factorization of the (2+1)-dimensional blp integrable system by the periodic fixed points of its...

Physics Letters A 160 ( 1991 ) 161-165 North-Holland PHYSICS LETTERS A

Factorization of the (2 + 1 )-dimensional BLP integrable system by the periodic fixed points of its B icklund transformations

John Weiss 49 Grandview Road, Arlington, MA 02174, USA

Received 2 September 1991; accepted for publication 12 September 1991 Communicated by D.D. Holm

Previously, we have found a faetorization of the ( 1 + 1 )-dimensional Toda lattice by the periodic fixed points of its B~icklund transformations. The Toda flow is realized by two commuting, one-dimensional Hamiltonian flows. By a result of Konopel- chenko, the Laplace-Darboux transformation is a B~icklund transformation for the (2 + 1 )-dimensional Boiti-Leon-Pempinelli (BLP) equation. A periodic fixed point of the Laplace transformation is an invariant manifold of the BLP flow. This manifold is determined by solutions of the ( l + 1 )-dimensional Toda lattice equations. From these results we find that the 2 + 1 BLP flow is factored by three commuting, one-dimensional Hamiltonian flows that are the periodic fixed points of its B~icklund transformations.

The two-dimensional Toda lattice equations were first studied in the works of Laplace, Moutard and Darboux [ 1 ] in connection with their classification of surfaces and factorizations of linear differential operators. For instance, Moutard essentially solved the free-end Toda lattice in the form of Wronskian determinants [ 1 ]. The two-dimensional Toda lattice was derived by Laplace in the eighteenth century through results relating factorizations of linear differential operators with certain gauge invariants. This Laplace transformation was proposed by Darboux as a fundamental method for the classification of certain surfaces in space related as focal, or caustic surfaces.

By a result of Konopelchenko [ 2 ], the BLP equation is factored by the periodic fixed point of the La- place transformation onto an invariant manifold determined by solutions of the two-dimensional periodic Toda lattice.

Previously, we have found [ 3 ] an infinite set of distinct B~icklund transformations for the two-dimensional Toda lattice. Although simple to describe the resulting systems of integrable ordinary differential equations have a rich structure that depends strongly on the length and number theoretic prop- erties of the period of the lattice. We emphasize that the t ime-space dependence of the lattice is factored

by commuting, finite-dimensional Hamiltonian flows. Surprisingly, these systems arise naturally from the periodic fixed points of the simple B~icklund transformations.

Using the above set of results we find a factorization of the BLP equation by three commuting, one- dimensional Hamiltonian flows that are the periodic fixed points of its B~icklund transformations.

By way of introduction we present the relevant results for the KdV system and the derivation of the Toda lattice from the Laplace transformation of focal surfaces. Then, we show how a simple generali- zation of the KdV result obtains the set of B~icklund transformations for the Toda lattice.

We then present the factorization of the BLP equation by the periodic fixed points of its B~icklund transformations.

The Korteweg-de Vries equation,

0 u,+

has the B~icklund transformation [ 4 ]

0 2 u=4 -~5x2 In ¢ + u 2 ,

where u2= -(axxx/epx+ ½ (~xx/¢x) 2 and

0375-9601/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved. 161

Volume 160, number 2 PHYSICS LETTERS A 11 November 1991

(9~ +{(9; x} =,z. (1) (9.r

Eq. ( 1 ) is sometimes known as the singular manifold equation for the KdV equation and itself has the two B~icklund transformations [4,6]

a~u+b ad-bc=l , (2) (9= c~,+d'

and

0,- = ~u?~ l ,

where, for both transformations,

(3)

~u_ + {~ x} =X.

The expression

0 (9.~.~ 1 (~xx~ 2 {(9;x}= Ox (9.,- 2\(9_,./

is the Schwarzian derivative, which is invariant under the Moebius group (2) [4,5].

By itself, transformation (2) is a point symmetry that does not lead to new forms of solution, and transformation (3) by itself is in involution. The ef- fective B~icklund transformation (BT) for ( 1 ) is the composition of (2) and (3). We find that [ 6 ]

' 0 ~ ( 4 ) (9,+l.x- (9..., ,

2 2

(9,+~.x 0..- k(9,..,-) ox

is a BT for ( 1 ). The periodic fixed points of the BT are defined by eqs. (4) and (5) with

n = 1 , 2 , 3 , 4 .... ( m o d N ) .

The periodic fixed points continue to define a strong BT for ( 1 ). That' is, the integrability conditions

(gnq" l ,x t ~ (gn÷ I J x

continue to imply that (9, satisfy ( 1 ), and, by the pe- riodicity (rood N), the set {(9,, n= 1, 2, ... (modN)} are solutions of ( 1 ).

We have found [ 6 ] that if

0 j '

then

~j.,,x + ~j.x = ¢ j _ ~ . , .

The KdV and Boussinesq systems are instances of the general system in component form [ 7 ]

_ _ ~ ~+. t , ,x ~.x + ~j.,.., +. . .+ = ~ _ ~ + p , (6)

where j = 1, 2, ... (modN) . The KdV systems correspond to p = 1 and the Boussinesq to p = 2. Let the circulant forward shift matrix [8] be

C=circ[0, 1, 0, 0 ..... 0 ] .

In the N-vector form eqs. (6) are

/ ¢,..,./¢, 'k (7)

with

A=I+C+.. .+C p, B = I - C p.

The Casimir integrals of (6) correspond to the null vectors of B. The null vectors of A produce the constraints.

Associated with the principal Casimir, for any N

N

/4~= 1-I ¢J, j = l

we find the principal integrals of (6)

HN_p . . . . . =L"oHN, (8)

where m = 0, 1, 2, ... and

N

L = ~ O~O~+,...O~+,.

The systems (7) have a Hamiltonian structure,

A( ' i ' x / ' l / . : g ( " - . . ~ , a l , (9,

\~N..U~,,U = \ ~ u )

where Hi = ~jv= 1 ~j. The higher-order equations associated with the in-

tegrals (8) are

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:, _ . . . & , . J ~ u = " "

When A is invertible, then

Q = A - I B

is an antisymmetric circulant matrix. We have the systems

~,:, =M¢V¢H, ( 11 )

and

~.x = M$V$HN_v,,, . . . . ( 12 )

where

-,:(" 5°( ̀.... 5 is the co-symplectic form.

With p = 1 the flow of the KdV equation is factored by the commuting, Hamiltonian systems [6]

~.x = M s V s H , , $., =M~VsH3.

Now, Darboux [ 1 ] has shown that the parameters (x, y) for surfaces in three dimensions can be defined so the coordinates (zj) of the surface satisfy a partial differential equation of the form

OZz Oz +b Oz + c z = 0 , (13) OxO---y + a Ux Oy

where (a, b, c) are functionals of the first fundamental form in the (x, y) parameters. Under the gauge transformation z-~2z, the form of ( 13 ) is pre- served and

Oa Ob h= -~x + a b - c , k= ~y + a b - c

are invariant. The Laplace transformation of a surface is a par-

tial factorization of (13 ) in terms of the invariants [11

Oz Oz~ zl = -~y + az , -~x + bZl =hz . (14)

Eqs. (14) imply that z satisfy (13) and z~ satisfy the system

OZz~ Ozl Oz, OxOy +al ~ +b, -~y +clz l = 0 ,

where

Olnh al = a - - - bl =b ,

Oy '

Oa Ob _ b O ln h c' =C- ~x + ~ Oy (15)

From ( 15 ) the Laplace transformation of the invariants is

OZln h hi = 2 h - k - - - kl = h . (16)

OxOy '

Darboux [ 1 ] studied the periodic points of the Laplace transformation and found that these surfaces are related as a sequence of focal surfaces. From (16), the periodic fixed points are

0 2 In hj _ -h j+ l + 2 h i - h i _ , , (17) OxOy

where j = 1, 2, 3, ... (mod n) and n is the order of the fixed point. The substitution

hj =eOJ+ t-oJ

obtains the two-dimensional periodic Toda lattice

Oj.x~= -e°J+'-°J+e °:-°j-~ • (18)

A result of Konopelchenko [2 ] exhibits the La- place transformation as a B~icklund transformation for the BLP equation [ 9 ]. For this purpose it is im- portant to note that the condition b = 0 is invariant under the Laplace transformation. The Lax pair for the BLP equation can be written as

x

~gxy+aqlx+c~=O , ~l, lt-Jv~t, gyy-{-(f Cy)~¢mO , and, in terms of the invariants of the Laplace transformation, h = a x - c and k = - c ,

0 x

x

kt-kyy-[-m-~y(k ~ (k-h))=O. ( 2 0 )

These equations are equivalent to the BLP equation.

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The periodic fixed points of Laplace transform, with h = hj and k= hi_ 1, factor the flow into the two- dimensional periodic Toda lattice

0 2 In hg _ -hj+~ + 2 h j - h j _ l (21) OxOy

and the system

hj.t+ ~ ( h j i (hg_,-hj+l))=O. (22)

Now, the B~icklund transformations for the Dar- boux equations (17) and the Toda lattice equations (18) were found in ref. [ 3 ]. With reference to systems ( 11 ) a n d ( 12 ), without loss of generality nor- malize the Casimir, HN= 1, and set

~.~=M~V~H, , (23)

~,.v =M4V¢(Hu_p_ l/Hu) , (24)

where HN-p- l =LoHN. Then, let 5 = e ~'j- ~'j+' and find that (19), (20) imply

u/j,~v = e CJ+"- ~'J- e ~'j- Cj-p , (25)

where j = 1, 2, 3 .... (modN) . T o see this let ~j=e °: and find

O,x=g2VoHI, O y=g2VoG,

where

G= ~, e -Oj-Oj+l . . . . . oj+p

1 HN_p_ 1 = E -

It can be shown that I

( e - ° ' - ~ e -o,,-0 . . . . . . . . . . . o,_°"/ O,x~, = C - P ( I - C p) ( I - C) ,] .

Let 0= (1-C)(u and find (24). When p= 1 (25) is the Toda lattice of period N.

I f N and p are relatively prime (25) is again a Toda lattice of length N. If N= rnp (25) is p distinct lat- tices of length m. When N and p have common fac- tors there is one lattice for each distinct orbit of translation byp (mod N). In all cases the set of fields

are directly related to the set of invariants hi. When A is not invertible we find for eqs. (9) and (10) a similar connection with the Toda lattice. In this case

one must take into account the constraints that apply to these systems to obtain a valid correspondence. See, for instance, ref. [7].

Consideration of the form of (23), (24) and the possible relations between p and N determine that for a lattice of fixed length m there will exist an infinite sequence of distince B~icklund transformations. For instance, we have a B~icklund transformation for a lattice of length m when N=pm fo rp= 1, 2 ,3 .....

Systems (23) and (24) have a rich structure. As a B~icklund transformation for a Toda lattice of fixed length these systems relate flows from different hierarchies of equations with the flows in the sequence through the Toda lattice. In other words, as de- scribed by (23) and (24) the hierarchies of flows through KdV, Boussinesq, etc. are interrelated.

The BLP system is related to the above construc- tion when p = 1. In this case we have

h j= - l /~ j , (26)

L = ~ 0~0~+,, 1

and

H n = f i ~ j . 1

The periodic fixed points of the KdV type, p = 1, B~icklund transformations factor the flow of the BLP system into three commuting Hamiltonian systems

=M~V~H,, (27)

~y=M¢V~H_2, (28)

~t =M~V~H_,, (29)

where

H~ = ~ 5, H_2 = (LoH,,)/H,,,

H_4=(L2oHn)/Hn .

From eqs. (26) and (27)

hj,x hi+ l.x 1 1 hj + hi+, = - h: + hj+, ' - -

and substitution in eq. (22) obtains

O [hj(hjhj+, +hj_,hj)]=O (30) h. + Ty

164


Eqs. (26) and (28) imply

2 2 h j . e = h j h j + 1 - h j _ , h j .

Subst i tut ion into eq. (30) , using (26) , obtains

1 1 1 1

#_,¢j

This equat ion may be wri t ten in the form

=a'sven2.

By the dual Hami l ton ian structure found in ref. [ 6 ], eq. (2.53) , we obta in eq. (29) and by the results o f ref. [6] the flows (27) , (28) and (29) commute .

Surprisingly, the factor izat ion o f the Toda lat t ice for p = 2 does not lead to a corresponding factori-

zation o f the BLP equation. It is possible that the case p = I is unique in this regard.

R e f e r e n c e s

[ 1 ] G. Darboux, Th6orie g6n~rale des surfaces, Vol. 2 (Chelsea, New York, 1972).

[2] B.G. Konopelchenko, Phys. Lett. A 156 ( 1991 ) 221. [3] J. Weiss, Phys. Len. A 137 (1989) 365. [4] J. Weiss, J. Math. Phys. 24 (1983) 1405. [ 5 ] E. Hiile, Ordinary differential equations in the complex plane

(Wiley, New York, 1976). [6] J. Weiss, J. Math. Phys. 27 (1986) 2647. [ 7 ] J. Weiss, J. Math. Phys. 28 ( 1987 ) 2025. [8 ] P. Davis, Circulant matrices (Wiley, New York, 1983). [9] M. Boiti, J.J.P. Leon and F. Pempinelli, Inverse Probl. 3

(1987) 25.

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