethat 119885119911119898minus1 satisfy Lax equations similar to 119885

120597119894(119885119911119898minus1) = [119863

119894 119885119911119898minus1] 119894 ⩾ 1 (36)

Now we substitute in the zero curvature relations for 119862119898

everywhere the relation 119862119898= 119863119898+ 119885119911119898minus1 and use the Lax

equations above and the fact that all 119885119911119898minus1 commute Thisgives the desired result

To clarify the link with the AKNS equation considerrelation (23) for119898

1= 2 and119898

2= 1

1205972(1198760119911 + 119876

1) = 1205971(11987601199112+ 1198761119911 + 119876

2)

+ [1198762 1198760119911 + 119876

1]

(37)

Then this identity reduces in sl2(119877)[119911 119911

minus1)⩾0 since 119876

0is

constant to the following two equalities

1205971(1198761) = [119876

0 1198762]

1205972(1198761) = 1205971(1198762) + [119876

2 1198761]

(38)

The first gives an expression of the off-diagonal terms of 1198762

in the coefficients 119902 and 119903 of 1198761 that is

11990212=119894

21205971(119902)

11990221= minus

119894

21205971(119903)

(39)

and the second equation becomes AKNS equations (4) if onehas 1205971= 120597120597119909 and 120597

2= 120597120597119905

3 The Linearization of Both Hierarchies

The zero curvature form of both hierarchies points at thepossible existence of a linear system of which the zerocurvature equations form the compatibility conditions Wepresent here such a system for each hierarchy For theAKNS hierarchy this system the linearization of the AKNShierarchy is as follows take a potential solution119876 of the form(17) and consider the system

119876120595 = 1205951198760

120597119898(120595) = 119861

119898120595 forall119898 ⩾ 0 119861

119898= (119876119911

119898)⩾0

(40)

Likewise for a potential solution 119885 of the strict AKNShierarchy of the form (19) the linearization of the strict AKNShierarchy is given by

119885120593 = 1205931198760119911

120597119898(120593) = 119862

119898120593 forall119898 ⩾ 1 119862

119898= (119885119911

119898minus1)gt0

(41)

Before specifying120595 and120593 we show themanipulations neededto derive the Lax equations of the hierarchy from theirlinearizationWedo this for the strictAKNShierarchy for the

6 Advances in Mathematical Physics

AKNS hierarchy one proceeds similarly Apply 120597119898to the first

equation in (41) and use both equalities in (41) in the sequel

120597119898(119885120593 minus 120593119876

0119911) = 120597

119898(119885) 120593 + 119885120597

119898(120593) minus 120597

119898(120593)1198760119911

= 0 = 120597119898(119885) 120593 + 119885119862

119898120593 minus 119862

1198981205931198760119911

= 120597119898(119885) minus [119862

119898 119885] 120593 = 0

(42)

Hence if we can scratch 120593 from (42) then we obtain the Laxequations of the strict AKNS hierarchy To get the propersetup where all these manipulations make sense we first havea look at the linearization for the trivial solutions119876 = 119876

0and

119885 = 1198760119911 In the AKNS case we have then

11987601205950= 12059501198760

120597119898(1205950) = 119876

01199111198981205950 forall119898 ⩾ 0

(43)

and for its strict version

11987601199111205930= 12059301198760119911

120597119898(1205930) = 119876

01199111198981205930 forall119898 ⩾ 1

(44)

Assuming that each derivation 120597119898equals 120597120597119905

119898 one arrives

for (43) at the solution

1205950= 1205950(119905 119911) = exp(

infin

sum

119898=0

1199051198981198760119911119898) (45)

where 119905 is the short hand notation for 119905119898| 119898 ⩾ 0 and for

(44) it leads to

1205930= 1205930(119905 119911) = exp(

infin

sum

119898=1

1199051198981198760119911119898)

with 119905 = 119905119898| 119898 ⩾ 1

(46)

In general one needs in the linearizations (120595) and (120593)respectively a left action with elements like 119876 119861

119898and 119885

119862119898from gl

2(119877)[119911 119911

minus1) respectively an action of all the 120597

119898

and a right action of 1198760and 119876

0119911 respectively This can

all be realized in suitable gl2(119877)[119911 119911

minus1) modules that are

perturbations of the trivial solutions 1205950and 120593

0 Consider in

a AKNS setting

M0= 119892 (119911) 120595

0=

119873

sum

119894=minusinfin

1198921198941199111198941205950| 119892 (119911)

=

119873

sum

119894=minusinfin

119892119894119911119894isin gl2(119877) [119911 119911

minus1)

(47)

and in a strict AKNS setting

M1= 119892 (119911) 120593

0=

119873

sum

119894=minusinfin

1198921198941199111198941205930| 119892 (119911)

=

119873

sum

119894=minusinfin

119892119894119911119894isin gl2(119877) [119911 119911

minus1)

(48)

where the products 119892(119911)1205950and 119892(119911)120593

0should be seen as

formal and both factors should be kept separate On bothM0

andM1 one can define the required actions for each ℎ(119911) isin

gl2(119877)[119911 119911

minus1) define

ℎ (119911) sdot 119892 (119911) 1205950fl ℎ (119911) 119892 (119911) 120595

0

respectively ℎ (119911) sdot 119892 (119911) 1205930fl ℎ (119911) 119892 (119911) 120593

0

(49)

We define the right-hand action of 1198760and 119876

0119911 respectively

by

119892 (119911) 12059501198760fl 119892 (119911)119876

0 1205950

respectively 119892 (119911) 12059301198760119911 fl 119892 (119911)119876

0119911 1205930

(50)

and the action of each 120597119898by

120597119898(119892 (119911) 120595

0)

=

119873

sum

119894=minusinfin

120597119898(119892119894) 119911119894+

119873

sum

119894=minusinfin

1198921198941198760119911119894+1198981205950

120597119898(119892 (119911) 120593

0)

=

119873

sum

119894=minusinfin

120597119898(119892119894) 119911119894+

119873

sum

119894=minusinfin

1198921198941198760119911119894+1198981205930

(51)

Analogous to the terminology in the scalar case (see [5]) wecall the elements of M

0and M

1oscillating matrices Note

that both M0and M

1are free gl

2(119877)[119911 119911

minus1) modules with

generators 1205950and 120593

0 respectively because for each ℎ(119911) isin

gl2(119877)[119911 119911

minus1) we have

ℎ (119911) sdot 1205950= ℎ (119911) sdot 1 120595

0= ℎ (119911) 120595

0

respectively ℎ (119911) sdot 1205930= ℎ (119911) sdot 1 120595

0= ℎ (119911) 120593

0

(52)

Hence in order to be able to perform legally the scratching of120595 = ℎ(119911)120595

0or 120593 = ℎ(119911)120593

0 it is enough to find oscillating

matrices such that ℎ(119911) is invertible in gl2(119877)[119911 119911

minus1) We will

now introduce a collection of such elements that will occur atthe construction of solutions of both hierarchies

For119898 = (1198981 1198982) isin Z2 let 120575(119898) isin gl

2(119877)[119911 119911

minus1) be given

by

120575 (119898) = (1199111198981 0

0 1199111198982) (53)

Then 120575(119898) has 120575(minus119898) as its inverse in gl2(119877)[119911 119911

minus1) and the

collection Δ = 120575(119898) | 119898 isin Z2 forms a group An element120595 isinM

0is called an oscillating matrix of type 120575(119898) if it has the

form

120595 = ℎ (119911) 120575 (119898) 1205950 with ℎ (119911) isin 119866

lt0 (54)

These oscillatingmatrices are examples of elements ofM0for

which the scratching procedure is valid Let 119876 be a potentialsolution of the AKNS hierarchy of the form (17) If there isan oscillating function 120595 of type 120575(119898) such that for 119876 and 120595

Advances in Mathematical Physics 7

(40) hold then we call120595 awave matrix of the AKNS hierarchyof type 120575(119898) In particular 119876 is then a solution of the AKNShierarchy and the first equation in (40) implies

119876ℎ (119911) 120575 (119898) = ℎ (119911)1198760120575 (119898) 997904rArr

119876 = ℎ (119911)1198760ℎ (119911)minus1

(55)

in other words the solution 119876 is totally determined by thewave matrix Similarly we call an element 120593 isin M

1an

oscillating matrix of type 120575(119898) if it has the form

120593 = ℎ (119911) 120575 (119898) 1205930 with ℎ (119911) isin 119866

⩽0 (56)

This type of oscillating matrices are examples of elements ofM1for which the scratching procedure is valid Let 119885 be a

potential solution of the strict AKNS hierarchy of the form(19) If there is an oscillating function 120593 of type 120575(119898) in M

1

such that for119885 and120593 (41) hold thenwe call120593 awavematrix ofthe strict AKNS hierarchy of type 120575(119898) In particular119885 is thena solution of the strict AKNS hierarchy and the first equationin (41) implies

119885ℎ (119911) 120575 (119898) = ℎ (119911)1198760119911120575 (119898) 997904rArr

119885 = ℎ (119911)1198760119911ℎ (119911)

minus1

(57)

so that also here the wave matrix fully determines thesolution

For both hierarchies there is a milder condition that is tobe satisfied by oscillating matrices of a certain type in orderthat they become a wave matrix of that hierarchy

Proposition 6 Let120595 = ℎ(119911)120575(119898)1205950be an oscillatingmatrix

of type 120575(119898) in M0and 119876 = ℎ(119911)119876

0ℎ(119911)minus1 the corresponding

potential solution of the AKNS hierarchy Similarly let 120593 =

ℎ(119911)120575(119898)1205930be such a matrix in M

1with potential solution

119885 = ℎ(119911)1198760119911ℎ(119911)minus1

(a) If there exists for each 119898 ⩾ 0 an element 119872119898isin

gl2(119877)[119911 119911

minus1)⩾0

such that

120597119898(120595) = 119872

119898120595 (58)

then each119872119898= (119876119911

119898)⩾0

and 120595 is a wave matrix oftype 120575(119898) for the AKNS hierarchy

(b) If there exists for each 119898 ⩾ 1 an element 119873119898isin

gl2(119877)[119911 119911

minus1)gt0

such that

120597119898(120593) = 119873

119898120593 (59)

then each 119873119898= (119885119911

119898minus1)gt0

and 120593 is a wave matrix oftype 120575(119898) for the strict AKNS hierarchy

Proof We give the proof again for the strict AKNS casethe other one is similar By using the fact that M

1is a free

gl2(119877)[119911 119911

minus1)module with generator 120593

0we can translate the

relations 120597119898(120593) = 119873

119898120593 into equations in gl

2(119877)[119911 119911

minus1) This

yields

120597119898(ℎ (119911)) + ℎ (119911)119876

0119911119898= 119873119898ℎ (119911) 997904rArr

120597119898(ℎ (119911)) ℎ (119911)

minus1+ 119885119911119898minus1

= 119873119898

(60)

Projecting the right-hand side on gl2(119877)[119911 119911

minus1)gt0

gives us theidentity

(119885119911119898minus1)gt0= 119873119898 (61)

we are looking for

In the next section we produce a context where we canconstruct wave matrices for both hierarchies in which theproduct is real

4 The Construction of the Solutions

In this section we will show how to construct a wide class ofsolutions of both hierarchies This is done in the style of [6]We first describe the group of loops we will work with Foreach 0 lt 119903 lt 1 let 119860

119903be the annulus

119911 | 119911 isin C 119903 ⩽ |119911| ⩽1

119903 (62)

Following [7] we use the notation 119871anGL2(C) for the col-lection of holomorphic maps from some annulus 119860

119903into

GL2(C) It is a group with respect to pointwise multiplication

and contains in a natural way GL2(C) as a subgroup as the

collection of constant maps into GL2(C) Other examples

of elements in 119871anGL2(C) are the elements of Δ However119871anGL2(C) is more than just a group it is an infinitedimensional Lie group Its manifold structure can be read offfrom its Lie algebra 119871angl2(C) consisting of all holomorphicmaps 120574 119880 rarr gl

2(C) where 119880 is an open neighborhood of

some annulus 119860119903 0 lt 119903 lt 1 Since gl

2(C) is a Lie algebra

the space 119871angl2(C) becomes a Lie algebra with respect to thepointwise commutator Topologically the space 119871angl2(C) isthe direct limit of all the spaces 119871an119903gl2(C) where this lastspace consists of all 120574 corresponding to the fixed annulus119860119903 One gives each 119871an119903gl2(C) the topology of uniform

convergence and with that topology it becomes a Banachspace In this way 119871angl2(C) becomes a Frechet space Thepointwise exponential map defines a local diffeomorphismaround zero in 119871angl2(C) (see eg [8])

Now each 120574 isin 119871angl2(C) possesses an expansion in aFourier series

120574 =

infin

sum

119896=minusinfin

120574119896119911119896 with each 120574

119896isin gl2(C) (63)

that converges absolutely on the annulus 119860119903

infin

sum

119896=minusinfin

10038171003817100381710038171205741198961003817100381710038171003817 119903minus|119896|

lt infin (64)

This Fourier expansion is used to make two different decom-positions of the Lie algebra119871angl2(C)Thefirst is the analoguefor the present Lie algebra 119871angl2(C) of decomposition (8) ofgl2(119877)[119911 119911

minus1) that lies at the basis of the Lax equations of the

AKNS hierarchy Namely consider the subspaces

119871angl2 (C)⩾0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=0

120574119896119911119896

8 Advances in Mathematical Physics

119871angl2 (C)lt0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =minus1

sum

119896=minusinfin

120574119896119911119896

(65)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra The first Lie algebra consists ofthe elements in 119871angl2(C) that extend to holomorphic mapsdefined on some disk 119911 isin C | |119911| ⩽ 1119903 0 lt 119903 lt 1 andthe second Lie algebra corresponds to the maps in 119871angl2(C)that have a holomorphic extension towards some disk aroundinfinity 119911 isin P1(C) | |119911| ⩾ 119903 0 lt 119903 lt 1 and that arezero at infinity To each of the two Lie subalgebras belongsa subgroup of 119871anGL2(C) The pointwise exponential mapapplied to elements of 119871angl2(C)lt0 yields elements of

119880minus= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

minus1

sum

119896=minusinfin

120574119896119911119896 (66)

and the exponential map applied to elements of 119871angl2(C)⩾0maps them into

119875+= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740 +

infin

sum

119896=1

120574119896119911119896 with 120574

0

isin GL2(C)

(67)

Both119880minusand 119875+are easily seen to be subgroups of 119871anGL2(C)

and since the direct sum of their Lie algebras is 119871angl2(C)their product

Ω = 119880minus119875+ (68)

is open in 119871anGL2(C) and is called like in the finite dimen-sional case the big cell with respect to 119880

minusand 119875

+

The second decomposition is the analogue of the splitting(11) of gl

2(119877)[119911 119911

minus1) that led to the Lax equations of the strict

AKNS hierarchy Now we consider the subspaces

119871angl2 (C)gt0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=1

120574119896119911119896

119871angl2 (C)⩽0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =0

sum

119896=minusinfin

120574119896119911119896

(69)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra 119871angl2(C)gt0 consists of mapswhose holomorphic extension to 119911 = 0 equals zero inthat point and 119871angl2(C)⩽0 is the set of maps that extendhomomorphically to a neighborhood of infinity To eachof them belongs a subgroup of 119871anGL2(C) The pointwiseexponential map applied to elements of 119871angl2(C)gt0 yieldselements of

119880+= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

infin

sum

119896=1

120574119896119911119896 (70)

and the exponential map applied to elements of 119871angl2(C)⩽0maps them into

119875minus= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740

+

minus1

sum

119896=minusinfin

120574119896119911119896 with 120574

0isin GL2(C)

(71)

Both119875minusand119880

+are easily seen to be subgroups of 119871anGL2(C)

Since the direct sum of their Lie algebras is 119871angl2(C) theirproduct 119875

minus119880+is open and because 119875

+= GL

2(C)119880+=

119880+GL2(C) and 119875

minus= GL2(C)119880minus= 119880minusGL2(C) this gives the

equality

119875minus119880+= Ω = 119880

minus119875+ (72)

the big cell in 119871anGL2(C)The next two subgroups of 119871anGL2(C) correspond to

the exponential factors in both linearizations The group ofcommuting flows relevant for the AKNS hierarchy is thegroup

Γ0= 1205740(119905) = exp(

infin

sum

119894=0

1199051198941198760119911119894) | 1205740isin 119875+ (73)

and similarly for the strict AKNS hierarchy we use thecommuting flows from

Γ1= 1205741(119905) = exp(

infin

sum

119894=1

1199051198941198760119911119894) | 1205741isin 119880+ (74)

Besides the groups Δ Γ0 and Γ

1 there are more subgroups in

119871anGL2(C) that commute with those three groups and theyare in a sense complimentary to Δ Γ

0and Δ Γ

1 respectively

That is why we use the following notations for them in 119880minus

there is

Γ119888

0= 120574119888

0(119905) = exp(

infin

sum

119896=1

1199041198961198760119911minus119896) | 120574119888

0isin 119880minus (75)

and in 119875minuswe have

Γ119888

1= 120574119888

1(119905) = exp(

infin

sum

119896=0

1199041198961198760119911minus119896) | 120574119888

1isin 119875minus (76)

We have now all ingredients to describe the constructionof the solutions to each hierarchy and we start with theAKNS hierarchy Take inside the product 119871anGL2(C) times Δthe collection 119878

0of pairs (119892 120575(119898)) such that there exists a

1205740(119905) isin Γ

0satisfying

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898) isin Ω = 119880

minus119875+ (77)

For such a pair (119892 120575(119898)) we take the collection Γ0(119892 120575(119898))

of all 1205740(119905) satisfying the condition (77) This is an open

nonempty subset of Γ0 Let 119877

0(119892 120575(119898)) be the algebra of

analytic functions Γ0(119892 120575(119898)) rarr C This is the algebra of

Advances in Mathematical Physics 9

functions 119877 that we associate with the point (119892 120575(119898)) isin 1198780

and for the commuting derivations of 1198770(119892 120575(119898)) we choose

120597119894fl 120597120597119905

119894 119894 ⩾ 0 By property (77) we have for all 120574

0(119905) isin

Γ0(119892 120575(119898))

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898)

= 119906minus(119892 120575 (119898))

minus1

119901+(119892 120575 (119898))

(78)

with 119906minus(119892 120575(119898)) isin 119880

minus119901+(119892 120575(119898)) isin 119875

+Then all thematrix

coefficients in the Fourier expansions of the elements 119906minus(119892

120575(119898)) and 119901+(119892 120575(119898)) belong to the algebra 119877

0(119892 120575(119898)) It is

convenient to write relation (78) in the form

Ψ119892120575(119898)

fl 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905)

= 119901+(119892 120575 (119898)) 120575 (119898) 120574

0(119905) 119892minus1

= 119901+(119892 120575 (119898)) 120574

0(119905) 120575 (119898) 119892

minus1

= 119902+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(79)

with 119902+(119892 120575(119898)) isin 119875

+ Clearly Ψ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M0 where all the products in the

decomposition 119906minus(119892 120575(119898))120575(119898)120574

0(119905) are no longer formal

but real This is our potential wave matrix for the AKNShierarchy

To get the potential wave function for the strict AKNShierarchy we proceed similarly only this time we use thedecomposition Ω = 119875

minus119880+ Consider again the product

119871anGL2(C) times Δ and the subset 1198781of pairs (119892 120575(119898)) such that

there exists a 1205741(119905) isin Γ

1satisfying

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898) isin Ω = 119875

minus119880+ (80)

For such a pair (119892 120575(119898)) we define Γ1(119892 120575(119898)) as the set of

all 1205741(119905) satisfying condition (77) This is an open nonempty

subset of Γ1 Let 119877

1(119892 120575(119898)) be the algebra of analytic

functions Γ1(119892 120575(119898)) rarr C This is the algebra of functions

119877 that we associate with the point (119892 120575(119898)) isin 1198781and for the

commuting derivations of 1198771(119892 120575(119898)) we choose 120597

119894fl 120597120597119905

119894

119894 gt 0 By property (80) we have for all 1205741(119905) isin Γ

1(119892 120575(119898))

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898)

= 119901minus(119892 120575 (119898))

minus1

119906+(119892 120575 (119898))

(81)

with 119901minus(119892 120575(119898)) isin 119875

minus 119906+(119892 120575(119898)) isin 119880

+ Then all

the matrix coefficients in the Fourier expansions of theelements 119901

minus(119892 120575(119898)) and 119906

+(119892 120575(119898)) belong to the algebra

1198771(119892 120575(119898)) It is convenient to write relation (81) in the form

Φ119892120575(119898)

fl 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905)

= 119906+(119892 120575 (119898)) 120575 (119898) 120574

1(119905) 119892minus1

= 119906+(119892 120575 (119898)) 120574

1(119905) 120575 (119898) 119892

minus1

= 119908+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(82)

with 119908+(119892 120575(119898)) isin 119880

+ Clearly Φ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M1where all the products in the

decomposition 119901minus(119892 120575(119898))120575(119898)120574

1(119905) are no longer formal

but realTo show that Ψ

119892120575(119898)is a wave matrix for the AKNS

hierarchy of type 120575(119898) and Φ119892120575(119898)

one for the strict AKNShierarchy we use the fact that it suffices to prove the relationsin Proposition 6 We treat first Ψ

119892120575(119898) In (79) we presented

two different decompositions of Ψ119892120575(119898)

that we will use tocompute 120597

119894(Ψ119892120575(119898)

) With respect to 119906minus(119892 120575(119898))120575(119898)120574

0(119905)

there holds120597119894(Ψ119892120575(119898)

)

= 120597119894(119906minus(119892 120575 (119898))) 119906

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905) = 119872

119894Ψ119892120575(119898)

(83)

with119872119894isin gl2(119877(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911 If we take

the decomposition 119902+(119892 120575(119898))120575(119898)119892

minus1 then we get

120597119894(Ψ119892120575(119898)

)

= 120597119894(119902+(119892 120575 (119898))) 119902

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(84)

with 120597119894(119902+(119892 120575(119898)))119902

+(119892 120575(119898))

minus1 of the formsum119895⩾0119902119895119911119895 with

all 119902119895isin gl2(119877(119892 120575(119898))) As Ψ

119892120575(119898)is invertible we obtain

119872119894= sum119895⩾0119902119895119911119895 and thus 119872

119894= sum119894

119895=0119902119895119911119895 and that is the

desired resultWe handle the case of Φ

119892120575(119898)in a similar way Also in

(79) you can find two different decompositions of Φ119892120575(119898)

that we will use to compute 120597119894(Φ119892120575(119898)

) With respect to theexpression 119901

minus(119892 120575(119898))120575(119898)120574

1(119905) we get for all 119894 ⩾ 1

120597119894(Φ119892120575(119898)

)

= 120597119894(119901minus(119892 120575 (119898))) 119901

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905) = 119873

119894Φ119892120575(119898)

(85)

again with 119873119894isin gl2(1198771(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911

Next we take the decomposition 119908+(119892 120575(119898))120575(119898)119892

minus1 andthat yields

120597119894(Φ119892120575(119898)

)

= 120597119894(119908+(119892 120575 (119898)))119908

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(86)

Since the zeroth order term of 119908+(119892 120575(119898)) is Id we note

that the element 120597119894(119908+(119892 120575(119898)))119908

+(119892 120575(119898))

minus1 has the formsum119895⩾1119908119895119911119895 with all 119908

119895isin gl2(1198771(119892 120575(119898))) As Φ

119892120575(119898)is also

invertible we obtain119873119894= sum119895⩾1119908119895119911119895 and thus119873

119894= sum119894

119895=1119908119895119911119895

and that is the result we were looking for in the strict caseThe abovementioned constructions are not affected seri-

ously by a number of transformations For example if wewrite 120575 = 120575((1 1)) then all 120575119896 119896 isin Z are central elementsand we have for all (119892 120575(119898)) isin 119878

0and 119878

1respectively and all

119896 isin Z

Ψ119892120575(119898)120575

119896 = Ψ119892120575(119898)

120575119896

respectively Φ119892120575(119898)120575

119896 = Φ119892120575(119898)

120575119896

(87)

10 Advances in Mathematical Physics

Also for any 1199011isin 119875+ each 120574119888

0(119904) isin Γ

119888

0 and all (119892 120575(119898)) isin

1198780 we see that the element (120574119888

0(119904)119892120575(minus119898)119901

1120575(119898) 120575(119898)) also

belongs to 1198780 for

120575 (119898) 1205740(119905) 120574119888

0(119904) 119892120575 (minus119898) 119901

1120575 (119898) 120574

0(119905)minus1120575 (minus119898)

= 120574119888

0(119904) 119906minus1

minus119901+1205740(119905) 11990111205740(119905)minus1

(88)

In particular we see that

Ψ120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575(119898)

= 119906minus120574119888

0(119904)minus1120575 (119898) 120574

0(119905) (89)

Its analogue in the strict case is as follows for any 1199061isin

119880+ each 120574119888

1(119904) isin Γ

119888

1 and all (119892 120575(119898)) isin 119878

1 the element

(120574119888

1(119904)119892120575(minus119898)119906

1120575(119898) 120575(119898)) also belongs to 119878

1and there

holds

Φ120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575(119898)

= 119901minus120574119888

1(119904)minus1120575 (119898) 120574

1(119905) (90)

We resume the foregoing results in the following theorem

Theorem 7 Consider the product spaceΠ fl 119871anGL2(C) times ΔThen the following holds

(a) For each point (119892 120575(119898)) isin Π that satisfies condition(77) we have a wave matrix of type 120575(119898) of the AKNShierarchyΨ

119892120575(119898) defined by (79) and this determines a

solution 119876119892120575(119898)

= 119906minus(119892 120575(119898))119876

0119906minus(119892 120575(119898))

minus1 of theAKNS hierarchy For each 119901

1isin 119875+ every 120574119888

0(119904) isin Γ

119888

0

and all powers of 120575 one has

119876120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575

119896120575(119898)

= 119876119892120575(119898)

(91)

(b) For each point (119892 120575(119898)) isin Π that satisfies condition(80) we have a wave matrix of type 120575(119898) of thestrict AKNS hierarchy Φ

119892120575(119898) defined by (82) and it

determines a solution 119885119892120575(119898)

= 119901minus(119892 120575(119898))119876

0119911119901minus(119892

120575(119898))minus1 of the strict AKNS hierarchy For each 119906

1isin 119880+

every 1205741198881(119904) isin Γ

119888

1 and all powers of 120575 one has

119885120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575

119896120575(119898)

= 119885119892120575(119898)

(92)

Competing Interests

The author declares that they have no competing interests

References

[1] M Adler P vanMoerbeke and P VanhaeckeAlgebraic Integra-bility Painleve Geometry and Lie Algebras vol 47 of Ergebnisseder Mathematik und ihrer Grenzgebiete Springer 2004

[2] G F Helminck A G Helminck and E A PanasenkoldquoIntegrable deformations in the algebra of pseudodifferentialoperators from a Lie algebraic perspectiverdquo Theoretical andMathematical Physics vol 174 no 1 pp 134ndash153 2013

[3] M J Ablowitz D J Kaup A C Newell and H Segur ldquoTheinverse scattering transform-Fourier analysis for nonlinearproblemsrdquo Studies in Applied Mathematics vol 53 no 4 pp249ndash315 1974

[4] H Flashka A CNewell and T Ratiu ldquoKac-Moody Lie algebrasand soliton equations II Lax equations associated with 119860

1

(1)rdquoPhysicaDNonlinear Phenomena vol 9 no 3 pp 300ndash323 1983

[5] EDateM JimboMKashiwara andTMiwa ldquoTransformationgroups for soliton equationsrdquo in Proceedings of the RIMSSymposium on Nonlinear Integrable SystemsmdashClassical Theoryand Quantum Theory T Jimbo and T Miwa Eds WorldScience Publisher Kyoto Japan 1981

[6] G Segal and G Wilson ldquoLoop groups and equations of KdVtyperdquo Publications Mathematiques de lrsquoIHES vol 61 pp 5ndash651985

[7] A Pressley and G Segal Loop Groups Oxford MathematicalMonographs Clarendon Press 1986

[8] R S Hamilton ldquoThe inverse function theorem of Nash andMoserrdquoBulletin of the AmericanMathematical Society vol 7 no1 pp 65ndash222 1982

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Mathematical Physics

Now we are looking for deformations 119876 of the form (17)such that the evolution with respect to the 120597

119898 satisfies the

following for all119898 ⩾ 0

120597119898(119876) = [(119876119911

119898)⩾0 119876] = minus [(119876119911

119898)lt0 119876] (20)

where the second identity follows from the fact that all 119876119911119898commute Equations (20) are called the Lax equations ofthe AKNS hierarchy and the deformation 119876 satisfying theseequations is called a solution of the hierarchy Note that119876 = 119876

0is a solution of the AKNS hierarchy and it is

called the trivial one AKNS equation (4) occurs among thecompatibility equations of this system as we will see furtheron Note that (20) for 119898 = 0 is simply 120597

0(119876) = [119876

0 119876]

Therefore if 1205970= 120597120597119905

0and both 119890minus1198941199050 and 1198901198941199050 belong to the

algebra 119877 of matrix coefficients then we can introduce theloop isin sl

2(119877)[119911 119911

minus1) given by

fl exp (minus11990501198760) 119876 exp (119905

01198760)

= sum

119896⩾0

exp (minus11990501198760) 119876119896exp (119905

01198760) 119911minus119896

(21)

which is easily seen to satisfy 1205970() = 0 This handles then

the dependence of 119876 of 1199050

Among the deformations 119885 of the form (19) we look for119885 such that their evolution with respect to 120597

119898 satisfies the

following for all119898 ⩾ 1

120597119898(119885) = [(119885119911

119898minus1)gt0 119885] = minus [(119885119911

119898minus1)⩽0 119885] (22)

where the second identity follows from the fact that all119885119911119898minus1 commute Since (22) correspond to the strict cut-

off (10) they are called the Lax equations of the strict AKNShierarchy and the deformation 119885 is called a solution of thishierarchy Again there is at least one solution 119885 = 119876

0119911 It is

called the trivial solution of the hierarchyFor both systems (20) and (22) one can speak of compat-

ibility Namely there holds the following result

Proposition 3 Both sets of Lax equations (20) and (22) arethe so-called compatible systems that is the projections 119861

119898fl

(119876119911119898)⩾0| 119898 ⩾ 0 of a solution 119876 to (20) satisfy the zero

curvature relations

1205971198981(1198611198982) minus 1205971198982(1198611198981) minus [119861

1198981 1198611198982] = 0 (23)

and the projections 119862119898fl (119885119911

119898minus1)gt0| 119898 ⩾ 1 corresponding

to a solution 119885 to (22) satisfy the zero curvature relations

1205971198981(1198621198982) minus 1205971198982(1198621198981) minus [119862

1198981 1198621198982] = 0 (24)

Proof The idea is to show that the left-hand side of (23) and(24) respectively belongs to

sl2(119877) [119911 119911

minus1)⩾0cap sl2(119877) [119911 119911

minus1)lt0

respectively sl2(119877) [119911 119911

minus1)gt0cap sl2(119877) [119911 119911

minus1)⩽0

(25)

and thus has to be zero We give the proof for 119862119898 that for

119861119898 is similar and is left to the reader The inclusion in the

first factor is clear as both 119862119898and 120597119899(119862119898) belong to the Lie

subalgebra sl2(119877)[119911 119911

minus1)gt0To show the other one we use Lax

equations (22) Note that the same Lax equations hold for allthe 119911119873119885 | 119873 ⩾ 0

120597119898(119911119873119885) = [(119885119911

119898minus1)gt0 119911119873119885] (26)

By substituting 119862119898119894

= 119911119898119894minus1119885 minus (119911

119898119894minus1119885)lt0 we get the

identities for

1205971198981(1198621198982) minus 1205971198982(1198621198981)

= 1205971198981(1199111198982minus1119885) minus 120597

1198981((1199111198982minus1119885)

⩽0)

minus 1205971198982(1199111198981minus1119885) + 120597

1198982((1199111198981minus1119885)

⩽0)

= [1198621198981 1199111198982minus1119885] minus [119862

1198982 1199111198981minus1119885]

minus 1205971198981((1199111198982minus1119885)

⩽0) + 1205971198982((1199111198981minus1119885)

⩽0)

(27)

and

[1198621198981 1198621198982]

= [1199111198981minus1119885 minus (119911

1198981minus1119885)⩽0 1199111198982minus1119885 minus (119911

1198982minus1119885)⩽0]

= minus [(1199111198981minus1119885)

⩽0 1199111198982minus1119885]

+ [(1199111198982minus1119885)

⩽0 1199111198981minus1119885]

+ [(1199111198981minus1119885)

⩽0 (1199111198982minus1119885)

⩽0]

(28)

Taking into account the second identity in (22) we see thatthe left-hand side of (24) is equal to

minus 1205971198981((1199111198982minus1119885)

⩽0) + 1205971198982((1199111198981minus1119885)

⩽0)

minus [(1199111198981minus1119885)

⩽0 (1199111198982minus1119885)

⩽0]

(29)

This element belongs to the Lie subalgebra sl2(119877)[119911 119911

minus1)⩽0

and that proves the claim

Reversely we have the following

Proposition 4 Suppose we have a deformation 119876 of the type(17) or a deformation 119885 of the form (19) Then there holds thefollowing

(1) Assume that the projections 119861119898fl (119876119911

119898)⩾0| 119898 ⩾ 0

satisfy the zero curvature relations (23) Then 119876 is asolution of the AKNS hierarchy

(2) Similarly if the projections 119862119898

fl (119885119911119898minus1)⩾0| 119898 ⩾

1 satisfy the zero curvature relations (24) then 119885 is asolution of the strict AKNS hierarchy

Advances in Mathematical Physics 5

Proof Againwe prove the statement for119885 that for119876 is shownin a similarway So assume that there is one Lax equation (22)that does not hold Then there is a119898 ⩾ 1 such that

120597119898(119885) minus [119862

119898 119885] = sum

119895⩽119896(119898)

119883119895119911119895 with 119883

119896(119898)= 0 (30)

Since both 120597119898(119885) and [119862

119898 119885] are of order smaller than or

equal to one in 119911 we know that 119896(119898) ⩽ 1 Further we can saythat for all119873 ⩾ 0

120597119898(119911119873119885) minus [119862

119898 119911119873119885] = sum

119895⩽119896(119898)

119883119895119911119895+119873

with 119883119896(119898)

= 0

(31)

and we see by letting 119873 go to infinity that the right-handside can obtain any sufficiently large order in 119911 By the zerocurvature relation for119873 and119898 we get for the left-hand side

120597119898(119911119873119885) minus [119862

119898 119911119873119885]

= 120597119898(119862119873) minus [119862

119898 119862119873] + 120597119898((119911119873119885)⩽0)

minus [119862119898 (119911119873119885)⩽0]

= 120597119873(119862119898) + 120597119898((119911119873119885)⩽0) minus [119862

119898 (119911119873119885)⩽0]

(32)

and this last expression is of order smaller or equal to 119898 in119911 This contradicts the unlimited growth in orders of 119911 of theright-hand side Hence all Lax equations (22) have to holdfor 119885

Because of the equivalence between Lax equations (20)for 119876 and the zero curvature relations (23) for 119861

119898 we call

this last set of equations also the zero curvature form of theAKNS hierarchy Similarly the zero curvature relations (24)for 119862

119898 are called the zero curvature form of the strict AKNS

hierarchyBesides the zero curvature relations for the cut-off rsquos

119861119898 and 119862

119898 respectively corresponding to respectively

a solution 119876 of the AKNS hierarchy and a solution 119885 of thestrict AKNS hierarchy also other parts satisfy such relationsDefine

119860119898fl 119861119898minus 119876119911119898 119898 ⩾ 0

119863119898fl 119862119898minus 119885119911119898minus1 119898 ⩾ 1

(33)

Then we have the following result

Corollary 5 The following relations hold

(i) The parts 119860119898| 119898 ⩾ 0 of a solution 119876 of the AKNS

hierarchy satisfy

1205971198981(1198601198982) minus 1205971198982(1198601198981) minus [119860

1198981 1198601198982] = 0 (34)

(ii) The parts 119863119898| 119898 ⩾ 0 of a solution 119885 of the strict

AKNS hierarchy satisfy

1205971198981(1198631198982) minus 1205971198982(1198631198981) minus [119863

1198981 1198631198982] = 0 (35)

Proof Again we show the result only in the strict case Notethat 119885119911119898minus1 satisfy Lax equations similar to 119885

120597119894(119885119911119898minus1) = [119863

119894 119885119911119898minus1] 119894 ⩾ 1 (36)

Now we substitute in the zero curvature relations for 119862119898

everywhere the relation 119862119898= 119863119898+ 119885119911119898minus1 and use the Lax

equations above and the fact that all 119885119911119898minus1 commute Thisgives the desired result

To clarify the link with the AKNS equation considerrelation (23) for119898

1= 2 and119898

2= 1

1205972(1198760119911 + 119876

1) = 1205971(11987601199112+ 1198761119911 + 119876

2)

+ [1198762 1198760119911 + 119876

1]

(37)

Then this identity reduces in sl2(119877)[119911 119911

minus1)⩾0 since 119876

0is

constant to the following two equalities

1205971(1198761) = [119876

0 1198762]

1205972(1198761) = 1205971(1198762) + [119876

2 1198761]

(38)

The first gives an expression of the off-diagonal terms of 1198762

in the coefficients 119902 and 119903 of 1198761 that is

11990212=119894

21205971(119902)

11990221= minus

119894

21205971(119903)

(39)

and the second equation becomes AKNS equations (4) if onehas 1205971= 120597120597119909 and 120597

2= 120597120597119905

3 The Linearization of Both Hierarchies

The zero curvature form of both hierarchies points at thepossible existence of a linear system of which the zerocurvature equations form the compatibility conditions Wepresent here such a system for each hierarchy For theAKNS hierarchy this system the linearization of the AKNShierarchy is as follows take a potential solution119876 of the form(17) and consider the system

119876120595 = 1205951198760

120597119898(120595) = 119861

119898120595 forall119898 ⩾ 0 119861

119898= (119876119911

119898)⩾0

(40)

Likewise for a potential solution 119885 of the strict AKNShierarchy of the form (19) the linearization of the strict AKNShierarchy is given by

119885120593 = 1205931198760119911

120597119898(120593) = 119862

119898120593 forall119898 ⩾ 1 119862

119898= (119885119911

119898minus1)gt0

(41)

Before specifying120595 and120593 we show themanipulations neededto derive the Lax equations of the hierarchy from theirlinearizationWedo this for the strictAKNShierarchy for the

6 Advances in Mathematical Physics

AKNS hierarchy one proceeds similarly Apply 120597119898to the first

equation in (41) and use both equalities in (41) in the sequel

120597119898(119885120593 minus 120593119876

0119911) = 120597

119898(119885) 120593 + 119885120597

119898(120593) minus 120597

119898(120593)1198760119911

= 0 = 120597119898(119885) 120593 + 119885119862

119898120593 minus 119862

1198981205931198760119911

= 120597119898(119885) minus [119862

119898 119885] 120593 = 0

(42)

Hence if we can scratch 120593 from (42) then we obtain the Laxequations of the strict AKNS hierarchy To get the propersetup where all these manipulations make sense we first havea look at the linearization for the trivial solutions119876 = 119876

0and

119885 = 1198760119911 In the AKNS case we have then

11987601205950= 12059501198760

120597119898(1205950) = 119876

01199111198981205950 forall119898 ⩾ 0

(43)

and for its strict version

11987601199111205930= 12059301198760119911

120597119898(1205930) = 119876

01199111198981205930 forall119898 ⩾ 1

(44)

Assuming that each derivation 120597119898equals 120597120597119905

119898 one arrives

for (43) at the solution

1205950= 1205950(119905 119911) = exp(

infin

sum

119898=0

1199051198981198760119911119898) (45)

where 119905 is the short hand notation for 119905119898| 119898 ⩾ 0 and for

(44) it leads to

1205930= 1205930(119905 119911) = exp(

infin

sum

119898=1

1199051198981198760119911119898)

with 119905 = 119905119898| 119898 ⩾ 1

(46)

In general one needs in the linearizations (120595) and (120593)respectively a left action with elements like 119876 119861

119898and 119885

119862119898from gl

2(119877)[119911 119911

minus1) respectively an action of all the 120597

119898

and a right action of 1198760and 119876

0119911 respectively This can

all be realized in suitable gl2(119877)[119911 119911

minus1) modules that are

perturbations of the trivial solutions 1205950and 120593

0 Consider in

a AKNS setting

M0= 119892 (119911) 120595

0=

119873

sum

119894=minusinfin

1198921198941199111198941205950| 119892 (119911)

=

119873

sum

119894=minusinfin

119892119894119911119894isin gl2(119877) [119911 119911

minus1)

(47)

and in a strict AKNS setting

M1= 119892 (119911) 120593

0=

119873

sum

119894=minusinfin

1198921198941199111198941205930| 119892 (119911)

=

119873

sum

119894=minusinfin

119892119894119911119894isin gl2(119877) [119911 119911

minus1)

(48)

where the products 119892(119911)1205950and 119892(119911)120593

0should be seen as

formal and both factors should be kept separate On bothM0

andM1 one can define the required actions for each ℎ(119911) isin

gl2(119877)[119911 119911

minus1) define

ℎ (119911) sdot 119892 (119911) 1205950fl ℎ (119911) 119892 (119911) 120595

0

respectively ℎ (119911) sdot 119892 (119911) 1205930fl ℎ (119911) 119892 (119911) 120593

0

(49)

We define the right-hand action of 1198760and 119876

0119911 respectively

by

119892 (119911) 12059501198760fl 119892 (119911)119876

0 1205950

respectively 119892 (119911) 12059301198760119911 fl 119892 (119911)119876

0119911 1205930

(50)

and the action of each 120597119898by

120597119898(119892 (119911) 120595

0)

=

119873

sum

119894=minusinfin

120597119898(119892119894) 119911119894+

119873

sum

119894=minusinfin

1198921198941198760119911119894+1198981205950

120597119898(119892 (119911) 120593

0)

=

119873

sum

119894=minusinfin

120597119898(119892119894) 119911119894+

119873

sum

119894=minusinfin

1198921198941198760119911119894+1198981205930

(51)

Analogous to the terminology in the scalar case (see [5]) wecall the elements of M

0and M

1oscillating matrices Note

that both M0and M

1are free gl

2(119877)[119911 119911

minus1) modules with

generators 1205950and 120593

0 respectively because for each ℎ(119911) isin

gl2(119877)[119911 119911

minus1) we have

ℎ (119911) sdot 1205950= ℎ (119911) sdot 1 120595

0= ℎ (119911) 120595

0

respectively ℎ (119911) sdot 1205930= ℎ (119911) sdot 1 120595

0= ℎ (119911) 120593

0

(52)

Hence in order to be able to perform legally the scratching of120595 = ℎ(119911)120595

0or 120593 = ℎ(119911)120593

0 it is enough to find oscillating

matrices such that ℎ(119911) is invertible in gl2(119877)[119911 119911

minus1) We will

now introduce a collection of such elements that will occur atthe construction of solutions of both hierarchies

For119898 = (1198981 1198982) isin Z2 let 120575(119898) isin gl

2(119877)[119911 119911

minus1) be given

by

120575 (119898) = (1199111198981 0

0 1199111198982) (53)

Then 120575(119898) has 120575(minus119898) as its inverse in gl2(119877)[119911 119911

minus1) and the

collection Δ = 120575(119898) | 119898 isin Z2 forms a group An element120595 isinM

0is called an oscillating matrix of type 120575(119898) if it has the

form

120595 = ℎ (119911) 120575 (119898) 1205950 with ℎ (119911) isin 119866

lt0 (54)

These oscillatingmatrices are examples of elements ofM0for

which the scratching procedure is valid Let 119876 be a potentialsolution of the AKNS hierarchy of the form (17) If there isan oscillating function 120595 of type 120575(119898) such that for 119876 and 120595

Advances in Mathematical Physics 7

(40) hold then we call120595 awave matrix of the AKNS hierarchyof type 120575(119898) In particular 119876 is then a solution of the AKNShierarchy and the first equation in (40) implies

119876ℎ (119911) 120575 (119898) = ℎ (119911)1198760120575 (119898) 997904rArr

119876 = ℎ (119911)1198760ℎ (119911)minus1

(55)

in other words the solution 119876 is totally determined by thewave matrix Similarly we call an element 120593 isin M

1an

oscillating matrix of type 120575(119898) if it has the form

120593 = ℎ (119911) 120575 (119898) 1205930 with ℎ (119911) isin 119866

⩽0 (56)

This type of oscillating matrices are examples of elements ofM1for which the scratching procedure is valid Let 119885 be a

potential solution of the strict AKNS hierarchy of the form(19) If there is an oscillating function 120593 of type 120575(119898) in M

1

such that for119885 and120593 (41) hold thenwe call120593 awavematrix ofthe strict AKNS hierarchy of type 120575(119898) In particular119885 is thena solution of the strict AKNS hierarchy and the first equationin (41) implies

119885ℎ (119911) 120575 (119898) = ℎ (119911)1198760119911120575 (119898) 997904rArr

119885 = ℎ (119911)1198760119911ℎ (119911)

minus1

(57)

so that also here the wave matrix fully determines thesolution

For both hierarchies there is a milder condition that is tobe satisfied by oscillating matrices of a certain type in orderthat they become a wave matrix of that hierarchy

Proposition 6 Let120595 = ℎ(119911)120575(119898)1205950be an oscillatingmatrix

of type 120575(119898) in M0and 119876 = ℎ(119911)119876

0ℎ(119911)minus1 the corresponding

potential solution of the AKNS hierarchy Similarly let 120593 =

ℎ(119911)120575(119898)1205930be such a matrix in M

1with potential solution

119885 = ℎ(119911)1198760119911ℎ(119911)minus1

(a) If there exists for each 119898 ⩾ 0 an element 119872119898isin

gl2(119877)[119911 119911

minus1)⩾0

such that

120597119898(120595) = 119872

119898120595 (58)

then each119872119898= (119876119911

119898)⩾0

and 120595 is a wave matrix oftype 120575(119898) for the AKNS hierarchy

(b) If there exists for each 119898 ⩾ 1 an element 119873119898isin

gl2(119877)[119911 119911

minus1)gt0

such that

120597119898(120593) = 119873

119898120593 (59)

then each 119873119898= (119885119911

119898minus1)gt0

and 120593 is a wave matrix oftype 120575(119898) for the strict AKNS hierarchy

Proof We give the proof again for the strict AKNS casethe other one is similar By using the fact that M

1is a free

gl2(119877)[119911 119911

minus1)module with generator 120593

0we can translate the

relations 120597119898(120593) = 119873

119898120593 into equations in gl

2(119877)[119911 119911

minus1) This

yields

120597119898(ℎ (119911)) + ℎ (119911)119876

0119911119898= 119873119898ℎ (119911) 997904rArr

120597119898(ℎ (119911)) ℎ (119911)

minus1+ 119885119911119898minus1

= 119873119898

(60)

Projecting the right-hand side on gl2(119877)[119911 119911

minus1)gt0

gives us theidentity

(119885119911119898minus1)gt0= 119873119898 (61)

we are looking for

In the next section we produce a context where we canconstruct wave matrices for both hierarchies in which theproduct is real

4 The Construction of the Solutions

In this section we will show how to construct a wide class ofsolutions of both hierarchies This is done in the style of [6]We first describe the group of loops we will work with Foreach 0 lt 119903 lt 1 let 119860

119903be the annulus

119911 | 119911 isin C 119903 ⩽ |119911| ⩽1

119903 (62)

Following [7] we use the notation 119871anGL2(C) for the col-lection of holomorphic maps from some annulus 119860

119903into

GL2(C) It is a group with respect to pointwise multiplication

and contains in a natural way GL2(C) as a subgroup as the

collection of constant maps into GL2(C) Other examples

of elements in 119871anGL2(C) are the elements of Δ However119871anGL2(C) is more than just a group it is an infinitedimensional Lie group Its manifold structure can be read offfrom its Lie algebra 119871angl2(C) consisting of all holomorphicmaps 120574 119880 rarr gl

2(C) where 119880 is an open neighborhood of

some annulus 119860119903 0 lt 119903 lt 1 Since gl

2(C) is a Lie algebra

the space 119871angl2(C) becomes a Lie algebra with respect to thepointwise commutator Topologically the space 119871angl2(C) isthe direct limit of all the spaces 119871an119903gl2(C) where this lastspace consists of all 120574 corresponding to the fixed annulus119860119903 One gives each 119871an119903gl2(C) the topology of uniform

convergence and with that topology it becomes a Banachspace In this way 119871angl2(C) becomes a Frechet space Thepointwise exponential map defines a local diffeomorphismaround zero in 119871angl2(C) (see eg [8])

Now each 120574 isin 119871angl2(C) possesses an expansion in aFourier series

120574 =

infin

sum

119896=minusinfin

120574119896119911119896 with each 120574

119896isin gl2(C) (63)

that converges absolutely on the annulus 119860119903

infin

sum

119896=minusinfin

10038171003817100381710038171205741198961003817100381710038171003817 119903minus|119896|

lt infin (64)

This Fourier expansion is used to make two different decom-positions of the Lie algebra119871angl2(C)Thefirst is the analoguefor the present Lie algebra 119871angl2(C) of decomposition (8) ofgl2(119877)[119911 119911

minus1) that lies at the basis of the Lax equations of the

AKNS hierarchy Namely consider the subspaces

119871angl2 (C)⩾0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=0

120574119896119911119896

8 Advances in Mathematical Physics

119871angl2 (C)lt0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =minus1

sum

119896=minusinfin

120574119896119911119896

(65)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra The first Lie algebra consists ofthe elements in 119871angl2(C) that extend to holomorphic mapsdefined on some disk 119911 isin C | |119911| ⩽ 1119903 0 lt 119903 lt 1 andthe second Lie algebra corresponds to the maps in 119871angl2(C)that have a holomorphic extension towards some disk aroundinfinity 119911 isin P1(C) | |119911| ⩾ 119903 0 lt 119903 lt 1 and that arezero at infinity To each of the two Lie subalgebras belongsa subgroup of 119871anGL2(C) The pointwise exponential mapapplied to elements of 119871angl2(C)lt0 yields elements of

119880minus= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

minus1

sum

119896=minusinfin

120574119896119911119896 (66)

and the exponential map applied to elements of 119871angl2(C)⩾0maps them into

119875+= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740 +

infin

sum

119896=1

120574119896119911119896 with 120574

0

isin GL2(C)

(67)

Both119880minusand 119875+are easily seen to be subgroups of 119871anGL2(C)

and since the direct sum of their Lie algebras is 119871angl2(C)their product

Ω = 119880minus119875+ (68)

is open in 119871anGL2(C) and is called like in the finite dimen-sional case the big cell with respect to 119880

minusand 119875

+

The second decomposition is the analogue of the splitting(11) of gl

2(119877)[119911 119911

minus1) that led to the Lax equations of the strict

AKNS hierarchy Now we consider the subspaces

119871angl2 (C)gt0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=1

120574119896119911119896

119871angl2 (C)⩽0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =0

sum

119896=minusinfin

120574119896119911119896

(69)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra 119871angl2(C)gt0 consists of mapswhose holomorphic extension to 119911 = 0 equals zero inthat point and 119871angl2(C)⩽0 is the set of maps that extendhomomorphically to a neighborhood of infinity To eachof them belongs a subgroup of 119871anGL2(C) The pointwiseexponential map applied to elements of 119871angl2(C)gt0 yieldselements of

119880+= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

infin

sum

119896=1

120574119896119911119896 (70)

and the exponential map applied to elements of 119871angl2(C)⩽0maps them into

119875minus= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740

+

minus1

sum

119896=minusinfin

120574119896119911119896 with 120574

0isin GL2(C)

(71)

Both119875minusand119880

+are easily seen to be subgroups of 119871anGL2(C)

Since the direct sum of their Lie algebras is 119871angl2(C) theirproduct 119875

minus119880+is open and because 119875

+= GL

2(C)119880+=

119880+GL2(C) and 119875

minus= GL2(C)119880minus= 119880minusGL2(C) this gives the

equality

119875minus119880+= Ω = 119880

minus119875+ (72)

the big cell in 119871anGL2(C)The next two subgroups of 119871anGL2(C) correspond to

the exponential factors in both linearizations The group ofcommuting flows relevant for the AKNS hierarchy is thegroup

Γ0= 1205740(119905) = exp(

infin

sum

119894=0

1199051198941198760119911119894) | 1205740isin 119875+ (73)

and similarly for the strict AKNS hierarchy we use thecommuting flows from

Γ1= 1205741(119905) = exp(

infin

sum

119894=1

1199051198941198760119911119894) | 1205741isin 119880+ (74)

Besides the groups Δ Γ0 and Γ

1 there are more subgroups in

119871anGL2(C) that commute with those three groups and theyare in a sense complimentary to Δ Γ

0and Δ Γ

1 respectively

That is why we use the following notations for them in 119880minus

there is

Γ119888

0= 120574119888

0(119905) = exp(

infin

sum

119896=1

1199041198961198760119911minus119896) | 120574119888

0isin 119880minus (75)

and in 119875minuswe have

Γ119888

1= 120574119888

1(119905) = exp(

infin

sum

119896=0

1199041198961198760119911minus119896) | 120574119888

1isin 119875minus (76)

We have now all ingredients to describe the constructionof the solutions to each hierarchy and we start with theAKNS hierarchy Take inside the product 119871anGL2(C) times Δthe collection 119878

0of pairs (119892 120575(119898)) such that there exists a

1205740(119905) isin Γ

0satisfying

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898) isin Ω = 119880

minus119875+ (77)

For such a pair (119892 120575(119898)) we take the collection Γ0(119892 120575(119898))

of all 1205740(119905) satisfying the condition (77) This is an open

nonempty subset of Γ0 Let 119877

0(119892 120575(119898)) be the algebra of

analytic functions Γ0(119892 120575(119898)) rarr C This is the algebra of

Advances in Mathematical Physics 9

functions 119877 that we associate with the point (119892 120575(119898)) isin 1198780

and for the commuting derivations of 1198770(119892 120575(119898)) we choose

120597119894fl 120597120597119905

119894 119894 ⩾ 0 By property (77) we have for all 120574

0(119905) isin

Γ0(119892 120575(119898))

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898)

= 119906minus(119892 120575 (119898))

minus1

119901+(119892 120575 (119898))

(78)

with 119906minus(119892 120575(119898)) isin 119880

minus119901+(119892 120575(119898)) isin 119875

+Then all thematrix

coefficients in the Fourier expansions of the elements 119906minus(119892

120575(119898)) and 119901+(119892 120575(119898)) belong to the algebra 119877

0(119892 120575(119898)) It is

convenient to write relation (78) in the form

Ψ119892120575(119898)

fl 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905)

= 119901+(119892 120575 (119898)) 120575 (119898) 120574

0(119905) 119892minus1

= 119901+(119892 120575 (119898)) 120574

0(119905) 120575 (119898) 119892

minus1

= 119902+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(79)

with 119902+(119892 120575(119898)) isin 119875

+ Clearly Ψ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M0 where all the products in the

decomposition 119906minus(119892 120575(119898))120575(119898)120574

0(119905) are no longer formal

but real This is our potential wave matrix for the AKNShierarchy

To get the potential wave function for the strict AKNShierarchy we proceed similarly only this time we use thedecomposition Ω = 119875

minus119880+ Consider again the product

119871anGL2(C) times Δ and the subset 1198781of pairs (119892 120575(119898)) such that

there exists a 1205741(119905) isin Γ

1satisfying

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898) isin Ω = 119875

minus119880+ (80)

For such a pair (119892 120575(119898)) we define Γ1(119892 120575(119898)) as the set of

all 1205741(119905) satisfying condition (77) This is an open nonempty

subset of Γ1 Let 119877

1(119892 120575(119898)) be the algebra of analytic

functions Γ1(119892 120575(119898)) rarr C This is the algebra of functions

119877 that we associate with the point (119892 120575(119898)) isin 1198781and for the

commuting derivations of 1198771(119892 120575(119898)) we choose 120597

119894fl 120597120597119905

119894

119894 gt 0 By property (80) we have for all 1205741(119905) isin Γ

1(119892 120575(119898))

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898)

= 119901minus(119892 120575 (119898))

minus1

119906+(119892 120575 (119898))

(81)

with 119901minus(119892 120575(119898)) isin 119875

minus 119906+(119892 120575(119898)) isin 119880

+ Then all

the matrix coefficients in the Fourier expansions of theelements 119901

minus(119892 120575(119898)) and 119906

+(119892 120575(119898)) belong to the algebra

1198771(119892 120575(119898)) It is convenient to write relation (81) in the form

Φ119892120575(119898)

fl 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905)

= 119906+(119892 120575 (119898)) 120575 (119898) 120574

1(119905) 119892minus1

= 119906+(119892 120575 (119898)) 120574

1(119905) 120575 (119898) 119892

minus1

= 119908+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(82)

with 119908+(119892 120575(119898)) isin 119880

+ Clearly Φ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M1where all the products in the

decomposition 119901minus(119892 120575(119898))120575(119898)120574

1(119905) are no longer formal

but realTo show that Ψ

119892120575(119898)is a wave matrix for the AKNS

hierarchy of type 120575(119898) and Φ119892120575(119898)

one for the strict AKNShierarchy we use the fact that it suffices to prove the relationsin Proposition 6 We treat first Ψ

119892120575(119898) In (79) we presented

two different decompositions of Ψ119892120575(119898)

that we will use tocompute 120597

119894(Ψ119892120575(119898)

) With respect to 119906minus(119892 120575(119898))120575(119898)120574

0(119905)

there holds120597119894(Ψ119892120575(119898)

)

= 120597119894(119906minus(119892 120575 (119898))) 119906

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905) = 119872

119894Ψ119892120575(119898)

(83)

with119872119894isin gl2(119877(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911 If we take

the decomposition 119902+(119892 120575(119898))120575(119898)119892

minus1 then we get

120597119894(Ψ119892120575(119898)

)

= 120597119894(119902+(119892 120575 (119898))) 119902

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(84)

with 120597119894(119902+(119892 120575(119898)))119902

+(119892 120575(119898))

minus1 of the formsum119895⩾0119902119895119911119895 with

all 119902119895isin gl2(119877(119892 120575(119898))) As Ψ

119892120575(119898)is invertible we obtain

119872119894= sum119895⩾0119902119895119911119895 and thus 119872

119894= sum119894

119895=0119902119895119911119895 and that is the

desired resultWe handle the case of Φ

119892120575(119898)in a similar way Also in

(79) you can find two different decompositions of Φ119892120575(119898)

that we will use to compute 120597119894(Φ119892120575(119898)

) With respect to theexpression 119901

minus(119892 120575(119898))120575(119898)120574

1(119905) we get for all 119894 ⩾ 1

120597119894(Φ119892120575(119898)

)

= 120597119894(119901minus(119892 120575 (119898))) 119901

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905) = 119873

119894Φ119892120575(119898)

(85)

again with 119873119894isin gl2(1198771(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911

Next we take the decomposition 119908+(119892 120575(119898))120575(119898)119892

minus1 andthat yields

120597119894(Φ119892120575(119898)

)

= 120597119894(119908+(119892 120575 (119898)))119908

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(86)

Since the zeroth order term of 119908+(119892 120575(119898)) is Id we note

that the element 120597119894(119908+(119892 120575(119898)))119908

+(119892 120575(119898))

minus1 has the formsum119895⩾1119908119895119911119895 with all 119908

119895isin gl2(1198771(119892 120575(119898))) As Φ

119892120575(119898)is also

invertible we obtain119873119894= sum119895⩾1119908119895119911119895 and thus119873

119894= sum119894

119895=1119908119895119911119895

and that is the result we were looking for in the strict caseThe abovementioned constructions are not affected seri-

ously by a number of transformations For example if wewrite 120575 = 120575((1 1)) then all 120575119896 119896 isin Z are central elementsand we have for all (119892 120575(119898)) isin 119878

0and 119878

1respectively and all

119896 isin Z

Ψ119892120575(119898)120575

119896 = Ψ119892120575(119898)

120575119896

respectively Φ119892120575(119898)120575

119896 = Φ119892120575(119898)

120575119896

(87)

10 Advances in Mathematical Physics

Also for any 1199011isin 119875+ each 120574119888

0(119904) isin Γ

119888

0 and all (119892 120575(119898)) isin

1198780 we see that the element (120574119888

0(119904)119892120575(minus119898)119901

1120575(119898) 120575(119898)) also

belongs to 1198780 for

120575 (119898) 1205740(119905) 120574119888

0(119904) 119892120575 (minus119898) 119901

1120575 (119898) 120574

0(119905)minus1120575 (minus119898)

= 120574119888

0(119904) 119906minus1

minus119901+1205740(119905) 11990111205740(119905)minus1

(88)

In particular we see that

Ψ120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575(119898)

= 119906minus120574119888

0(119904)minus1120575 (119898) 120574

0(119905) (89)

Its analogue in the strict case is as follows for any 1199061isin

119880+ each 120574119888

1(119904) isin Γ

119888

1 and all (119892 120575(119898)) isin 119878

1 the element

(120574119888

1(119904)119892120575(minus119898)119906

1120575(119898) 120575(119898)) also belongs to 119878

1and there

holds

Φ120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575(119898)

= 119901minus120574119888

1(119904)minus1120575 (119898) 120574

1(119905) (90)

We resume the foregoing results in the following theorem

Theorem 7 Consider the product spaceΠ fl 119871anGL2(C) times ΔThen the following holds

(a) For each point (119892 120575(119898)) isin Π that satisfies condition(77) we have a wave matrix of type 120575(119898) of the AKNShierarchyΨ

119892120575(119898) defined by (79) and this determines a

solution 119876119892120575(119898)

= 119906minus(119892 120575(119898))119876

0119906minus(119892 120575(119898))

minus1 of theAKNS hierarchy For each 119901

1isin 119875+ every 120574119888

0(119904) isin Γ

119888

0

and all powers of 120575 one has

119876120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575

119896120575(119898)

= 119876119892120575(119898)

(91)

(b) For each point (119892 120575(119898)) isin Π that satisfies condition(80) we have a wave matrix of type 120575(119898) of thestrict AKNS hierarchy Φ

119892120575(119898) defined by (82) and it

determines a solution 119885119892120575(119898)

= 119901minus(119892 120575(119898))119876

0119911119901minus(119892

120575(119898))minus1 of the strict AKNS hierarchy For each 119906

1isin 119880+

every 1205741198881(119904) isin Γ

119888

1 and all powers of 120575 one has

119885120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575

119896120575(119898)

= 119885119892120575(119898)

(92)

Competing Interests

The author declares that they have no competing interests

References

[1] M Adler P vanMoerbeke and P VanhaeckeAlgebraic Integra-bility Painleve Geometry and Lie Algebras vol 47 of Ergebnisseder Mathematik und ihrer Grenzgebiete Springer 2004

[2] G F Helminck A G Helminck and E A PanasenkoldquoIntegrable deformations in the algebra of pseudodifferentialoperators from a Lie algebraic perspectiverdquo Theoretical andMathematical Physics vol 174 no 1 pp 134ndash153 2013

[3] M J Ablowitz D J Kaup A C Newell and H Segur ldquoTheinverse scattering transform-Fourier analysis for nonlinearproblemsrdquo Studies in Applied Mathematics vol 53 no 4 pp249ndash315 1974

[4] H Flashka A CNewell and T Ratiu ldquoKac-Moody Lie algebrasand soliton equations II Lax equations associated with 119860

1

(1)rdquoPhysicaDNonlinear Phenomena vol 9 no 3 pp 300ndash323 1983

[5] EDateM JimboMKashiwara andTMiwa ldquoTransformationgroups for soliton equationsrdquo in Proceedings of the RIMSSymposium on Nonlinear Integrable SystemsmdashClassical Theoryand Quantum Theory T Jimbo and T Miwa Eds WorldScience Publisher Kyoto Japan 1981

[6] G Segal and G Wilson ldquoLoop groups and equations of KdVtyperdquo Publications Mathematiques de lrsquoIHES vol 61 pp 5ndash651985

[7] A Pressley and G Segal Loop Groups Oxford MathematicalMonographs Clarendon Press 1986

[8] R S Hamilton ldquoThe inverse function theorem of Nash andMoserrdquoBulletin of the AmericanMathematical Society vol 7 no1 pp 65ndash222 1982

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 5

Proof Againwe prove the statement for119885 that for119876 is shownin a similarway So assume that there is one Lax equation (22)that does not hold Then there is a119898 ⩾ 1 such that

120597119898(119885) minus [119862

119898 119885] = sum

119895⩽119896(119898)

119883119895119911119895 with 119883

119896(119898)= 0 (30)

Since both 120597119898(119885) and [119862

119898 119885] are of order smaller than or

equal to one in 119911 we know that 119896(119898) ⩽ 1 Further we can saythat for all119873 ⩾ 0

120597119898(119911119873119885) minus [119862

119898 119911119873119885] = sum

119895⩽119896(119898)

119883119895119911119895+119873

with 119883119896(119898)

= 0

(31)

and we see by letting 119873 go to infinity that the right-handside can obtain any sufficiently large order in 119911 By the zerocurvature relation for119873 and119898 we get for the left-hand side

120597119898(119911119873119885) minus [119862

119898 119911119873119885]

= 120597119898(119862119873) minus [119862

119898 119862119873] + 120597119898((119911119873119885)⩽0)

minus [119862119898 (119911119873119885)⩽0]

= 120597119873(119862119898) + 120597119898((119911119873119885)⩽0) minus [119862

119898 (119911119873119885)⩽0]

(32)

and this last expression is of order smaller or equal to 119898 in119911 This contradicts the unlimited growth in orders of 119911 of theright-hand side Hence all Lax equations (22) have to holdfor 119885

Because of the equivalence between Lax equations (20)for 119876 and the zero curvature relations (23) for 119861

119898 we call

this last set of equations also the zero curvature form of theAKNS hierarchy Similarly the zero curvature relations (24)for 119862

119898 are called the zero curvature form of the strict AKNS

hierarchyBesides the zero curvature relations for the cut-off rsquos

119861119898 and 119862

119898 respectively corresponding to respectively

a solution 119876 of the AKNS hierarchy and a solution 119885 of thestrict AKNS hierarchy also other parts satisfy such relationsDefine

119860119898fl 119861119898minus 119876119911119898 119898 ⩾ 0

119863119898fl 119862119898minus 119885119911119898minus1 119898 ⩾ 1

(33)

Then we have the following result

Corollary 5 The following relations hold

(i) The parts 119860119898| 119898 ⩾ 0 of a solution 119876 of the AKNS

hierarchy satisfy

1205971198981(1198601198982) minus 1205971198982(1198601198981) minus [119860

1198981 1198601198982] = 0 (34)

(ii) The parts 119863119898| 119898 ⩾ 0 of a solution 119885 of the strict

AKNS hierarchy satisfy

1205971198981(1198631198982) minus 1205971198982(1198631198981) minus [119863

1198981 1198631198982] = 0 (35)

Proof Again we show the result only in the strict case Notethat 119885119911119898minus1 satisfy Lax equations similar to 119885

120597119894(119885119911119898minus1) = [119863

119894 119885119911119898minus1] 119894 ⩾ 1 (36)

Now we substitute in the zero curvature relations for 119862119898

everywhere the relation 119862119898= 119863119898+ 119885119911119898minus1 and use the Lax

equations above and the fact that all 119885119911119898minus1 commute Thisgives the desired result

To clarify the link with the AKNS equation considerrelation (23) for119898

1= 2 and119898

2= 1

1205972(1198760119911 + 119876

1) = 1205971(11987601199112+ 1198761119911 + 119876

2)

+ [1198762 1198760119911 + 119876

1]

(37)

Then this identity reduces in sl2(119877)[119911 119911

minus1)⩾0 since 119876

0is

constant to the following two equalities

1205971(1198761) = [119876

0 1198762]

1205972(1198761) = 1205971(1198762) + [119876

2 1198761]

(38)

The first gives an expression of the off-diagonal terms of 1198762

in the coefficients 119902 and 119903 of 1198761 that is

11990212=119894

21205971(119902)

11990221= minus

119894

21205971(119903)

(39)

and the second equation becomes AKNS equations (4) if onehas 1205971= 120597120597119909 and 120597

2= 120597120597119905

3 The Linearization of Both Hierarchies

The zero curvature form of both hierarchies points at thepossible existence of a linear system of which the zerocurvature equations form the compatibility conditions Wepresent here such a system for each hierarchy For theAKNS hierarchy this system the linearization of the AKNShierarchy is as follows take a potential solution119876 of the form(17) and consider the system

119876120595 = 1205951198760

120597119898(120595) = 119861

119898120595 forall119898 ⩾ 0 119861

119898= (119876119911

119898)⩾0

(40)

Likewise for a potential solution 119885 of the strict AKNShierarchy of the form (19) the linearization of the strict AKNShierarchy is given by

119885120593 = 1205931198760119911

120597119898(120593) = 119862

119898120593 forall119898 ⩾ 1 119862

119898= (119885119911

119898minus1)gt0

(41)

Before specifying120595 and120593 we show themanipulations neededto derive the Lax equations of the hierarchy from theirlinearizationWedo this for the strictAKNShierarchy for the

6 Advances in Mathematical Physics

AKNS hierarchy one proceeds similarly Apply 120597119898to the first

equation in (41) and use both equalities in (41) in the sequel

120597119898(119885120593 minus 120593119876

0119911) = 120597

119898(119885) 120593 + 119885120597

119898(120593) minus 120597

119898(120593)1198760119911

= 0 = 120597119898(119885) 120593 + 119885119862

119898120593 minus 119862

1198981205931198760119911

= 120597119898(119885) minus [119862

119898 119885] 120593 = 0

(42)

Hence if we can scratch 120593 from (42) then we obtain the Laxequations of the strict AKNS hierarchy To get the propersetup where all these manipulations make sense we first havea look at the linearization for the trivial solutions119876 = 119876

0and

119885 = 1198760119911 In the AKNS case we have then

11987601205950= 12059501198760

120597119898(1205950) = 119876

01199111198981205950 forall119898 ⩾ 0

(43)

and for its strict version

11987601199111205930= 12059301198760119911

120597119898(1205930) = 119876

01199111198981205930 forall119898 ⩾ 1

(44)

Assuming that each derivation 120597119898equals 120597120597119905

119898 one arrives

for (43) at the solution

1205950= 1205950(119905 119911) = exp(

infin

sum

119898=0

1199051198981198760119911119898) (45)

where 119905 is the short hand notation for 119905119898| 119898 ⩾ 0 and for

(44) it leads to

1205930= 1205930(119905 119911) = exp(

infin

sum

119898=1

1199051198981198760119911119898)

with 119905 = 119905119898| 119898 ⩾ 1

(46)

In general one needs in the linearizations (120595) and (120593)respectively a left action with elements like 119876 119861

119898and 119885

119862119898from gl

2(119877)[119911 119911

minus1) respectively an action of all the 120597

119898

and a right action of 1198760and 119876

0119911 respectively This can

all be realized in suitable gl2(119877)[119911 119911

minus1) modules that are

perturbations of the trivial solutions 1205950and 120593

0 Consider in

a AKNS setting

M0= 119892 (119911) 120595

0=

119873

sum

119894=minusinfin

1198921198941199111198941205950| 119892 (119911)

=

119873

sum

119894=minusinfin

119892119894119911119894isin gl2(119877) [119911 119911

minus1)

(47)

and in a strict AKNS setting

M1= 119892 (119911) 120593

0=

119873

sum

119894=minusinfin

1198921198941199111198941205930| 119892 (119911)

=

119873

sum

119894=minusinfin

119892119894119911119894isin gl2(119877) [119911 119911

minus1)

(48)

where the products 119892(119911)1205950and 119892(119911)120593

0should be seen as

formal and both factors should be kept separate On bothM0

andM1 one can define the required actions for each ℎ(119911) isin

gl2(119877)[119911 119911

minus1) define

ℎ (119911) sdot 119892 (119911) 1205950fl ℎ (119911) 119892 (119911) 120595

0

respectively ℎ (119911) sdot 119892 (119911) 1205930fl ℎ (119911) 119892 (119911) 120593

0

(49)

We define the right-hand action of 1198760and 119876

0119911 respectively

by

119892 (119911) 12059501198760fl 119892 (119911)119876

0 1205950

respectively 119892 (119911) 12059301198760119911 fl 119892 (119911)119876

0119911 1205930

(50)

and the action of each 120597119898by

120597119898(119892 (119911) 120595

0)

=

119873

sum

119894=minusinfin

120597119898(119892119894) 119911119894+

119873

sum

119894=minusinfin

1198921198941198760119911119894+1198981205950

120597119898(119892 (119911) 120593

0)

=

119873

sum

119894=minusinfin

120597119898(119892119894) 119911119894+

119873

sum

119894=minusinfin

1198921198941198760119911119894+1198981205930

(51)

Analogous to the terminology in the scalar case (see [5]) wecall the elements of M

0and M

1oscillating matrices Note

that both M0and M

1are free gl

2(119877)[119911 119911

minus1) modules with

generators 1205950and 120593

0 respectively because for each ℎ(119911) isin

gl2(119877)[119911 119911

minus1) we have

ℎ (119911) sdot 1205950= ℎ (119911) sdot 1 120595

0= ℎ (119911) 120595

0

respectively ℎ (119911) sdot 1205930= ℎ (119911) sdot 1 120595

0= ℎ (119911) 120593

0

(52)

Hence in order to be able to perform legally the scratching of120595 = ℎ(119911)120595

0or 120593 = ℎ(119911)120593

0 it is enough to find oscillating

matrices such that ℎ(119911) is invertible in gl2(119877)[119911 119911

minus1) We will

now introduce a collection of such elements that will occur atthe construction of solutions of both hierarchies

For119898 = (1198981 1198982) isin Z2 let 120575(119898) isin gl

2(119877)[119911 119911

minus1) be given

by

120575 (119898) = (1199111198981 0

0 1199111198982) (53)

Then 120575(119898) has 120575(minus119898) as its inverse in gl2(119877)[119911 119911

minus1) and the

collection Δ = 120575(119898) | 119898 isin Z2 forms a group An element120595 isinM

0is called an oscillating matrix of type 120575(119898) if it has the

form

120595 = ℎ (119911) 120575 (119898) 1205950 with ℎ (119911) isin 119866

lt0 (54)

These oscillatingmatrices are examples of elements ofM0for

which the scratching procedure is valid Let 119876 be a potentialsolution of the AKNS hierarchy of the form (17) If there isan oscillating function 120595 of type 120575(119898) such that for 119876 and 120595

Advances in Mathematical Physics 7

(40) hold then we call120595 awave matrix of the AKNS hierarchyof type 120575(119898) In particular 119876 is then a solution of the AKNShierarchy and the first equation in (40) implies

119876ℎ (119911) 120575 (119898) = ℎ (119911)1198760120575 (119898) 997904rArr

119876 = ℎ (119911)1198760ℎ (119911)minus1

(55)

in other words the solution 119876 is totally determined by thewave matrix Similarly we call an element 120593 isin M

1an

oscillating matrix of type 120575(119898) if it has the form

120593 = ℎ (119911) 120575 (119898) 1205930 with ℎ (119911) isin 119866

⩽0 (56)

This type of oscillating matrices are examples of elements ofM1for which the scratching procedure is valid Let 119885 be a

potential solution of the strict AKNS hierarchy of the form(19) If there is an oscillating function 120593 of type 120575(119898) in M

1

such that for119885 and120593 (41) hold thenwe call120593 awavematrix ofthe strict AKNS hierarchy of type 120575(119898) In particular119885 is thena solution of the strict AKNS hierarchy and the first equationin (41) implies

119885ℎ (119911) 120575 (119898) = ℎ (119911)1198760119911120575 (119898) 997904rArr

119885 = ℎ (119911)1198760119911ℎ (119911)

minus1

(57)

so that also here the wave matrix fully determines thesolution

For both hierarchies there is a milder condition that is tobe satisfied by oscillating matrices of a certain type in orderthat they become a wave matrix of that hierarchy

Proposition 6 Let120595 = ℎ(119911)120575(119898)1205950be an oscillatingmatrix

of type 120575(119898) in M0and 119876 = ℎ(119911)119876

0ℎ(119911)minus1 the corresponding

potential solution of the AKNS hierarchy Similarly let 120593 =

ℎ(119911)120575(119898)1205930be such a matrix in M

1with potential solution

119885 = ℎ(119911)1198760119911ℎ(119911)minus1

(a) If there exists for each 119898 ⩾ 0 an element 119872119898isin

gl2(119877)[119911 119911

minus1)⩾0

such that

120597119898(120595) = 119872

119898120595 (58)

then each119872119898= (119876119911

119898)⩾0

and 120595 is a wave matrix oftype 120575(119898) for the AKNS hierarchy

(b) If there exists for each 119898 ⩾ 1 an element 119873119898isin

gl2(119877)[119911 119911

minus1)gt0

such that

120597119898(120593) = 119873

119898120593 (59)

then each 119873119898= (119885119911

119898minus1)gt0

and 120593 is a wave matrix oftype 120575(119898) for the strict AKNS hierarchy

Proof We give the proof again for the strict AKNS casethe other one is similar By using the fact that M

1is a free

gl2(119877)[119911 119911

minus1)module with generator 120593

0we can translate the

relations 120597119898(120593) = 119873

119898120593 into equations in gl

2(119877)[119911 119911

minus1) This

yields

120597119898(ℎ (119911)) + ℎ (119911)119876

0119911119898= 119873119898ℎ (119911) 997904rArr

120597119898(ℎ (119911)) ℎ (119911)

minus1+ 119885119911119898minus1

= 119873119898

(60)

Projecting the right-hand side on gl2(119877)[119911 119911

minus1)gt0

gives us theidentity

(119885119911119898minus1)gt0= 119873119898 (61)

we are looking for

In the next section we produce a context where we canconstruct wave matrices for both hierarchies in which theproduct is real

4 The Construction of the Solutions

In this section we will show how to construct a wide class ofsolutions of both hierarchies This is done in the style of [6]We first describe the group of loops we will work with Foreach 0 lt 119903 lt 1 let 119860

119903be the annulus

119911 | 119911 isin C 119903 ⩽ |119911| ⩽1

119903 (62)

Following [7] we use the notation 119871anGL2(C) for the col-lection of holomorphic maps from some annulus 119860

119903into

GL2(C) It is a group with respect to pointwise multiplication

and contains in a natural way GL2(C) as a subgroup as the

collection of constant maps into GL2(C) Other examples

of elements in 119871anGL2(C) are the elements of Δ However119871anGL2(C) is more than just a group it is an infinitedimensional Lie group Its manifold structure can be read offfrom its Lie algebra 119871angl2(C) consisting of all holomorphicmaps 120574 119880 rarr gl

2(C) where 119880 is an open neighborhood of

some annulus 119860119903 0 lt 119903 lt 1 Since gl

2(C) is a Lie algebra

the space 119871angl2(C) becomes a Lie algebra with respect to thepointwise commutator Topologically the space 119871angl2(C) isthe direct limit of all the spaces 119871an119903gl2(C) where this lastspace consists of all 120574 corresponding to the fixed annulus119860119903 One gives each 119871an119903gl2(C) the topology of uniform

convergence and with that topology it becomes a Banachspace In this way 119871angl2(C) becomes a Frechet space Thepointwise exponential map defines a local diffeomorphismaround zero in 119871angl2(C) (see eg [8])

Now each 120574 isin 119871angl2(C) possesses an expansion in aFourier series

120574 =

infin

sum

119896=minusinfin

120574119896119911119896 with each 120574

119896isin gl2(C) (63)

that converges absolutely on the annulus 119860119903

infin

sum

119896=minusinfin

10038171003817100381710038171205741198961003817100381710038171003817 119903minus|119896|

lt infin (64)

This Fourier expansion is used to make two different decom-positions of the Lie algebra119871angl2(C)Thefirst is the analoguefor the present Lie algebra 119871angl2(C) of decomposition (8) ofgl2(119877)[119911 119911

minus1) that lies at the basis of the Lax equations of the

AKNS hierarchy Namely consider the subspaces

119871angl2 (C)⩾0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=0

120574119896119911119896

8 Advances in Mathematical Physics

119871angl2 (C)lt0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =minus1

sum

119896=minusinfin

120574119896119911119896

(65)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra The first Lie algebra consists ofthe elements in 119871angl2(C) that extend to holomorphic mapsdefined on some disk 119911 isin C | |119911| ⩽ 1119903 0 lt 119903 lt 1 andthe second Lie algebra corresponds to the maps in 119871angl2(C)that have a holomorphic extension towards some disk aroundinfinity 119911 isin P1(C) | |119911| ⩾ 119903 0 lt 119903 lt 1 and that arezero at infinity To each of the two Lie subalgebras belongsa subgroup of 119871anGL2(C) The pointwise exponential mapapplied to elements of 119871angl2(C)lt0 yields elements of

119880minus= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

minus1

sum

119896=minusinfin

120574119896119911119896 (66)

and the exponential map applied to elements of 119871angl2(C)⩾0maps them into

119875+= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740 +

infin

sum

119896=1

120574119896119911119896 with 120574

0

isin GL2(C)

(67)

Both119880minusand 119875+are easily seen to be subgroups of 119871anGL2(C)

and since the direct sum of their Lie algebras is 119871angl2(C)their product

Ω = 119880minus119875+ (68)

is open in 119871anGL2(C) and is called like in the finite dimen-sional case the big cell with respect to 119880

minusand 119875

+

The second decomposition is the analogue of the splitting(11) of gl

2(119877)[119911 119911

minus1) that led to the Lax equations of the strict

AKNS hierarchy Now we consider the subspaces

119871angl2 (C)gt0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=1

120574119896119911119896

119871angl2 (C)⩽0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =0

sum

119896=minusinfin

120574119896119911119896

(69)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra 119871angl2(C)gt0 consists of mapswhose holomorphic extension to 119911 = 0 equals zero inthat point and 119871angl2(C)⩽0 is the set of maps that extendhomomorphically to a neighborhood of infinity To eachof them belongs a subgroup of 119871anGL2(C) The pointwiseexponential map applied to elements of 119871angl2(C)gt0 yieldselements of

119880+= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

infin

sum

119896=1

120574119896119911119896 (70)

and the exponential map applied to elements of 119871angl2(C)⩽0maps them into

119875minus= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740

+

minus1

sum

119896=minusinfin

120574119896119911119896 with 120574

0isin GL2(C)

(71)

Both119875minusand119880

+are easily seen to be subgroups of 119871anGL2(C)

Since the direct sum of their Lie algebras is 119871angl2(C) theirproduct 119875

minus119880+is open and because 119875

+= GL

2(C)119880+=

119880+GL2(C) and 119875

minus= GL2(C)119880minus= 119880minusGL2(C) this gives the

equality

119875minus119880+= Ω = 119880

minus119875+ (72)

the big cell in 119871anGL2(C)The next two subgroups of 119871anGL2(C) correspond to

the exponential factors in both linearizations The group ofcommuting flows relevant for the AKNS hierarchy is thegroup

Γ0= 1205740(119905) = exp(

infin

sum

119894=0

1199051198941198760119911119894) | 1205740isin 119875+ (73)

and similarly for the strict AKNS hierarchy we use thecommuting flows from

Γ1= 1205741(119905) = exp(

infin

sum

119894=1

1199051198941198760119911119894) | 1205741isin 119880+ (74)

Besides the groups Δ Γ0 and Γ

1 there are more subgroups in

119871anGL2(C) that commute with those three groups and theyare in a sense complimentary to Δ Γ

0and Δ Γ

1 respectively

That is why we use the following notations for them in 119880minus

there is

Γ119888

0= 120574119888

0(119905) = exp(

infin

sum

119896=1

1199041198961198760119911minus119896) | 120574119888

0isin 119880minus (75)

and in 119875minuswe have

Γ119888

1= 120574119888

1(119905) = exp(

infin

sum

119896=0

1199041198961198760119911minus119896) | 120574119888

1isin 119875minus (76)

We have now all ingredients to describe the constructionof the solutions to each hierarchy and we start with theAKNS hierarchy Take inside the product 119871anGL2(C) times Δthe collection 119878

0of pairs (119892 120575(119898)) such that there exists a

1205740(119905) isin Γ

0satisfying

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898) isin Ω = 119880

minus119875+ (77)

For such a pair (119892 120575(119898)) we take the collection Γ0(119892 120575(119898))

of all 1205740(119905) satisfying the condition (77) This is an open

nonempty subset of Γ0 Let 119877

0(119892 120575(119898)) be the algebra of

analytic functions Γ0(119892 120575(119898)) rarr C This is the algebra of

Advances in Mathematical Physics 9

functions 119877 that we associate with the point (119892 120575(119898)) isin 1198780

and for the commuting derivations of 1198770(119892 120575(119898)) we choose

120597119894fl 120597120597119905

119894 119894 ⩾ 0 By property (77) we have for all 120574

0(119905) isin

Γ0(119892 120575(119898))

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898)

= 119906minus(119892 120575 (119898))

minus1

119901+(119892 120575 (119898))

(78)

with 119906minus(119892 120575(119898)) isin 119880

minus119901+(119892 120575(119898)) isin 119875

+Then all thematrix

coefficients in the Fourier expansions of the elements 119906minus(119892

120575(119898)) and 119901+(119892 120575(119898)) belong to the algebra 119877

0(119892 120575(119898)) It is

convenient to write relation (78) in the form

Ψ119892120575(119898)

fl 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905)

= 119901+(119892 120575 (119898)) 120575 (119898) 120574

0(119905) 119892minus1

= 119901+(119892 120575 (119898)) 120574

0(119905) 120575 (119898) 119892

minus1

= 119902+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(79)

with 119902+(119892 120575(119898)) isin 119875

+ Clearly Ψ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M0 where all the products in the

decomposition 119906minus(119892 120575(119898))120575(119898)120574

0(119905) are no longer formal

but real This is our potential wave matrix for the AKNShierarchy

To get the potential wave function for the strict AKNShierarchy we proceed similarly only this time we use thedecomposition Ω = 119875

minus119880+ Consider again the product

119871anGL2(C) times Δ and the subset 1198781of pairs (119892 120575(119898)) such that

there exists a 1205741(119905) isin Γ

1satisfying

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898) isin Ω = 119875

minus119880+ (80)

For such a pair (119892 120575(119898)) we define Γ1(119892 120575(119898)) as the set of

all 1205741(119905) satisfying condition (77) This is an open nonempty

subset of Γ1 Let 119877

1(119892 120575(119898)) be the algebra of analytic

functions Γ1(119892 120575(119898)) rarr C This is the algebra of functions

119877 that we associate with the point (119892 120575(119898)) isin 1198781and for the

commuting derivations of 1198771(119892 120575(119898)) we choose 120597

119894fl 120597120597119905

119894

119894 gt 0 By property (80) we have for all 1205741(119905) isin Γ

1(119892 120575(119898))

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898)

= 119901minus(119892 120575 (119898))

minus1

119906+(119892 120575 (119898))

(81)

with 119901minus(119892 120575(119898)) isin 119875

minus 119906+(119892 120575(119898)) isin 119880

+ Then all

the matrix coefficients in the Fourier expansions of theelements 119901

minus(119892 120575(119898)) and 119906

+(119892 120575(119898)) belong to the algebra

1198771(119892 120575(119898)) It is convenient to write relation (81) in the form

Φ119892120575(119898)

fl 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905)

= 119906+(119892 120575 (119898)) 120575 (119898) 120574

1(119905) 119892minus1

= 119906+(119892 120575 (119898)) 120574

1(119905) 120575 (119898) 119892

minus1

= 119908+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(82)

with 119908+(119892 120575(119898)) isin 119880

+ Clearly Φ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M1where all the products in the

decomposition 119901minus(119892 120575(119898))120575(119898)120574

1(119905) are no longer formal

but realTo show that Ψ

119892120575(119898)is a wave matrix for the AKNS

hierarchy of type 120575(119898) and Φ119892120575(119898)

one for the strict AKNShierarchy we use the fact that it suffices to prove the relationsin Proposition 6 We treat first Ψ

119892120575(119898) In (79) we presented

two different decompositions of Ψ119892120575(119898)

that we will use tocompute 120597

119894(Ψ119892120575(119898)

) With respect to 119906minus(119892 120575(119898))120575(119898)120574

0(119905)

there holds120597119894(Ψ119892120575(119898)

)

= 120597119894(119906minus(119892 120575 (119898))) 119906

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905) = 119872

119894Ψ119892120575(119898)

(83)

with119872119894isin gl2(119877(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911 If we take

the decomposition 119902+(119892 120575(119898))120575(119898)119892

minus1 then we get

120597119894(Ψ119892120575(119898)

)

= 120597119894(119902+(119892 120575 (119898))) 119902

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(84)

with 120597119894(119902+(119892 120575(119898)))119902

+(119892 120575(119898))

minus1 of the formsum119895⩾0119902119895119911119895 with

all 119902119895isin gl2(119877(119892 120575(119898))) As Ψ

119892120575(119898)is invertible we obtain

119872119894= sum119895⩾0119902119895119911119895 and thus 119872

119894= sum119894

119895=0119902119895119911119895 and that is the

desired resultWe handle the case of Φ

119892120575(119898)in a similar way Also in

(79) you can find two different decompositions of Φ119892120575(119898)

that we will use to compute 120597119894(Φ119892120575(119898)

) With respect to theexpression 119901

minus(119892 120575(119898))120575(119898)120574

1(119905) we get for all 119894 ⩾ 1

120597119894(Φ119892120575(119898)

)

= 120597119894(119901minus(119892 120575 (119898))) 119901

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905) = 119873

119894Φ119892120575(119898)

(85)

again with 119873119894isin gl2(1198771(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911

Next we take the decomposition 119908+(119892 120575(119898))120575(119898)119892

minus1 andthat yields

120597119894(Φ119892120575(119898)

)

= 120597119894(119908+(119892 120575 (119898)))119908

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(86)

Since the zeroth order term of 119908+(119892 120575(119898)) is Id we note

that the element 120597119894(119908+(119892 120575(119898)))119908

+(119892 120575(119898))

minus1 has the formsum119895⩾1119908119895119911119895 with all 119908

119895isin gl2(1198771(119892 120575(119898))) As Φ

119892120575(119898)is also

invertible we obtain119873119894= sum119895⩾1119908119895119911119895 and thus119873

119894= sum119894

119895=1119908119895119911119895

and that is the result we were looking for in the strict caseThe abovementioned constructions are not affected seri-

ously by a number of transformations For example if wewrite 120575 = 120575((1 1)) then all 120575119896 119896 isin Z are central elementsand we have for all (119892 120575(119898)) isin 119878

0and 119878

1respectively and all

119896 isin Z

Ψ119892120575(119898)120575

119896 = Ψ119892120575(119898)

120575119896

respectively Φ119892120575(119898)120575

119896 = Φ119892120575(119898)

120575119896

(87)

10 Advances in Mathematical Physics

Also for any 1199011isin 119875+ each 120574119888

0(119904) isin Γ

119888

0 and all (119892 120575(119898)) isin

1198780 we see that the element (120574119888

0(119904)119892120575(minus119898)119901

1120575(119898) 120575(119898)) also

belongs to 1198780 for

120575 (119898) 1205740(119905) 120574119888

0(119904) 119892120575 (minus119898) 119901

1120575 (119898) 120574

0(119905)minus1120575 (minus119898)

= 120574119888

0(119904) 119906minus1

minus119901+1205740(119905) 11990111205740(119905)minus1

(88)

In particular we see that

Ψ120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575(119898)

= 119906minus120574119888

0(119904)minus1120575 (119898) 120574

0(119905) (89)

Its analogue in the strict case is as follows for any 1199061isin

119880+ each 120574119888

1(119904) isin Γ

119888

1 and all (119892 120575(119898)) isin 119878

1 the element

(120574119888

1(119904)119892120575(minus119898)119906

1120575(119898) 120575(119898)) also belongs to 119878

1and there

holds

Φ120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575(119898)

= 119901minus120574119888

1(119904)minus1120575 (119898) 120574

1(119905) (90)

We resume the foregoing results in the following theorem

Theorem 7 Consider the product spaceΠ fl 119871anGL2(C) times ΔThen the following holds

(a) For each point (119892 120575(119898)) isin Π that satisfies condition(77) we have a wave matrix of type 120575(119898) of the AKNShierarchyΨ

119892120575(119898) defined by (79) and this determines a

solution 119876119892120575(119898)

= 119906minus(119892 120575(119898))119876

0119906minus(119892 120575(119898))

minus1 of theAKNS hierarchy For each 119901

1isin 119875+ every 120574119888

0(119904) isin Γ

119888

0

and all powers of 120575 one has

119876120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575

119896120575(119898)

= 119876119892120575(119898)

(91)

(b) For each point (119892 120575(119898)) isin Π that satisfies condition(80) we have a wave matrix of type 120575(119898) of thestrict AKNS hierarchy Φ

119892120575(119898) defined by (82) and it

determines a solution 119885119892120575(119898)

= 119901minus(119892 120575(119898))119876

0119911119901minus(119892

120575(119898))minus1 of the strict AKNS hierarchy For each 119906

1isin 119880+

every 1205741198881(119904) isin Γ

119888

1 and all powers of 120575 one has

119885120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575

119896120575(119898)

= 119885119892120575(119898)

(92)

Competing Interests

The author declares that they have no competing interests

References

[1] M Adler P vanMoerbeke and P VanhaeckeAlgebraic Integra-bility Painleve Geometry and Lie Algebras vol 47 of Ergebnisseder Mathematik und ihrer Grenzgebiete Springer 2004

[2] G F Helminck A G Helminck and E A PanasenkoldquoIntegrable deformations in the algebra of pseudodifferentialoperators from a Lie algebraic perspectiverdquo Theoretical andMathematical Physics vol 174 no 1 pp 134ndash153 2013

[3] M J Ablowitz D J Kaup A C Newell and H Segur ldquoTheinverse scattering transform-Fourier analysis for nonlinearproblemsrdquo Studies in Applied Mathematics vol 53 no 4 pp249ndash315 1974

[4] H Flashka A CNewell and T Ratiu ldquoKac-Moody Lie algebrasand soliton equations II Lax equations associated with 119860

1

(1)rdquoPhysicaDNonlinear Phenomena vol 9 no 3 pp 300ndash323 1983

[5] EDateM JimboMKashiwara andTMiwa ldquoTransformationgroups for soliton equationsrdquo in Proceedings of the RIMSSymposium on Nonlinear Integrable SystemsmdashClassical Theoryand Quantum Theory T Jimbo and T Miwa Eds WorldScience Publisher Kyoto Japan 1981

[6] G Segal and G Wilson ldquoLoop groups and equations of KdVtyperdquo Publications Mathematiques de lrsquoIHES vol 61 pp 5ndash651985

[7] A Pressley and G Segal Loop Groups Oxford MathematicalMonographs Clarendon Press 1986

[8] R S Hamilton ldquoThe inverse function theorem of Nash andMoserrdquoBulletin of the AmericanMathematical Society vol 7 no1 pp 65ndash222 1982

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Mathematical Physics

AKNS hierarchy one proceeds similarly Apply 120597119898to the first

equation in (41) and use both equalities in (41) in the sequel

120597119898(119885120593 minus 120593119876

0119911) = 120597

119898(119885) 120593 + 119885120597

119898(120593) minus 120597

119898(120593)1198760119911

= 0 = 120597119898(119885) 120593 + 119885119862

119898120593 minus 119862

1198981205931198760119911

= 120597119898(119885) minus [119862

119898 119885] 120593 = 0

(42)

Hence if we can scratch 120593 from (42) then we obtain the Laxequations of the strict AKNS hierarchy To get the propersetup where all these manipulations make sense we first havea look at the linearization for the trivial solutions119876 = 119876

0and

119885 = 1198760119911 In the AKNS case we have then

11987601205950= 12059501198760

120597119898(1205950) = 119876

01199111198981205950 forall119898 ⩾ 0

(43)

and for its strict version

11987601199111205930= 12059301198760119911

120597119898(1205930) = 119876

01199111198981205930 forall119898 ⩾ 1

(44)

Assuming that each derivation 120597119898equals 120597120597119905

119898 one arrives

for (43) at the solution

1205950= 1205950(119905 119911) = exp(

infin

sum

119898=0

1199051198981198760119911119898) (45)

where 119905 is the short hand notation for 119905119898| 119898 ⩾ 0 and for

(44) it leads to

1205930= 1205930(119905 119911) = exp(

infin

sum

119898=1

1199051198981198760119911119898)

with 119905 = 119905119898| 119898 ⩾ 1

(46)

In general one needs in the linearizations (120595) and (120593)respectively a left action with elements like 119876 119861

119898and 119885

119862119898from gl

2(119877)[119911 119911

minus1) respectively an action of all the 120597

119898

and a right action of 1198760and 119876

0119911 respectively This can

all be realized in suitable gl2(119877)[119911 119911

minus1) modules that are

perturbations of the trivial solutions 1205950and 120593

0 Consider in

a AKNS setting

M0= 119892 (119911) 120595

0=

119873

sum

119894=minusinfin

1198921198941199111198941205950| 119892 (119911)

=

119873

sum

119894=minusinfin

119892119894119911119894isin gl2(119877) [119911 119911

minus1)

(47)

and in a strict AKNS setting

M1= 119892 (119911) 120593

0=

119873

sum

119894=minusinfin

1198921198941199111198941205930| 119892 (119911)

=

119873

sum

119894=minusinfin

119892119894119911119894isin gl2(119877) [119911 119911

minus1)

(48)

where the products 119892(119911)1205950and 119892(119911)120593

0should be seen as

formal and both factors should be kept separate On bothM0

andM1 one can define the required actions for each ℎ(119911) isin

gl2(119877)[119911 119911

minus1) define

ℎ (119911) sdot 119892 (119911) 1205950fl ℎ (119911) 119892 (119911) 120595

0

respectively ℎ (119911) sdot 119892 (119911) 1205930fl ℎ (119911) 119892 (119911) 120593

0

(49)

We define the right-hand action of 1198760and 119876

0119911 respectively

by

119892 (119911) 12059501198760fl 119892 (119911)119876

0 1205950

respectively 119892 (119911) 12059301198760119911 fl 119892 (119911)119876

0119911 1205930

(50)

and the action of each 120597119898by

120597119898(119892 (119911) 120595

0)

=

119873

sum

119894=minusinfin

120597119898(119892119894) 119911119894+

119873

sum

119894=minusinfin

1198921198941198760119911119894+1198981205950

120597119898(119892 (119911) 120593

0)

=

119873

sum

119894=minusinfin

120597119898(119892119894) 119911119894+

119873

sum

119894=minusinfin

1198921198941198760119911119894+1198981205930

(51)

Analogous to the terminology in the scalar case (see [5]) wecall the elements of M

0and M

1oscillating matrices Note

that both M0and M

1are free gl

2(119877)[119911 119911

minus1) modules with

generators 1205950and 120593

0 respectively because for each ℎ(119911) isin

gl2(119877)[119911 119911

minus1) we have

ℎ (119911) sdot 1205950= ℎ (119911) sdot 1 120595

0= ℎ (119911) 120595

0

respectively ℎ (119911) sdot 1205930= ℎ (119911) sdot 1 120595

0= ℎ (119911) 120593

0

(52)

Hence in order to be able to perform legally the scratching of120595 = ℎ(119911)120595

0or 120593 = ℎ(119911)120593

0 it is enough to find oscillating

matrices such that ℎ(119911) is invertible in gl2(119877)[119911 119911

minus1) We will

now introduce a collection of such elements that will occur atthe construction of solutions of both hierarchies

For119898 = (1198981 1198982) isin Z2 let 120575(119898) isin gl

2(119877)[119911 119911

minus1) be given

by

120575 (119898) = (1199111198981 0

0 1199111198982) (53)

Then 120575(119898) has 120575(minus119898) as its inverse in gl2(119877)[119911 119911

minus1) and the

collection Δ = 120575(119898) | 119898 isin Z2 forms a group An element120595 isinM

0is called an oscillating matrix of type 120575(119898) if it has the

form

120595 = ℎ (119911) 120575 (119898) 1205950 with ℎ (119911) isin 119866

lt0 (54)

These oscillatingmatrices are examples of elements ofM0for

which the scratching procedure is valid Let 119876 be a potentialsolution of the AKNS hierarchy of the form (17) If there isan oscillating function 120595 of type 120575(119898) such that for 119876 and 120595

Advances in Mathematical Physics 7

(40) hold then we call120595 awave matrix of the AKNS hierarchyof type 120575(119898) In particular 119876 is then a solution of the AKNShierarchy and the first equation in (40) implies

119876ℎ (119911) 120575 (119898) = ℎ (119911)1198760120575 (119898) 997904rArr

119876 = ℎ (119911)1198760ℎ (119911)minus1

(55)

in other words the solution 119876 is totally determined by thewave matrix Similarly we call an element 120593 isin M

1an

oscillating matrix of type 120575(119898) if it has the form

120593 = ℎ (119911) 120575 (119898) 1205930 with ℎ (119911) isin 119866

⩽0 (56)

This type of oscillating matrices are examples of elements ofM1for which the scratching procedure is valid Let 119885 be a

potential solution of the strict AKNS hierarchy of the form(19) If there is an oscillating function 120593 of type 120575(119898) in M

1

such that for119885 and120593 (41) hold thenwe call120593 awavematrix ofthe strict AKNS hierarchy of type 120575(119898) In particular119885 is thena solution of the strict AKNS hierarchy and the first equationin (41) implies

119885ℎ (119911) 120575 (119898) = ℎ (119911)1198760119911120575 (119898) 997904rArr

119885 = ℎ (119911)1198760119911ℎ (119911)

minus1

(57)

so that also here the wave matrix fully determines thesolution

For both hierarchies there is a milder condition that is tobe satisfied by oscillating matrices of a certain type in orderthat they become a wave matrix of that hierarchy

Proposition 6 Let120595 = ℎ(119911)120575(119898)1205950be an oscillatingmatrix

of type 120575(119898) in M0and 119876 = ℎ(119911)119876

0ℎ(119911)minus1 the corresponding

potential solution of the AKNS hierarchy Similarly let 120593 =

ℎ(119911)120575(119898)1205930be such a matrix in M

1with potential solution

119885 = ℎ(119911)1198760119911ℎ(119911)minus1

(a) If there exists for each 119898 ⩾ 0 an element 119872119898isin

gl2(119877)[119911 119911

minus1)⩾0

such that

120597119898(120595) = 119872

119898120595 (58)

then each119872119898= (119876119911

119898)⩾0

and 120595 is a wave matrix oftype 120575(119898) for the AKNS hierarchy

(b) If there exists for each 119898 ⩾ 1 an element 119873119898isin

gl2(119877)[119911 119911

minus1)gt0

such that

120597119898(120593) = 119873

119898120593 (59)

then each 119873119898= (119885119911

119898minus1)gt0

and 120593 is a wave matrix oftype 120575(119898) for the strict AKNS hierarchy

Proof We give the proof again for the strict AKNS casethe other one is similar By using the fact that M

1is a free

gl2(119877)[119911 119911

minus1)module with generator 120593

0we can translate the

relations 120597119898(120593) = 119873

119898120593 into equations in gl

2(119877)[119911 119911

minus1) This

yields

120597119898(ℎ (119911)) + ℎ (119911)119876

0119911119898= 119873119898ℎ (119911) 997904rArr

120597119898(ℎ (119911)) ℎ (119911)

minus1+ 119885119911119898minus1

= 119873119898

(60)

Projecting the right-hand side on gl2(119877)[119911 119911

minus1)gt0

gives us theidentity

(119885119911119898minus1)gt0= 119873119898 (61)

we are looking for

In the next section we produce a context where we canconstruct wave matrices for both hierarchies in which theproduct is real

4 The Construction of the Solutions

In this section we will show how to construct a wide class ofsolutions of both hierarchies This is done in the style of [6]We first describe the group of loops we will work with Foreach 0 lt 119903 lt 1 let 119860

119903be the annulus

119911 | 119911 isin C 119903 ⩽ |119911| ⩽1

119903 (62)

Following [7] we use the notation 119871anGL2(C) for the col-lection of holomorphic maps from some annulus 119860

119903into

GL2(C) It is a group with respect to pointwise multiplication

and contains in a natural way GL2(C) as a subgroup as the

collection of constant maps into GL2(C) Other examples

of elements in 119871anGL2(C) are the elements of Δ However119871anGL2(C) is more than just a group it is an infinitedimensional Lie group Its manifold structure can be read offfrom its Lie algebra 119871angl2(C) consisting of all holomorphicmaps 120574 119880 rarr gl

2(C) where 119880 is an open neighborhood of

some annulus 119860119903 0 lt 119903 lt 1 Since gl

2(C) is a Lie algebra

the space 119871angl2(C) becomes a Lie algebra with respect to thepointwise commutator Topologically the space 119871angl2(C) isthe direct limit of all the spaces 119871an119903gl2(C) where this lastspace consists of all 120574 corresponding to the fixed annulus119860119903 One gives each 119871an119903gl2(C) the topology of uniform

convergence and with that topology it becomes a Banachspace In this way 119871angl2(C) becomes a Frechet space Thepointwise exponential map defines a local diffeomorphismaround zero in 119871angl2(C) (see eg [8])

Now each 120574 isin 119871angl2(C) possesses an expansion in aFourier series

120574 =

infin

sum

119896=minusinfin

120574119896119911119896 with each 120574

119896isin gl2(C) (63)

that converges absolutely on the annulus 119860119903

infin

sum

119896=minusinfin

10038171003817100381710038171205741198961003817100381710038171003817 119903minus|119896|

lt infin (64)

This Fourier expansion is used to make two different decom-positions of the Lie algebra119871angl2(C)Thefirst is the analoguefor the present Lie algebra 119871angl2(C) of decomposition (8) ofgl2(119877)[119911 119911

minus1) that lies at the basis of the Lax equations of the

AKNS hierarchy Namely consider the subspaces

119871angl2 (C)⩾0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=0

120574119896119911119896

8 Advances in Mathematical Physics

119871angl2 (C)lt0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =minus1

sum

119896=minusinfin

120574119896119911119896

(65)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra The first Lie algebra consists ofthe elements in 119871angl2(C) that extend to holomorphic mapsdefined on some disk 119911 isin C | |119911| ⩽ 1119903 0 lt 119903 lt 1 andthe second Lie algebra corresponds to the maps in 119871angl2(C)that have a holomorphic extension towards some disk aroundinfinity 119911 isin P1(C) | |119911| ⩾ 119903 0 lt 119903 lt 1 and that arezero at infinity To each of the two Lie subalgebras belongsa subgroup of 119871anGL2(C) The pointwise exponential mapapplied to elements of 119871angl2(C)lt0 yields elements of

119880minus= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

minus1

sum

119896=minusinfin

120574119896119911119896 (66)

and the exponential map applied to elements of 119871angl2(C)⩾0maps them into

119875+= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740 +

infin

sum

119896=1

120574119896119911119896 with 120574

0

isin GL2(C)

(67)

Both119880minusand 119875+are easily seen to be subgroups of 119871anGL2(C)

and since the direct sum of their Lie algebras is 119871angl2(C)their product

Ω = 119880minus119875+ (68)

is open in 119871anGL2(C) and is called like in the finite dimen-sional case the big cell with respect to 119880

minusand 119875

+

The second decomposition is the analogue of the splitting(11) of gl

2(119877)[119911 119911

minus1) that led to the Lax equations of the strict

AKNS hierarchy Now we consider the subspaces

119871angl2 (C)gt0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=1

120574119896119911119896

119871angl2 (C)⩽0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =0

sum

119896=minusinfin

120574119896119911119896

(69)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra 119871angl2(C)gt0 consists of mapswhose holomorphic extension to 119911 = 0 equals zero inthat point and 119871angl2(C)⩽0 is the set of maps that extendhomomorphically to a neighborhood of infinity To eachof them belongs a subgroup of 119871anGL2(C) The pointwiseexponential map applied to elements of 119871angl2(C)gt0 yieldselements of

119880+= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

infin

sum

119896=1

120574119896119911119896 (70)

and the exponential map applied to elements of 119871angl2(C)⩽0maps them into

119875minus= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740

+

minus1

sum

119896=minusinfin

120574119896119911119896 with 120574

0isin GL2(C)

(71)

Both119875minusand119880

+are easily seen to be subgroups of 119871anGL2(C)

Since the direct sum of their Lie algebras is 119871angl2(C) theirproduct 119875

minus119880+is open and because 119875

+= GL

2(C)119880+=

119880+GL2(C) and 119875

minus= GL2(C)119880minus= 119880minusGL2(C) this gives the

equality

119875minus119880+= Ω = 119880

minus119875+ (72)

the big cell in 119871anGL2(C)The next two subgroups of 119871anGL2(C) correspond to

the exponential factors in both linearizations The group ofcommuting flows relevant for the AKNS hierarchy is thegroup

Γ0= 1205740(119905) = exp(

infin

sum

119894=0

1199051198941198760119911119894) | 1205740isin 119875+ (73)

and similarly for the strict AKNS hierarchy we use thecommuting flows from

Γ1= 1205741(119905) = exp(

infin

sum

119894=1

1199051198941198760119911119894) | 1205741isin 119880+ (74)

Besides the groups Δ Γ0 and Γ

1 there are more subgroups in

119871anGL2(C) that commute with those three groups and theyare in a sense complimentary to Δ Γ

0and Δ Γ

1 respectively

That is why we use the following notations for them in 119880minus

there is

Γ119888

0= 120574119888

0(119905) = exp(

infin

sum

119896=1

1199041198961198760119911minus119896) | 120574119888

0isin 119880minus (75)

and in 119875minuswe have

Γ119888

1= 120574119888

1(119905) = exp(

infin

sum

119896=0

1199041198961198760119911minus119896) | 120574119888

1isin 119875minus (76)

We have now all ingredients to describe the constructionof the solutions to each hierarchy and we start with theAKNS hierarchy Take inside the product 119871anGL2(C) times Δthe collection 119878

0of pairs (119892 120575(119898)) such that there exists a

1205740(119905) isin Γ

0satisfying

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898) isin Ω = 119880

minus119875+ (77)

For such a pair (119892 120575(119898)) we take the collection Γ0(119892 120575(119898))

of all 1205740(119905) satisfying the condition (77) This is an open

nonempty subset of Γ0 Let 119877

0(119892 120575(119898)) be the algebra of

analytic functions Γ0(119892 120575(119898)) rarr C This is the algebra of

Advances in Mathematical Physics 9

functions 119877 that we associate with the point (119892 120575(119898)) isin 1198780

and for the commuting derivations of 1198770(119892 120575(119898)) we choose

120597119894fl 120597120597119905

119894 119894 ⩾ 0 By property (77) we have for all 120574

0(119905) isin

Γ0(119892 120575(119898))

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898)

= 119906minus(119892 120575 (119898))

minus1

119901+(119892 120575 (119898))

(78)

with 119906minus(119892 120575(119898)) isin 119880

minus119901+(119892 120575(119898)) isin 119875

+Then all thematrix

coefficients in the Fourier expansions of the elements 119906minus(119892

120575(119898)) and 119901+(119892 120575(119898)) belong to the algebra 119877

0(119892 120575(119898)) It is

convenient to write relation (78) in the form

Ψ119892120575(119898)

fl 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905)

= 119901+(119892 120575 (119898)) 120575 (119898) 120574

0(119905) 119892minus1

= 119901+(119892 120575 (119898)) 120574

0(119905) 120575 (119898) 119892

minus1

= 119902+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(79)

with 119902+(119892 120575(119898)) isin 119875

+ Clearly Ψ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M0 where all the products in the

decomposition 119906minus(119892 120575(119898))120575(119898)120574

0(119905) are no longer formal

but real This is our potential wave matrix for the AKNShierarchy

To get the potential wave function for the strict AKNShierarchy we proceed similarly only this time we use thedecomposition Ω = 119875

minus119880+ Consider again the product

119871anGL2(C) times Δ and the subset 1198781of pairs (119892 120575(119898)) such that

there exists a 1205741(119905) isin Γ

1satisfying

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898) isin Ω = 119875

minus119880+ (80)

For such a pair (119892 120575(119898)) we define Γ1(119892 120575(119898)) as the set of

all 1205741(119905) satisfying condition (77) This is an open nonempty

subset of Γ1 Let 119877

1(119892 120575(119898)) be the algebra of analytic

functions Γ1(119892 120575(119898)) rarr C This is the algebra of functions

119877 that we associate with the point (119892 120575(119898)) isin 1198781and for the

commuting derivations of 1198771(119892 120575(119898)) we choose 120597

119894fl 120597120597119905

119894

119894 gt 0 By property (80) we have for all 1205741(119905) isin Γ

1(119892 120575(119898))

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898)

= 119901minus(119892 120575 (119898))

minus1

119906+(119892 120575 (119898))

(81)

with 119901minus(119892 120575(119898)) isin 119875

minus 119906+(119892 120575(119898)) isin 119880

+ Then all

the matrix coefficients in the Fourier expansions of theelements 119901

minus(119892 120575(119898)) and 119906

+(119892 120575(119898)) belong to the algebra

1198771(119892 120575(119898)) It is convenient to write relation (81) in the form

Φ119892120575(119898)

fl 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905)

= 119906+(119892 120575 (119898)) 120575 (119898) 120574

1(119905) 119892minus1

= 119906+(119892 120575 (119898)) 120574

1(119905) 120575 (119898) 119892

minus1

= 119908+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(82)

with 119908+(119892 120575(119898)) isin 119880

+ Clearly Φ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M1where all the products in the

decomposition 119901minus(119892 120575(119898))120575(119898)120574

1(119905) are no longer formal

but realTo show that Ψ

119892120575(119898)is a wave matrix for the AKNS

hierarchy of type 120575(119898) and Φ119892120575(119898)

one for the strict AKNShierarchy we use the fact that it suffices to prove the relationsin Proposition 6 We treat first Ψ

119892120575(119898) In (79) we presented

two different decompositions of Ψ119892120575(119898)

that we will use tocompute 120597

119894(Ψ119892120575(119898)

) With respect to 119906minus(119892 120575(119898))120575(119898)120574

0(119905)

there holds120597119894(Ψ119892120575(119898)

)

= 120597119894(119906minus(119892 120575 (119898))) 119906

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905) = 119872

119894Ψ119892120575(119898)

(83)

with119872119894isin gl2(119877(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911 If we take

the decomposition 119902+(119892 120575(119898))120575(119898)119892

minus1 then we get

120597119894(Ψ119892120575(119898)

)

= 120597119894(119902+(119892 120575 (119898))) 119902

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(84)

with 120597119894(119902+(119892 120575(119898)))119902

+(119892 120575(119898))

minus1 of the formsum119895⩾0119902119895119911119895 with

all 119902119895isin gl2(119877(119892 120575(119898))) As Ψ

119892120575(119898)is invertible we obtain

119872119894= sum119895⩾0119902119895119911119895 and thus 119872

119894= sum119894

119895=0119902119895119911119895 and that is the

desired resultWe handle the case of Φ

119892120575(119898)in a similar way Also in

(79) you can find two different decompositions of Φ119892120575(119898)

that we will use to compute 120597119894(Φ119892120575(119898)

) With respect to theexpression 119901

minus(119892 120575(119898))120575(119898)120574

1(119905) we get for all 119894 ⩾ 1

120597119894(Φ119892120575(119898)

)

= 120597119894(119901minus(119892 120575 (119898))) 119901

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905) = 119873

119894Φ119892120575(119898)

(85)

again with 119873119894isin gl2(1198771(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911

Next we take the decomposition 119908+(119892 120575(119898))120575(119898)119892

minus1 andthat yields

120597119894(Φ119892120575(119898)

)

= 120597119894(119908+(119892 120575 (119898)))119908

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(86)

Since the zeroth order term of 119908+(119892 120575(119898)) is Id we note

that the element 120597119894(119908+(119892 120575(119898)))119908

+(119892 120575(119898))

minus1 has the formsum119895⩾1119908119895119911119895 with all 119908

119895isin gl2(1198771(119892 120575(119898))) As Φ

119892120575(119898)is also

invertible we obtain119873119894= sum119895⩾1119908119895119911119895 and thus119873

119894= sum119894

119895=1119908119895119911119895

and that is the result we were looking for in the strict caseThe abovementioned constructions are not affected seri-

ously by a number of transformations For example if wewrite 120575 = 120575((1 1)) then all 120575119896 119896 isin Z are central elementsand we have for all (119892 120575(119898)) isin 119878

0and 119878

1respectively and all

119896 isin Z

Ψ119892120575(119898)120575

119896 = Ψ119892120575(119898)

120575119896

respectively Φ119892120575(119898)120575

119896 = Φ119892120575(119898)

120575119896

(87)

10 Advances in Mathematical Physics

Also for any 1199011isin 119875+ each 120574119888

0(119904) isin Γ

119888

0 and all (119892 120575(119898)) isin

1198780 we see that the element (120574119888

0(119904)119892120575(minus119898)119901

1120575(119898) 120575(119898)) also

belongs to 1198780 for

120575 (119898) 1205740(119905) 120574119888

0(119904) 119892120575 (minus119898) 119901

1120575 (119898) 120574

0(119905)minus1120575 (minus119898)

= 120574119888

0(119904) 119906minus1

minus119901+1205740(119905) 11990111205740(119905)minus1

(88)

In particular we see that

Ψ120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575(119898)

= 119906minus120574119888

0(119904)minus1120575 (119898) 120574

0(119905) (89)

Its analogue in the strict case is as follows for any 1199061isin

119880+ each 120574119888

1(119904) isin Γ

119888

1 and all (119892 120575(119898)) isin 119878

1 the element

(120574119888

1(119904)119892120575(minus119898)119906

1120575(119898) 120575(119898)) also belongs to 119878

1and there

holds

Φ120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575(119898)

= 119901minus120574119888

1(119904)minus1120575 (119898) 120574

1(119905) (90)

We resume the foregoing results in the following theorem

Theorem 7 Consider the product spaceΠ fl 119871anGL2(C) times ΔThen the following holds

(a) For each point (119892 120575(119898)) isin Π that satisfies condition(77) we have a wave matrix of type 120575(119898) of the AKNShierarchyΨ

119892120575(119898) defined by (79) and this determines a

solution 119876119892120575(119898)

= 119906minus(119892 120575(119898))119876

0119906minus(119892 120575(119898))

minus1 of theAKNS hierarchy For each 119901

1isin 119875+ every 120574119888

0(119904) isin Γ

119888

0

and all powers of 120575 one has

119876120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575

119896120575(119898)

= 119876119892120575(119898)

(91)

(b) For each point (119892 120575(119898)) isin Π that satisfies condition(80) we have a wave matrix of type 120575(119898) of thestrict AKNS hierarchy Φ

119892120575(119898) defined by (82) and it

determines a solution 119885119892120575(119898)

= 119901minus(119892 120575(119898))119876

0119911119901minus(119892

120575(119898))minus1 of the strict AKNS hierarchy For each 119906

1isin 119880+

every 1205741198881(119904) isin Γ

119888

1 and all powers of 120575 one has

119885120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575

119896120575(119898)

= 119885119892120575(119898)

(92)

Competing Interests

The author declares that they have no competing interests

References

[1] M Adler P vanMoerbeke and P VanhaeckeAlgebraic Integra-bility Painleve Geometry and Lie Algebras vol 47 of Ergebnisseder Mathematik und ihrer Grenzgebiete Springer 2004

[2] G F Helminck A G Helminck and E A PanasenkoldquoIntegrable deformations in the algebra of pseudodifferentialoperators from a Lie algebraic perspectiverdquo Theoretical andMathematical Physics vol 174 no 1 pp 134ndash153 2013

[3] M J Ablowitz D J Kaup A C Newell and H Segur ldquoTheinverse scattering transform-Fourier analysis for nonlinearproblemsrdquo Studies in Applied Mathematics vol 53 no 4 pp249ndash315 1974

[4] H Flashka A CNewell and T Ratiu ldquoKac-Moody Lie algebrasand soliton equations II Lax equations associated with 119860

1

(1)rdquoPhysicaDNonlinear Phenomena vol 9 no 3 pp 300ndash323 1983

[5] EDateM JimboMKashiwara andTMiwa ldquoTransformationgroups for soliton equationsrdquo in Proceedings of the RIMSSymposium on Nonlinear Integrable SystemsmdashClassical Theoryand Quantum Theory T Jimbo and T Miwa Eds WorldScience Publisher Kyoto Japan 1981

[6] G Segal and G Wilson ldquoLoop groups and equations of KdVtyperdquo Publications Mathematiques de lrsquoIHES vol 61 pp 5ndash651985

[7] A Pressley and G Segal Loop Groups Oxford MathematicalMonographs Clarendon Press 1986

[8] R S Hamilton ldquoThe inverse function theorem of Nash andMoserrdquoBulletin of the AmericanMathematical Society vol 7 no1 pp 65ndash222 1982

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Advances in Mathematical Physics 7

(40) hold then we call120595 awave matrix of the AKNS hierarchyof type 120575(119898) In particular 119876 is then a solution of the AKNShierarchy and the first equation in (40) implies

119876ℎ (119911) 120575 (119898) = ℎ (119911)1198760120575 (119898) 997904rArr

119876 = ℎ (119911)1198760ℎ (119911)minus1

(55)

in other words the solution 119876 is totally determined by thewave matrix Similarly we call an element 120593 isin M

1an

oscillating matrix of type 120575(119898) if it has the form

120593 = ℎ (119911) 120575 (119898) 1205930 with ℎ (119911) isin 119866

⩽0 (56)

This type of oscillating matrices are examples of elements ofM1for which the scratching procedure is valid Let 119885 be a

potential solution of the strict AKNS hierarchy of the form(19) If there is an oscillating function 120593 of type 120575(119898) in M

1

such that for119885 and120593 (41) hold thenwe call120593 awavematrix ofthe strict AKNS hierarchy of type 120575(119898) In particular119885 is thena solution of the strict AKNS hierarchy and the first equationin (41) implies

119885ℎ (119911) 120575 (119898) = ℎ (119911)1198760119911120575 (119898) 997904rArr

119885 = ℎ (119911)1198760119911ℎ (119911)

minus1

(57)

so that also here the wave matrix fully determines thesolution

For both hierarchies there is a milder condition that is tobe satisfied by oscillating matrices of a certain type in orderthat they become a wave matrix of that hierarchy

Proposition 6 Let120595 = ℎ(119911)120575(119898)1205950be an oscillatingmatrix

of type 120575(119898) in M0and 119876 = ℎ(119911)119876

0ℎ(119911)minus1 the corresponding

potential solution of the AKNS hierarchy Similarly let 120593 =

ℎ(119911)120575(119898)1205930be such a matrix in M

1with potential solution

119885 = ℎ(119911)1198760119911ℎ(119911)minus1

(a) If there exists for each 119898 ⩾ 0 an element 119872119898isin

gl2(119877)[119911 119911

minus1)⩾0

such that

120597119898(120595) = 119872

119898120595 (58)

then each119872119898= (119876119911

119898)⩾0

and 120595 is a wave matrix oftype 120575(119898) for the AKNS hierarchy

(b) If there exists for each 119898 ⩾ 1 an element 119873119898isin

gl2(119877)[119911 119911

minus1)gt0

such that

120597119898(120593) = 119873

119898120593 (59)

then each 119873119898= (119885119911

119898minus1)gt0

and 120593 is a wave matrix oftype 120575(119898) for the strict AKNS hierarchy

Proof We give the proof again for the strict AKNS casethe other one is similar By using the fact that M

1is a free

gl2(119877)[119911 119911

minus1)module with generator 120593

0we can translate the

relations 120597119898(120593) = 119873

119898120593 into equations in gl

2(119877)[119911 119911

minus1) This

yields

120597119898(ℎ (119911)) + ℎ (119911)119876

0119911119898= 119873119898ℎ (119911) 997904rArr

120597119898(ℎ (119911)) ℎ (119911)

minus1+ 119885119911119898minus1

= 119873119898

(60)

Projecting the right-hand side on gl2(119877)[119911 119911

minus1)gt0

gives us theidentity

(119885119911119898minus1)gt0= 119873119898 (61)

we are looking for

In the next section we produce a context where we canconstruct wave matrices for both hierarchies in which theproduct is real

4 The Construction of the Solutions

In this section we will show how to construct a wide class ofsolutions of both hierarchies This is done in the style of [6]We first describe the group of loops we will work with Foreach 0 lt 119903 lt 1 let 119860

119903be the annulus

119911 | 119911 isin C 119903 ⩽ |119911| ⩽1

119903 (62)

Following [7] we use the notation 119871anGL2(C) for the col-lection of holomorphic maps from some annulus 119860

119903into

GL2(C) It is a group with respect to pointwise multiplication

and contains in a natural way GL2(C) as a subgroup as the

collection of constant maps into GL2(C) Other examples

of elements in 119871anGL2(C) are the elements of Δ However119871anGL2(C) is more than just a group it is an infinitedimensional Lie group Its manifold structure can be read offfrom its Lie algebra 119871angl2(C) consisting of all holomorphicmaps 120574 119880 rarr gl

2(C) where 119880 is an open neighborhood of

some annulus 119860119903 0 lt 119903 lt 1 Since gl

2(C) is a Lie algebra

the space 119871angl2(C) becomes a Lie algebra with respect to thepointwise commutator Topologically the space 119871angl2(C) isthe direct limit of all the spaces 119871an119903gl2(C) where this lastspace consists of all 120574 corresponding to the fixed annulus119860119903 One gives each 119871an119903gl2(C) the topology of uniform

convergence and with that topology it becomes a Banachspace In this way 119871angl2(C) becomes a Frechet space Thepointwise exponential map defines a local diffeomorphismaround zero in 119871angl2(C) (see eg [8])

Now each 120574 isin 119871angl2(C) possesses an expansion in aFourier series

120574 =

infin

sum

119896=minusinfin

120574119896119911119896 with each 120574

119896isin gl2(C) (63)

that converges absolutely on the annulus 119860119903

infin

sum

119896=minusinfin

10038171003817100381710038171205741198961003817100381710038171003817 119903minus|119896|

lt infin (64)

This Fourier expansion is used to make two different decom-positions of the Lie algebra119871angl2(C)Thefirst is the analoguefor the present Lie algebra 119871angl2(C) of decomposition (8) ofgl2(119877)[119911 119911

minus1) that lies at the basis of the Lax equations of the

AKNS hierarchy Namely consider the subspaces

119871angl2 (C)⩾0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=0

120574119896119911119896

8 Advances in Mathematical Physics

119871angl2 (C)lt0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =minus1

sum

119896=minusinfin

120574119896119911119896

(65)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra The first Lie algebra consists ofthe elements in 119871angl2(C) that extend to holomorphic mapsdefined on some disk 119911 isin C | |119911| ⩽ 1119903 0 lt 119903 lt 1 andthe second Lie algebra corresponds to the maps in 119871angl2(C)that have a holomorphic extension towards some disk aroundinfinity 119911 isin P1(C) | |119911| ⩾ 119903 0 lt 119903 lt 1 and that arezero at infinity To each of the two Lie subalgebras belongsa subgroup of 119871anGL2(C) The pointwise exponential mapapplied to elements of 119871angl2(C)lt0 yields elements of

119880minus= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

minus1

sum

119896=minusinfin

120574119896119911119896 (66)

and the exponential map applied to elements of 119871angl2(C)⩾0maps them into

119875+= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740 +

infin

sum

119896=1

120574119896119911119896 with 120574

0

isin GL2(C)

(67)

Both119880minusand 119875+are easily seen to be subgroups of 119871anGL2(C)

and since the direct sum of their Lie algebras is 119871angl2(C)their product

Ω = 119880minus119875+ (68)

is open in 119871anGL2(C) and is called like in the finite dimen-sional case the big cell with respect to 119880

minusand 119875

+

The second decomposition is the analogue of the splitting(11) of gl

2(119877)[119911 119911

minus1) that led to the Lax equations of the strict

AKNS hierarchy Now we consider the subspaces

119871angl2 (C)gt0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=1

120574119896119911119896

119871angl2 (C)⩽0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =0

sum

119896=minusinfin

120574119896119911119896

(69)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra 119871angl2(C)gt0 consists of mapswhose holomorphic extension to 119911 = 0 equals zero inthat point and 119871angl2(C)⩽0 is the set of maps that extendhomomorphically to a neighborhood of infinity To eachof them belongs a subgroup of 119871anGL2(C) The pointwiseexponential map applied to elements of 119871angl2(C)gt0 yieldselements of

119880+= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

infin

sum

119896=1

120574119896119911119896 (70)

and the exponential map applied to elements of 119871angl2(C)⩽0maps them into

119875minus= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740

+

minus1

sum

119896=minusinfin

120574119896119911119896 with 120574

0isin GL2(C)

(71)

Both119875minusand119880

+are easily seen to be subgroups of 119871anGL2(C)

Since the direct sum of their Lie algebras is 119871angl2(C) theirproduct 119875

minus119880+is open and because 119875

+= GL

2(C)119880+=

119880+GL2(C) and 119875

minus= GL2(C)119880minus= 119880minusGL2(C) this gives the

equality

119875minus119880+= Ω = 119880

minus119875+ (72)

the big cell in 119871anGL2(C)The next two subgroups of 119871anGL2(C) correspond to

the exponential factors in both linearizations The group ofcommuting flows relevant for the AKNS hierarchy is thegroup

Γ0= 1205740(119905) = exp(

infin

sum

119894=0

1199051198941198760119911119894) | 1205740isin 119875+ (73)

and similarly for the strict AKNS hierarchy we use thecommuting flows from

Γ1= 1205741(119905) = exp(

infin

sum

119894=1

1199051198941198760119911119894) | 1205741isin 119880+ (74)

Besides the groups Δ Γ0 and Γ

1 there are more subgroups in

119871anGL2(C) that commute with those three groups and theyare in a sense complimentary to Δ Γ

0and Δ Γ

1 respectively

That is why we use the following notations for them in 119880minus

there is

Γ119888

0= 120574119888

0(119905) = exp(

infin

sum

119896=1

1199041198961198760119911minus119896) | 120574119888

0isin 119880minus (75)

and in 119875minuswe have

Γ119888

1= 120574119888

1(119905) = exp(

infin

sum

119896=0

1199041198961198760119911minus119896) | 120574119888

1isin 119875minus (76)

We have now all ingredients to describe the constructionof the solutions to each hierarchy and we start with theAKNS hierarchy Take inside the product 119871anGL2(C) times Δthe collection 119878

0of pairs (119892 120575(119898)) such that there exists a

1205740(119905) isin Γ

0satisfying

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898) isin Ω = 119880

minus119875+ (77)

For such a pair (119892 120575(119898)) we take the collection Γ0(119892 120575(119898))

of all 1205740(119905) satisfying the condition (77) This is an open

nonempty subset of Γ0 Let 119877

0(119892 120575(119898)) be the algebra of

analytic functions Γ0(119892 120575(119898)) rarr C This is the algebra of

Advances in Mathematical Physics 9

functions 119877 that we associate with the point (119892 120575(119898)) isin 1198780

and for the commuting derivations of 1198770(119892 120575(119898)) we choose

120597119894fl 120597120597119905

119894 119894 ⩾ 0 By property (77) we have for all 120574

0(119905) isin

Γ0(119892 120575(119898))

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898)

= 119906minus(119892 120575 (119898))

minus1

119901+(119892 120575 (119898))

(78)

with 119906minus(119892 120575(119898)) isin 119880

minus119901+(119892 120575(119898)) isin 119875

+Then all thematrix

coefficients in the Fourier expansions of the elements 119906minus(119892

120575(119898)) and 119901+(119892 120575(119898)) belong to the algebra 119877

0(119892 120575(119898)) It is

convenient to write relation (78) in the form

Ψ119892120575(119898)

fl 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905)

= 119901+(119892 120575 (119898)) 120575 (119898) 120574

0(119905) 119892minus1

= 119901+(119892 120575 (119898)) 120574

0(119905) 120575 (119898) 119892

minus1

= 119902+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(79)

with 119902+(119892 120575(119898)) isin 119875

+ Clearly Ψ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M0 where all the products in the

decomposition 119906minus(119892 120575(119898))120575(119898)120574

0(119905) are no longer formal

but real This is our potential wave matrix for the AKNShierarchy

To get the potential wave function for the strict AKNShierarchy we proceed similarly only this time we use thedecomposition Ω = 119875

minus119880+ Consider again the product

119871anGL2(C) times Δ and the subset 1198781of pairs (119892 120575(119898)) such that

there exists a 1205741(119905) isin Γ

1satisfying

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898) isin Ω = 119875

minus119880+ (80)

For such a pair (119892 120575(119898)) we define Γ1(119892 120575(119898)) as the set of

all 1205741(119905) satisfying condition (77) This is an open nonempty

subset of Γ1 Let 119877

1(119892 120575(119898)) be the algebra of analytic

functions Γ1(119892 120575(119898)) rarr C This is the algebra of functions

119877 that we associate with the point (119892 120575(119898)) isin 1198781and for the

commuting derivations of 1198771(119892 120575(119898)) we choose 120597

119894fl 120597120597119905

119894

119894 gt 0 By property (80) we have for all 1205741(119905) isin Γ

1(119892 120575(119898))

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898)

= 119901minus(119892 120575 (119898))

minus1

119906+(119892 120575 (119898))

(81)

with 119901minus(119892 120575(119898)) isin 119875

minus 119906+(119892 120575(119898)) isin 119880

+ Then all

the matrix coefficients in the Fourier expansions of theelements 119901

minus(119892 120575(119898)) and 119906

+(119892 120575(119898)) belong to the algebra

1198771(119892 120575(119898)) It is convenient to write relation (81) in the form

Φ119892120575(119898)

fl 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905)

= 119906+(119892 120575 (119898)) 120575 (119898) 120574

1(119905) 119892minus1

= 119906+(119892 120575 (119898)) 120574

1(119905) 120575 (119898) 119892

minus1

= 119908+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(82)

with 119908+(119892 120575(119898)) isin 119880

+ Clearly Φ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M1where all the products in the

decomposition 119901minus(119892 120575(119898))120575(119898)120574

1(119905) are no longer formal

but realTo show that Ψ

119892120575(119898)is a wave matrix for the AKNS

hierarchy of type 120575(119898) and Φ119892120575(119898)

one for the strict AKNShierarchy we use the fact that it suffices to prove the relationsin Proposition 6 We treat first Ψ

119892120575(119898) In (79) we presented

two different decompositions of Ψ119892120575(119898)

that we will use tocompute 120597

119894(Ψ119892120575(119898)

) With respect to 119906minus(119892 120575(119898))120575(119898)120574

0(119905)

there holds120597119894(Ψ119892120575(119898)

)

= 120597119894(119906minus(119892 120575 (119898))) 119906

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905) = 119872

119894Ψ119892120575(119898)

(83)

with119872119894isin gl2(119877(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911 If we take

the decomposition 119902+(119892 120575(119898))120575(119898)119892

minus1 then we get

120597119894(Ψ119892120575(119898)

)

= 120597119894(119902+(119892 120575 (119898))) 119902

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(84)

with 120597119894(119902+(119892 120575(119898)))119902

+(119892 120575(119898))

minus1 of the formsum119895⩾0119902119895119911119895 with

all 119902119895isin gl2(119877(119892 120575(119898))) As Ψ

119892120575(119898)is invertible we obtain

119872119894= sum119895⩾0119902119895119911119895 and thus 119872

119894= sum119894

119895=0119902119895119911119895 and that is the

desired resultWe handle the case of Φ

119892120575(119898)in a similar way Also in

(79) you can find two different decompositions of Φ119892120575(119898)

that we will use to compute 120597119894(Φ119892120575(119898)

) With respect to theexpression 119901

minus(119892 120575(119898))120575(119898)120574

1(119905) we get for all 119894 ⩾ 1

120597119894(Φ119892120575(119898)

)

= 120597119894(119901minus(119892 120575 (119898))) 119901

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905) = 119873

119894Φ119892120575(119898)

(85)

again with 119873119894isin gl2(1198771(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911

Next we take the decomposition 119908+(119892 120575(119898))120575(119898)119892

minus1 andthat yields

120597119894(Φ119892120575(119898)

)

= 120597119894(119908+(119892 120575 (119898)))119908

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(86)

Since the zeroth order term of 119908+(119892 120575(119898)) is Id we note

that the element 120597119894(119908+(119892 120575(119898)))119908

+(119892 120575(119898))

minus1 has the formsum119895⩾1119908119895119911119895 with all 119908

119895isin gl2(1198771(119892 120575(119898))) As Φ

119892120575(119898)is also

invertible we obtain119873119894= sum119895⩾1119908119895119911119895 and thus119873

119894= sum119894

119895=1119908119895119911119895

and that is the result we were looking for in the strict caseThe abovementioned constructions are not affected seri-

ously by a number of transformations For example if wewrite 120575 = 120575((1 1)) then all 120575119896 119896 isin Z are central elementsand we have for all (119892 120575(119898)) isin 119878

0and 119878

1respectively and all

119896 isin Z

Ψ119892120575(119898)120575

119896 = Ψ119892120575(119898)

120575119896

respectively Φ119892120575(119898)120575

119896 = Φ119892120575(119898)

120575119896

(87)

10 Advances in Mathematical Physics

Also for any 1199011isin 119875+ each 120574119888

0(119904) isin Γ

119888

0 and all (119892 120575(119898)) isin

1198780 we see that the element (120574119888

0(119904)119892120575(minus119898)119901

1120575(119898) 120575(119898)) also

belongs to 1198780 for

120575 (119898) 1205740(119905) 120574119888

0(119904) 119892120575 (minus119898) 119901

1120575 (119898) 120574

0(119905)minus1120575 (minus119898)

= 120574119888

0(119904) 119906minus1

minus119901+1205740(119905) 11990111205740(119905)minus1

(88)

In particular we see that

Ψ120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575(119898)

= 119906minus120574119888

0(119904)minus1120575 (119898) 120574

0(119905) (89)

Its analogue in the strict case is as follows for any 1199061isin

119880+ each 120574119888

1(119904) isin Γ

119888

1 and all (119892 120575(119898)) isin 119878

1 the element

(120574119888

1(119904)119892120575(minus119898)119906

1120575(119898) 120575(119898)) also belongs to 119878

1and there

holds

Φ120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575(119898)

= 119901minus120574119888

1(119904)minus1120575 (119898) 120574

1(119905) (90)

We resume the foregoing results in the following theorem

Theorem 7 Consider the product spaceΠ fl 119871anGL2(C) times ΔThen the following holds

(a) For each point (119892 120575(119898)) isin Π that satisfies condition(77) we have a wave matrix of type 120575(119898) of the AKNShierarchyΨ

119892120575(119898) defined by (79) and this determines a

solution 119876119892120575(119898)

= 119906minus(119892 120575(119898))119876

0119906minus(119892 120575(119898))

minus1 of theAKNS hierarchy For each 119901

1isin 119875+ every 120574119888

0(119904) isin Γ

119888

0

and all powers of 120575 one has

119876120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575

119896120575(119898)

= 119876119892120575(119898)

(91)

(b) For each point (119892 120575(119898)) isin Π that satisfies condition(80) we have a wave matrix of type 120575(119898) of thestrict AKNS hierarchy Φ

119892120575(119898) defined by (82) and it

determines a solution 119885119892120575(119898)

= 119901minus(119892 120575(119898))119876

0119911119901minus(119892

120575(119898))minus1 of the strict AKNS hierarchy For each 119906

1isin 119880+

every 1205741198881(119904) isin Γ

119888

1 and all powers of 120575 one has

119885120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575

119896120575(119898)

= 119885119892120575(119898)

(92)

Competing Interests

The author declares that they have no competing interests

References

[1] M Adler P vanMoerbeke and P VanhaeckeAlgebraic Integra-bility Painleve Geometry and Lie Algebras vol 47 of Ergebnisseder Mathematik und ihrer Grenzgebiete Springer 2004

[2] G F Helminck A G Helminck and E A PanasenkoldquoIntegrable deformations in the algebra of pseudodifferentialoperators from a Lie algebraic perspectiverdquo Theoretical andMathematical Physics vol 174 no 1 pp 134ndash153 2013

[3] M J Ablowitz D J Kaup A C Newell and H Segur ldquoTheinverse scattering transform-Fourier analysis for nonlinearproblemsrdquo Studies in Applied Mathematics vol 53 no 4 pp249ndash315 1974

[4] H Flashka A CNewell and T Ratiu ldquoKac-Moody Lie algebrasand soliton equations II Lax equations associated with 119860

1

(1)rdquoPhysicaDNonlinear Phenomena vol 9 no 3 pp 300ndash323 1983

[5] EDateM JimboMKashiwara andTMiwa ldquoTransformationgroups for soliton equationsrdquo in Proceedings of the RIMSSymposium on Nonlinear Integrable SystemsmdashClassical Theoryand Quantum Theory T Jimbo and T Miwa Eds WorldScience Publisher Kyoto Japan 1981

[6] G Segal and G Wilson ldquoLoop groups and equations of KdVtyperdquo Publications Mathematiques de lrsquoIHES vol 61 pp 5ndash651985

[7] A Pressley and G Segal Loop Groups Oxford MathematicalMonographs Clarendon Press 1986

[8] R S Hamilton ldquoThe inverse function theorem of Nash andMoserrdquoBulletin of the AmericanMathematical Society vol 7 no1 pp 65ndash222 1982

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8 Advances in Mathematical Physics

119871angl2 (C)lt0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =minus1

sum

119896=minusinfin

120574119896119911119896

(65)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra The first Lie algebra consists ofthe elements in 119871angl2(C) that extend to holomorphic mapsdefined on some disk 119911 isin C | |119911| ⩽ 1119903 0 lt 119903 lt 1 andthe second Lie algebra corresponds to the maps in 119871angl2(C)that have a holomorphic extension towards some disk aroundinfinity 119911 isin P1(C) | |119911| ⩾ 119903 0 lt 119903 lt 1 and that arezero at infinity To each of the two Lie subalgebras belongsa subgroup of 119871anGL2(C) The pointwise exponential mapapplied to elements of 119871angl2(C)lt0 yields elements of

119880minus= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

minus1

sum

119896=minusinfin

120574119896119911119896 (66)

and the exponential map applied to elements of 119871angl2(C)⩾0maps them into

119875+= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740 +

infin

sum

119896=1

120574119896119911119896 with 120574

0

isin GL2(C)

(67)

Both119880minusand 119875+are easily seen to be subgroups of 119871anGL2(C)

and since the direct sum of their Lie algebras is 119871angl2(C)their product

Ω = 119880minus119875+ (68)

is open in 119871anGL2(C) and is called like in the finite dimen-sional case the big cell with respect to 119880

minusand 119875

+

The second decomposition is the analogue of the splitting(11) of gl

2(119877)[119911 119911

minus1) that led to the Lax equations of the strict

AKNS hierarchy Now we consider the subspaces

119871angl2 (C)gt0 fl 120574 | 120574 isin 119871angl2 (C) 120574 =infin

sum

119896=1

120574119896119911119896

119871angl2 (C)⩽0

fl 120574 | 120574 isin 119871angl2 (C) 120574 =0

sum

119896=minusinfin

120574119896119911119896

(69)

Both are Lie subalgebras of 119871angl2(C) and their direct sumequals the whole Lie algebra 119871angl2(C)gt0 consists of mapswhose holomorphic extension to 119911 = 0 equals zero inthat point and 119871angl2(C)⩽0 is the set of maps that extendhomomorphically to a neighborhood of infinity To eachof them belongs a subgroup of 119871anGL2(C) The pointwiseexponential map applied to elements of 119871angl2(C)gt0 yieldselements of

119880+= 120574 | 120574 isin 119871angl2 (C) 120574 = Id +

infin

sum

119896=1

120574119896119911119896 (70)

and the exponential map applied to elements of 119871angl2(C)⩽0maps them into

119875minus= 120574 | 120574 isin 119871angl2 (C) 120574 = 1205740

+

minus1

sum

119896=minusinfin

120574119896119911119896 with 120574

0isin GL2(C)

(71)

Both119875minusand119880

+are easily seen to be subgroups of 119871anGL2(C)

Since the direct sum of their Lie algebras is 119871angl2(C) theirproduct 119875

minus119880+is open and because 119875

+= GL

2(C)119880+=

119880+GL2(C) and 119875

minus= GL2(C)119880minus= 119880minusGL2(C) this gives the

equality

119875minus119880+= Ω = 119880

minus119875+ (72)

the big cell in 119871anGL2(C)The next two subgroups of 119871anGL2(C) correspond to

the exponential factors in both linearizations The group ofcommuting flows relevant for the AKNS hierarchy is thegroup

Γ0= 1205740(119905) = exp(

infin

sum

119894=0

1199051198941198760119911119894) | 1205740isin 119875+ (73)

and similarly for the strict AKNS hierarchy we use thecommuting flows from

Γ1= 1205741(119905) = exp(

infin

sum

119894=1

1199051198941198760119911119894) | 1205741isin 119880+ (74)

Besides the groups Δ Γ0 and Γ

1 there are more subgroups in

119871anGL2(C) that commute with those three groups and theyare in a sense complimentary to Δ Γ

0and Δ Γ

1 respectively

That is why we use the following notations for them in 119880minus

there is

Γ119888

0= 120574119888

0(119905) = exp(

infin

sum

119896=1

1199041198961198760119911minus119896) | 120574119888

0isin 119880minus (75)

and in 119875minuswe have

Γ119888

1= 120574119888

1(119905) = exp(

infin

sum

119896=0

1199041198961198760119911minus119896) | 120574119888

1isin 119875minus (76)

We have now all ingredients to describe the constructionof the solutions to each hierarchy and we start with theAKNS hierarchy Take inside the product 119871anGL2(C) times Δthe collection 119878

0of pairs (119892 120575(119898)) such that there exists a

1205740(119905) isin Γ

0satisfying

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898) isin Ω = 119880

minus119875+ (77)

For such a pair (119892 120575(119898)) we take the collection Γ0(119892 120575(119898))

of all 1205740(119905) satisfying the condition (77) This is an open

nonempty subset of Γ0 Let 119877

0(119892 120575(119898)) be the algebra of

analytic functions Γ0(119892 120575(119898)) rarr C This is the algebra of

Advances in Mathematical Physics 9

functions 119877 that we associate with the point (119892 120575(119898)) isin 1198780

and for the commuting derivations of 1198770(119892 120575(119898)) we choose

120597119894fl 120597120597119905

119894 119894 ⩾ 0 By property (77) we have for all 120574

0(119905) isin

Γ0(119892 120575(119898))

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898)

= 119906minus(119892 120575 (119898))

minus1

119901+(119892 120575 (119898))

(78)

with 119906minus(119892 120575(119898)) isin 119880

minus119901+(119892 120575(119898)) isin 119875

+Then all thematrix

coefficients in the Fourier expansions of the elements 119906minus(119892

120575(119898)) and 119901+(119892 120575(119898)) belong to the algebra 119877

0(119892 120575(119898)) It is

convenient to write relation (78) in the form

Ψ119892120575(119898)

fl 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905)

= 119901+(119892 120575 (119898)) 120575 (119898) 120574

0(119905) 119892minus1

= 119901+(119892 120575 (119898)) 120574

0(119905) 120575 (119898) 119892

minus1

= 119902+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(79)

with 119902+(119892 120575(119898)) isin 119875

+ Clearly Ψ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M0 where all the products in the

decomposition 119906minus(119892 120575(119898))120575(119898)120574

0(119905) are no longer formal

but real This is our potential wave matrix for the AKNShierarchy

To get the potential wave function for the strict AKNShierarchy we proceed similarly only this time we use thedecomposition Ω = 119875

minus119880+ Consider again the product

119871anGL2(C) times Δ and the subset 1198781of pairs (119892 120575(119898)) such that

there exists a 1205741(119905) isin Γ

1satisfying

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898) isin Ω = 119875

minus119880+ (80)

For such a pair (119892 120575(119898)) we define Γ1(119892 120575(119898)) as the set of

all 1205741(119905) satisfying condition (77) This is an open nonempty

subset of Γ1 Let 119877

1(119892 120575(119898)) be the algebra of analytic

functions Γ1(119892 120575(119898)) rarr C This is the algebra of functions

119877 that we associate with the point (119892 120575(119898)) isin 1198781and for the

commuting derivations of 1198771(119892 120575(119898)) we choose 120597

119894fl 120597120597119905

119894

119894 gt 0 By property (80) we have for all 1205741(119905) isin Γ

1(119892 120575(119898))

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898)

= 119901minus(119892 120575 (119898))

minus1

119906+(119892 120575 (119898))

(81)

with 119901minus(119892 120575(119898)) isin 119875

minus 119906+(119892 120575(119898)) isin 119880

+ Then all

the matrix coefficients in the Fourier expansions of theelements 119901

minus(119892 120575(119898)) and 119906

+(119892 120575(119898)) belong to the algebra

1198771(119892 120575(119898)) It is convenient to write relation (81) in the form

Φ119892120575(119898)

fl 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905)

= 119906+(119892 120575 (119898)) 120575 (119898) 120574

1(119905) 119892minus1

= 119906+(119892 120575 (119898)) 120574

1(119905) 120575 (119898) 119892

minus1

= 119908+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(82)

with 119908+(119892 120575(119898)) isin 119880

+ Clearly Φ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M1where all the products in the

decomposition 119901minus(119892 120575(119898))120575(119898)120574

1(119905) are no longer formal

but realTo show that Ψ

119892120575(119898)is a wave matrix for the AKNS

hierarchy of type 120575(119898) and Φ119892120575(119898)

one for the strict AKNShierarchy we use the fact that it suffices to prove the relationsin Proposition 6 We treat first Ψ

119892120575(119898) In (79) we presented

two different decompositions of Ψ119892120575(119898)

that we will use tocompute 120597

119894(Ψ119892120575(119898)

) With respect to 119906minus(119892 120575(119898))120575(119898)120574

0(119905)

there holds120597119894(Ψ119892120575(119898)

)

= 120597119894(119906minus(119892 120575 (119898))) 119906

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905) = 119872

119894Ψ119892120575(119898)

(83)

with119872119894isin gl2(119877(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911 If we take

the decomposition 119902+(119892 120575(119898))120575(119898)119892

minus1 then we get

120597119894(Ψ119892120575(119898)

)

= 120597119894(119902+(119892 120575 (119898))) 119902

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(84)

with 120597119894(119902+(119892 120575(119898)))119902

+(119892 120575(119898))

minus1 of the formsum119895⩾0119902119895119911119895 with

all 119902119895isin gl2(119877(119892 120575(119898))) As Ψ

119892120575(119898)is invertible we obtain

119872119894= sum119895⩾0119902119895119911119895 and thus 119872

119894= sum119894

119895=0119902119895119911119895 and that is the

desired resultWe handle the case of Φ

119892120575(119898)in a similar way Also in

(79) you can find two different decompositions of Φ119892120575(119898)

that we will use to compute 120597119894(Φ119892120575(119898)

) With respect to theexpression 119901

minus(119892 120575(119898))120575(119898)120574

1(119905) we get for all 119894 ⩾ 1

120597119894(Φ119892120575(119898)

)

= 120597119894(119901minus(119892 120575 (119898))) 119901

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905) = 119873

119894Φ119892120575(119898)

(85)

again with 119873119894isin gl2(1198771(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911

Next we take the decomposition 119908+(119892 120575(119898))120575(119898)119892

minus1 andthat yields

120597119894(Φ119892120575(119898)

)

= 120597119894(119908+(119892 120575 (119898)))119908

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(86)

Since the zeroth order term of 119908+(119892 120575(119898)) is Id we note

that the element 120597119894(119908+(119892 120575(119898)))119908

+(119892 120575(119898))

minus1 has the formsum119895⩾1119908119895119911119895 with all 119908

119895isin gl2(1198771(119892 120575(119898))) As Φ

119892120575(119898)is also

invertible we obtain119873119894= sum119895⩾1119908119895119911119895 and thus119873

119894= sum119894

119895=1119908119895119911119895

and that is the result we were looking for in the strict caseThe abovementioned constructions are not affected seri-

ously by a number of transformations For example if wewrite 120575 = 120575((1 1)) then all 120575119896 119896 isin Z are central elementsand we have for all (119892 120575(119898)) isin 119878

0and 119878

1respectively and all

119896 isin Z

Ψ119892120575(119898)120575

119896 = Ψ119892120575(119898)

120575119896

respectively Φ119892120575(119898)120575

119896 = Φ119892120575(119898)

120575119896

(87)

10 Advances in Mathematical Physics

Also for any 1199011isin 119875+ each 120574119888

0(119904) isin Γ

119888

0 and all (119892 120575(119898)) isin

1198780 we see that the element (120574119888

0(119904)119892120575(minus119898)119901

1120575(119898) 120575(119898)) also

belongs to 1198780 for

120575 (119898) 1205740(119905) 120574119888

0(119904) 119892120575 (minus119898) 119901

1120575 (119898) 120574

0(119905)minus1120575 (minus119898)

= 120574119888

0(119904) 119906minus1

minus119901+1205740(119905) 11990111205740(119905)minus1

(88)

In particular we see that

Ψ120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575(119898)

= 119906minus120574119888

0(119904)minus1120575 (119898) 120574

0(119905) (89)

Its analogue in the strict case is as follows for any 1199061isin

119880+ each 120574119888

1(119904) isin Γ

119888

1 and all (119892 120575(119898)) isin 119878

1 the element

(120574119888

1(119904)119892120575(minus119898)119906

1120575(119898) 120575(119898)) also belongs to 119878

1and there

holds

Φ120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575(119898)

= 119901minus120574119888

1(119904)minus1120575 (119898) 120574

1(119905) (90)

We resume the foregoing results in the following theorem

Theorem 7 Consider the product spaceΠ fl 119871anGL2(C) times ΔThen the following holds

(a) For each point (119892 120575(119898)) isin Π that satisfies condition(77) we have a wave matrix of type 120575(119898) of the AKNShierarchyΨ

119892120575(119898) defined by (79) and this determines a

solution 119876119892120575(119898)

= 119906minus(119892 120575(119898))119876

0119906minus(119892 120575(119898))

minus1 of theAKNS hierarchy For each 119901

1isin 119875+ every 120574119888

0(119904) isin Γ

119888

0

and all powers of 120575 one has

119876120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575

119896120575(119898)

= 119876119892120575(119898)

(91)

(b) For each point (119892 120575(119898)) isin Π that satisfies condition(80) we have a wave matrix of type 120575(119898) of thestrict AKNS hierarchy Φ

119892120575(119898) defined by (82) and it

determines a solution 119885119892120575(119898)

= 119901minus(119892 120575(119898))119876

0119911119901minus(119892

120575(119898))minus1 of the strict AKNS hierarchy For each 119906

1isin 119880+

every 1205741198881(119904) isin Γ

119888

1 and all powers of 120575 one has

119885120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575

119896120575(119898)

= 119885119892120575(119898)

(92)

Competing Interests

The author declares that they have no competing interests

References

[1] M Adler P vanMoerbeke and P VanhaeckeAlgebraic Integra-bility Painleve Geometry and Lie Algebras vol 47 of Ergebnisseder Mathematik und ihrer Grenzgebiete Springer 2004

[2] G F Helminck A G Helminck and E A PanasenkoldquoIntegrable deformations in the algebra of pseudodifferentialoperators from a Lie algebraic perspectiverdquo Theoretical andMathematical Physics vol 174 no 1 pp 134ndash153 2013

[3] M J Ablowitz D J Kaup A C Newell and H Segur ldquoTheinverse scattering transform-Fourier analysis for nonlinearproblemsrdquo Studies in Applied Mathematics vol 53 no 4 pp249ndash315 1974

[4] H Flashka A CNewell and T Ratiu ldquoKac-Moody Lie algebrasand soliton equations II Lax equations associated with 119860

1

(1)rdquoPhysicaDNonlinear Phenomena vol 9 no 3 pp 300ndash323 1983

[5] EDateM JimboMKashiwara andTMiwa ldquoTransformationgroups for soliton equationsrdquo in Proceedings of the RIMSSymposium on Nonlinear Integrable SystemsmdashClassical Theoryand Quantum Theory T Jimbo and T Miwa Eds WorldScience Publisher Kyoto Japan 1981

[6] G Segal and G Wilson ldquoLoop groups and equations of KdVtyperdquo Publications Mathematiques de lrsquoIHES vol 61 pp 5ndash651985

[7] A Pressley and G Segal Loop Groups Oxford MathematicalMonographs Clarendon Press 1986

[8] R S Hamilton ldquoThe inverse function theorem of Nash andMoserrdquoBulletin of the AmericanMathematical Society vol 7 no1 pp 65ndash222 1982

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 9

functions 119877 that we associate with the point (119892 120575(119898)) isin 1198780

and for the commuting derivations of 1198770(119892 120575(119898)) we choose

120597119894fl 120597120597119905

119894 119894 ⩾ 0 By property (77) we have for all 120574

0(119905) isin

Γ0(119892 120575(119898))

120575 (119898) 1205740(119905) 1198921205740(119905)minus1120575 (minus119898)

= 119906minus(119892 120575 (119898))

minus1

119901+(119892 120575 (119898))

(78)

with 119906minus(119892 120575(119898)) isin 119880

minus119901+(119892 120575(119898)) isin 119875

+Then all thematrix

coefficients in the Fourier expansions of the elements 119906minus(119892

120575(119898)) and 119901+(119892 120575(119898)) belong to the algebra 119877

0(119892 120575(119898)) It is

convenient to write relation (78) in the form

Ψ119892120575(119898)

fl 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905)

= 119901+(119892 120575 (119898)) 120575 (119898) 120574

0(119905) 119892minus1

= 119901+(119892 120575 (119898)) 120574

0(119905) 120575 (119898) 119892

minus1

= 119902+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(79)

with 119902+(119892 120575(119898)) isin 119875

+ Clearly Ψ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M0 where all the products in the

decomposition 119906minus(119892 120575(119898))120575(119898)120574

0(119905) are no longer formal

but real This is our potential wave matrix for the AKNShierarchy

To get the potential wave function for the strict AKNShierarchy we proceed similarly only this time we use thedecomposition Ω = 119875

minus119880+ Consider again the product

119871anGL2(C) times Δ and the subset 1198781of pairs (119892 120575(119898)) such that

there exists a 1205741(119905) isin Γ

1satisfying

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898) isin Ω = 119875

minus119880+ (80)

For such a pair (119892 120575(119898)) we define Γ1(119892 120575(119898)) as the set of

all 1205741(119905) satisfying condition (77) This is an open nonempty

subset of Γ1 Let 119877

1(119892 120575(119898)) be the algebra of analytic

functions Γ1(119892 120575(119898)) rarr C This is the algebra of functions

119877 that we associate with the point (119892 120575(119898)) isin 1198781and for the

commuting derivations of 1198771(119892 120575(119898)) we choose 120597

119894fl 120597120597119905

119894

119894 gt 0 By property (80) we have for all 1205741(119905) isin Γ

1(119892 120575(119898))

120575 (119898) 1205741(119905) 1198921205741(119905)minus1120575 (minus119898)

= 119901minus(119892 120575 (119898))

minus1

119906+(119892 120575 (119898))

(81)

with 119901minus(119892 120575(119898)) isin 119875

minus 119906+(119892 120575(119898)) isin 119880

+ Then all

the matrix coefficients in the Fourier expansions of theelements 119901

minus(119892 120575(119898)) and 119906

+(119892 120575(119898)) belong to the algebra

1198771(119892 120575(119898)) It is convenient to write relation (81) in the form

Φ119892120575(119898)

fl 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905)

= 119906+(119892 120575 (119898)) 120575 (119898) 120574

1(119905) 119892minus1

= 119906+(119892 120575 (119898)) 120574

1(119905) 120575 (119898) 119892

minus1

= 119908+(119892 120575 (119898)) 120575 (119898) 119892

minus1

(82)

with 119908+(119892 120575(119898)) isin 119880

+ Clearly Φ

119892120575(119898)is an oscillating

matrix of type 120575(119898) in M1where all the products in the

decomposition 119901minus(119892 120575(119898))120575(119898)120574

1(119905) are no longer formal

but realTo show that Ψ

119892120575(119898)is a wave matrix for the AKNS

hierarchy of type 120575(119898) and Φ119892120575(119898)

one for the strict AKNShierarchy we use the fact that it suffices to prove the relationsin Proposition 6 We treat first Ψ

119892120575(119898) In (79) we presented

two different decompositions of Ψ119892120575(119898)

that we will use tocompute 120597

119894(Ψ119892120575(119898)

) With respect to 119906minus(119892 120575(119898))120575(119898)120574

0(119905)

there holds120597119894(Ψ119892120575(119898)

)

= 120597119894(119906minus(119892 120575 (119898))) 119906

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119906minus(119892 120575 (119898)) 120575 (119898) 120574

0(119905) = 119872

119894Ψ119892120575(119898)

(83)

with119872119894isin gl2(119877(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911 If we take

the decomposition 119902+(119892 120575(119898))120575(119898)119892

minus1 then we get

120597119894(Ψ119892120575(119898)

)

= 120597119894(119902+(119892 120575 (119898))) 119902

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(84)

with 120597119894(119902+(119892 120575(119898)))119902

+(119892 120575(119898))

minus1 of the formsum119895⩾0119902119895119911119895 with

all 119902119895isin gl2(119877(119892 120575(119898))) As Ψ

119892120575(119898)is invertible we obtain

119872119894= sum119895⩾0119902119895119911119895 and thus 119872

119894= sum119894

119895=0119902119895119911119895 and that is the

desired resultWe handle the case of Φ

119892120575(119898)in a similar way Also in

(79) you can find two different decompositions of Φ119892120575(119898)

that we will use to compute 120597119894(Φ119892120575(119898)

) With respect to theexpression 119901

minus(119892 120575(119898))120575(119898)120574

1(119905) we get for all 119894 ⩾ 1

120597119894(Φ119892120575(119898)

)

= 120597119894(119901minus(119892 120575 (119898))) 119901

minus(119892 120575 (119898))

minus1

+ 119911119894

sdot 119901minus(119892 120575 (119898)) 120575 (119898) 120574

1(119905) = 119873

119894Φ119892120575(119898)

(85)

again with 119873119894isin gl2(1198771(119892 120575(119898)))[119911 119911

minus1) of order 119894 in 119911

Next we take the decomposition 119908+(119892 120575(119898))120575(119898)119892

minus1 andthat yields

120597119894(Φ119892120575(119898)

)

= 120597119894(119908+(119892 120575 (119898)))119908

+(119892 120575 (119898))

minus1

Ψ119892120575(119898)

(86)

Since the zeroth order term of 119908+(119892 120575(119898)) is Id we note

that the element 120597119894(119908+(119892 120575(119898)))119908

+(119892 120575(119898))

minus1 has the formsum119895⩾1119908119895119911119895 with all 119908

119895isin gl2(1198771(119892 120575(119898))) As Φ

119892120575(119898)is also

invertible we obtain119873119894= sum119895⩾1119908119895119911119895 and thus119873

119894= sum119894

119895=1119908119895119911119895

and that is the result we were looking for in the strict caseThe abovementioned constructions are not affected seri-

ously by a number of transformations For example if wewrite 120575 = 120575((1 1)) then all 120575119896 119896 isin Z are central elementsand we have for all (119892 120575(119898)) isin 119878

0and 119878

1respectively and all

119896 isin Z

Ψ119892120575(119898)120575

119896 = Ψ119892120575(119898)

120575119896

respectively Φ119892120575(119898)120575

119896 = Φ119892120575(119898)

120575119896

(87)

10 Advances in Mathematical Physics

Also for any 1199011isin 119875+ each 120574119888

0(119904) isin Γ

119888

0 and all (119892 120575(119898)) isin

1198780 we see that the element (120574119888

0(119904)119892120575(minus119898)119901

1120575(119898) 120575(119898)) also

belongs to 1198780 for

120575 (119898) 1205740(119905) 120574119888

0(119904) 119892120575 (minus119898) 119901

1120575 (119898) 120574

0(119905)minus1120575 (minus119898)

= 120574119888

0(119904) 119906minus1

minus119901+1205740(119905) 11990111205740(119905)minus1

(88)

In particular we see that

Ψ120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575(119898)

= 119906minus120574119888

0(119904)minus1120575 (119898) 120574

0(119905) (89)

Its analogue in the strict case is as follows for any 1199061isin

119880+ each 120574119888

1(119904) isin Γ

119888

1 and all (119892 120575(119898)) isin 119878

1 the element

(120574119888

1(119904)119892120575(minus119898)119906

1120575(119898) 120575(119898)) also belongs to 119878

1and there

holds

Φ120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575(119898)

= 119901minus120574119888

1(119904)minus1120575 (119898) 120574

1(119905) (90)

We resume the foregoing results in the following theorem

Theorem 7 Consider the product spaceΠ fl 119871anGL2(C) times ΔThen the following holds

(a) For each point (119892 120575(119898)) isin Π that satisfies condition(77) we have a wave matrix of type 120575(119898) of the AKNShierarchyΨ

119892120575(119898) defined by (79) and this determines a

solution 119876119892120575(119898)

= 119906minus(119892 120575(119898))119876

0119906minus(119892 120575(119898))

minus1 of theAKNS hierarchy For each 119901

1isin 119875+ every 120574119888

0(119904) isin Γ

119888

0

and all powers of 120575 one has

119876120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575

119896120575(119898)

= 119876119892120575(119898)

(91)

(b) For each point (119892 120575(119898)) isin Π that satisfies condition(80) we have a wave matrix of type 120575(119898) of thestrict AKNS hierarchy Φ

119892120575(119898) defined by (82) and it

determines a solution 119885119892120575(119898)

= 119901minus(119892 120575(119898))119876

0119911119901minus(119892

120575(119898))minus1 of the strict AKNS hierarchy For each 119906

1isin 119880+

every 1205741198881(119904) isin Γ

119888

1 and all powers of 120575 one has

119885120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575

119896120575(119898)

= 119885119892120575(119898)

(92)

Competing Interests

The author declares that they have no competing interests

References

[1] M Adler P vanMoerbeke and P VanhaeckeAlgebraic Integra-bility Painleve Geometry and Lie Algebras vol 47 of Ergebnisseder Mathematik und ihrer Grenzgebiete Springer 2004

[2] G F Helminck A G Helminck and E A PanasenkoldquoIntegrable deformations in the algebra of pseudodifferentialoperators from a Lie algebraic perspectiverdquo Theoretical andMathematical Physics vol 174 no 1 pp 134ndash153 2013

[3] M J Ablowitz D J Kaup A C Newell and H Segur ldquoTheinverse scattering transform-Fourier analysis for nonlinearproblemsrdquo Studies in Applied Mathematics vol 53 no 4 pp249ndash315 1974

[4] H Flashka A CNewell and T Ratiu ldquoKac-Moody Lie algebrasand soliton equations II Lax equations associated with 119860

1

(1)rdquoPhysicaDNonlinear Phenomena vol 9 no 3 pp 300ndash323 1983

[5] EDateM JimboMKashiwara andTMiwa ldquoTransformationgroups for soliton equationsrdquo in Proceedings of the RIMSSymposium on Nonlinear Integrable SystemsmdashClassical Theoryand Quantum Theory T Jimbo and T Miwa Eds WorldScience Publisher Kyoto Japan 1981

[6] G Segal and G Wilson ldquoLoop groups and equations of KdVtyperdquo Publications Mathematiques de lrsquoIHES vol 61 pp 5ndash651985

[7] A Pressley and G Segal Loop Groups Oxford MathematicalMonographs Clarendon Press 1986

[8] R S Hamilton ldquoThe inverse function theorem of Nash andMoserrdquoBulletin of the AmericanMathematical Society vol 7 no1 pp 65ndash222 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Mathematical Physics

Also for any 1199011isin 119875+ each 120574119888

0(119904) isin Γ

119888

0 and all (119892 120575(119898)) isin

1198780 we see that the element (120574119888

0(119904)119892120575(minus119898)119901

1120575(119898) 120575(119898)) also

belongs to 1198780 for

120575 (119898) 1205740(119905) 120574119888

0(119904) 119892120575 (minus119898) 119901

1120575 (119898) 120574

0(119905)minus1120575 (minus119898)

= 120574119888

0(119904) 119906minus1

minus119901+1205740(119905) 11990111205740(119905)minus1

(88)

In particular we see that

Ψ120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575(119898)

= 119906minus120574119888

0(119904)minus1120575 (119898) 120574

0(119905) (89)

Its analogue in the strict case is as follows for any 1199061isin

119880+ each 120574119888

1(119904) isin Γ

119888

1 and all (119892 120575(119898)) isin 119878

1 the element

(120574119888

1(119904)119892120575(minus119898)119906

1120575(119898) 120575(119898)) also belongs to 119878

1and there

holds

Φ120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575(119898)

= 119901minus120574119888

1(119904)minus1120575 (119898) 120574

1(119905) (90)

We resume the foregoing results in the following theorem

Theorem 7 Consider the product spaceΠ fl 119871anGL2(C) times ΔThen the following holds

(a) For each point (119892 120575(119898)) isin Π that satisfies condition(77) we have a wave matrix of type 120575(119898) of the AKNShierarchyΨ

119892120575(119898) defined by (79) and this determines a

solution 119876119892120575(119898)

= 119906minus(119892 120575(119898))119876

0119906minus(119892 120575(119898))

minus1 of theAKNS hierarchy For each 119901

1isin 119875+ every 120574119888

0(119904) isin Γ

119888

0

and all powers of 120575 one has

119876120574119888

0(119904)119892120575(minus119898)1199011120575(119898)120575

119896120575(119898)

= 119876119892120575(119898)

(91)

(b) For each point (119892 120575(119898)) isin Π that satisfies condition(80) we have a wave matrix of type 120575(119898) of thestrict AKNS hierarchy Φ

119892120575(119898) defined by (82) and it

determines a solution 119885119892120575(119898)

= 119901minus(119892 120575(119898))119876

0119911119901minus(119892

120575(119898))minus1 of the strict AKNS hierarchy For each 119906

1isin 119880+

every 1205741198881(119904) isin Γ

119888

1 and all powers of 120575 one has

119885120574119888

1(119904)119892120575(minus119898)1199061120575(119898)120575

119896120575(119898)

= 119885119892120575(119898)

(92)

Competing Interests

The author declares that they have no competing interests

References

[1] M Adler P vanMoerbeke and P VanhaeckeAlgebraic Integra-bility Painleve Geometry and Lie Algebras vol 47 of Ergebnisseder Mathematik und ihrer Grenzgebiete Springer 2004

[2] G F Helminck A G Helminck and E A PanasenkoldquoIntegrable deformations in the algebra of pseudodifferentialoperators from a Lie algebraic perspectiverdquo Theoretical andMathematical Physics vol 174 no 1 pp 134ndash153 2013

[3] M J Ablowitz D J Kaup A C Newell and H Segur ldquoTheinverse scattering transform-Fourier analysis for nonlinearproblemsrdquo Studies in Applied Mathematics vol 53 no 4 pp249ndash315 1974

[4] H Flashka A CNewell and T Ratiu ldquoKac-Moody Lie algebrasand soliton equations II Lax equations associated with 119860

1

(1)rdquoPhysicaDNonlinear Phenomena vol 9 no 3 pp 300ndash323 1983

[5] EDateM JimboMKashiwara andTMiwa ldquoTransformationgroups for soliton equationsrdquo in Proceedings of the RIMSSymposium on Nonlinear Integrable SystemsmdashClassical Theoryand Quantum Theory T Jimbo and T Miwa Eds WorldScience Publisher Kyoto Japan 1981

[6] G Segal and G Wilson ldquoLoop groups and equations of KdVtyperdquo Publications Mathematiques de lrsquoIHES vol 61 pp 5ndash651985

[7] A Pressley and G Segal Loop Groups Oxford MathematicalMonographs Clarendon Press 1986

[8] R S Hamilton ldquoThe inverse function theorem of Nash andMoserrdquoBulletin of the AmericanMathematical Society vol 7 no1 pp 65ndash222 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of