Research ArticleTwo Kinds of Darboux-Baumlcklund Transformations forthe 119902-Deformed KdV Hierarchy with Self-Consistent Sources

Hongxia Wu12 Liangjuan Gao1 Jingxin Liu1 and Yunbo Zeng3

1Department of Mathematics School of Sciences Jimei University Xiamen 361021 China2Department of Mathematics The University of Texas Rio Grande Valley Edinburg TX 78539 USA3Department of Mathematics Tsinghua University Beijing 100084 China

Correspondence should be addressed to Hongxia Wu wuhongxiajmueducn

Received 19 April 2016 Accepted 23 July 2016

Academic Editor Pavel Kurasov

Copyright copy 2016 Hongxia Wu et al 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

Two kinds of Darboux-Backlund transformations (DBTs) are constructed for the 119902-deformed 119873th KdV hierarchy with self-consistent sources (119902-NKdVHSCS) by using the 119902-deformed pseudodifferential operators Note that one of the DBTs providesa nonauto Backlund transformation for two 119902-deformed 119873th KdV equations with self-consistent sources (119902-NKdVESCS) withdifferent degree In addition the soliton solution to the first nontrivial equation of 119902-KdVHSCS is also obtained

1 Introduction

The 119902-deformed integrable systems are regarded as the 119902-deformation of the related classical ones The 119902-deformationis performed by using the 119902-derivative 120597119902 to take the placeof usual derivative 120597119909 and it reduces to a classical integrablesystem as 119902 rarr 1 In recent years some 119902-deformed integrablesystems especially the 119902-deformed 119873th KdV hierarchy (119902-NKdVH) and the 119902-deformed KP hierarchy (119902-KPH) haveattracted much interest both in mathematics and in physics[1ndash17] It was shown that 119902-NKdVH inherited some integrablestructures from the classical 119873th KdV hierarchy such asinfinite conservation law [2] bi-Hamiltonian structure [3 4]tau function [5 6] Darboux-Backlund transformation [7]and 119902-Miura transformation [8] In 1999 some elementaryDBTs of the 119902-NKdVH (also called 119902-deformed Gelfand-Dickey hierarchy) were constructed by using the 119902-deformedpseudodifferential operators The formula for the 119899-timesrepeated DBTs was also presented which produces the newsoliton solutions to the 119902-NKdVH [7] For 119902-KPH its bi-Hamiltonian structure tau function additional symmetries119902-effect in 119902-soliton Virasoro constraints of tau function andintegrable extension and 119902-Backlund transformation werealso explored in [9ndash17] In 2008 based on the symmetryconstraint for 119902-KPH the new extension of this hierarchy

was considered [16] Two kinds of reductions of this newextended 119902-KP hierarchy were also studied which give many1 + 1 dimensional 119902-deformed soliton equations with self-consistent sources [16] For example the 119899-reduction 119871

119899

lt0= 0

gives 119902-NKdVHSCS However to our knowledge the DBTsand the soliton solution for 119902-NKdVHSCS still remain unex-plored It is known that the DBT is an important propertyto characterize the integrability of the hierarchy Thus it isnecessary for us to explore the DBT for 119902-NKdVHSCS Wethink our research results will deepen our understanding onsoliton solutions of this hierarchy

The outline of this paper is as follows In Section 2some notations in the 119902-calculus and the definition of the 119902-NKdVHSCS are briefly reviewed In Section 3 we aim at theconstruction of auto DBTs for 119902-NKdVHSCS In Section 4the nonauto DBTs for 119902-NKdVHSCS are constructed In Sec-tion 5 one soliton solution to the first nontrivial equation of119902-NKdVHSCS is obtained by using nonauto DBTs Section 6is devoted to a brief summary

2 The 119902-Deformed 119873th KdV Hierarchy withSelf-Consistent Sources (119902-NKdVHSCS)

In this section we briefly review some notations in the 119902-calculus and the definition of the 119902-NKdVHSCS

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 8153752 11 pageshttpdxdoiorg10115520168153752

2 Advances in Mathematical Physics

The 119902-derivative operator 120597119902 and 119902-shift operator 120579 aredefined by

[120597119902119891 (119909)] =119891 (119902119909) minus 119891 (119909)

119909 (119902 minus 1) (1)

120579 (119891 (119909)) = 119891 (119902119909) (2)

In this paper we introduce twonotations [119875119891] and119875∘119891 = 119875119891in which 119875 is a 119902-pseudo-differential operator (119902-PDO) givenby

119875 =

119899

sum

119894=minusinfin

119901119894120597119894

119902 (3)

[119875119891] denotes 119875 acting on the function 119891 while 119875119891 indicatesthe multiplication of 119875 and 119891 that is 120597119902119891 = 120579(119891)120597119902 + [120597119902119891]

It can be easily shown from (1) that when 119902 rarr 1 120597119902

reduces to the ordinary differential operator 120597119909 and that 120579 and120597119902 do not commute but satisfy

[120597119902120579119896(119891)] = 119902

119896120579119896[120597119902119891] 119896 isin 119885 (4)

Let 120597minus1119902

be the formal inverse of 120597119902 such as 120597119902120597minus1

119902119891 = 120597

minus1

119902120597119902119891 =

119891 In general the 119902-deformed Leibnitz rule holds

120597119899

119902119891 = sum

119896ge0

(119899

119896)

119902

120579119899minus119896

(120597119896

119902119891) 120597119899minus119896

119902 119899 isin 119885 (5)

where 119902-number and 119902-binomial are defined by

(119899)119902 =119902119899minus 1

119902 minus 1

(119899

119896)

119902

=(119899)119902 (119899 minus 1)119902 sdot sdot sdot (119899 minus 119896 + 1)119902

(1)119902 (2)119902 sdot sdot sdot (119896)119902

(119899

0)

119902

= 1

(6)

For a 119902-PDO 119875 = sum119899

119894=minusinfin119901119894120597119894

119902 we separate 119875 into the

differential part 119875+ = sum119899

119894=0119901119894120597119894

119902and the integral part 119875minus =

sum119894leminus1

119901119894120597119894

119902 The conjugate operation 119875

lowast is given by

119875lowast

=

119899

sum

119894=minusinfin

(120597lowast

119902)119894

119901119894 (7)

where 120597lowast

119902= minus120597119902120579

minus1= minus(1119902)1205971119902 (120597

minus1

119902)lowast

= (120597lowast

119902)minus1

= minus120579120597minus1

119902

The 119902-exponential function 119864119902(119909) is defined as

119864119902 (119909) = exp(

infin

sum

119896=1

(1 minus 119902)119896

119896 (1 minus 119902119896)119909119896) (8)

satisfying [120597119896

119902119864119902(119909119911)] = 119911

119896119864119902(119909119911) 119896 isin 119885

The extended 119902-KPH was given by [16]

119871 119905119899

= [119861119899 119871] 119861119899 = (119871119899)ge0

(9a)

119871120591119896

= [

[

119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895 119871

]

]

119861119896 = (119871119896)ge0

119896 = 119899 (9b)

120601119895119905119899

= [119861119899120601119895]

120595119895119905119899

= minus [119861lowast

119899120595119895]

119895 = 1 119898

(9c)

where 119871 = 120597119902 + 1199060 + 1199061120597minus1

119902+ 1199062120597

minus2

119902+ sdot sdot sdot and the coefficients

119906119894 (119894 = 0 1 ) are the functions of 119905 = (119909 1199051 )The commutativity of (9a) (9b) and (9c) leads to the

zero-curvature representation of 119902-KPH ((9a) (9b) and(9c)) As the 119899-reduction of the extended 119902-KPH the 119902-NKdVHSCS is defined as follows [16]

119861119899120591119896

= [

[

119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895 119861119899

]

]

119861119896 = (119861119899)119896119899

ge0(10a)

[119861119899120601119895] = 120582119899

119895120601119895 (10b)

[119861lowast

119899120595119895] = 120583

119899

119895120595119895 119895 = 1 119898 (10c)

Under (10b) and (10c) the Lax representation for (10a) is

[119861119899120593] = 120582120593 (11a)

120593120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)120593]

]

(11b)

We find that when 119898 = 0 119902-NKdVHSCS ((10a) (10b) and(10c) and (11a) and (11b)) can be reduced to the 119902-NKdVHand its related Lax representation respectively In additionwhen 119899 = 2 119896 = 1 ((10a) (10b) and (10c)) becomes the firstnontrivial soliton equation of 119902-KdVHSCS given by

V11205911

+ 1198901 +

119898

sum

119895=1

1198921198951 = 0 (12a)

V01205913

+ 1198900 +

119898

sum

119895=1

1198921198950 = 0 (12b)

120579 (V1) minus V1 + 1199060 minus 1205792(1199060) = 0 (12c)

[(1205972

119902+ V1120597119902 + V0) 120593119895] minus 120582

2

119895120593119895 = 0 (12d)

[(1205972

119902+ V1120597119902 + V0)

lowast

120595119895] minus 1205832

119895120595119895 = 0 119895 = 1 119898 (12e)

3 The Auto Darboux-BaumlcklundTransformation (DBT) for 119902-NKdVHSCS

In this section we will focus on the construction of auto DBTfor 119902-NKdVHSCS

Advances in Mathematical Physics 3

Theorem 1 Assume 119861119899 120601119895 120595119895 (119895 = 1 119898) be the solutionof 119902-NKdVHSCS ((10a) (10b) and (10c)) and ℎ1 satisfies (11a)and (11b) with 120582 = 120582

119899

1 the DBT is defined by

119861119899 = 1198791119861119899119879minus1

1= 120597119899

119902+ V119899minus1120597

119899minus1

119902+ sdot sdot sdot + V1120597119902 + V0 (13a)

120601 = [1198791120601] =119882119902 [ℎ1 120601]

ℎ1

(13b)

120601119895= [1198791120601119895] =

119882119902 [ℎ1 120601119895]

ℎ1

(13c)

120595119895= [(119879

minus1

1)lowast

120595119895] = minus120579 (Ω (ℎ1 120595119895))

120579 (ℎ1)

119895 = 1 119898

(13d)

Then 119861119899 120601 120601119895 120595119895 (119895 = 1 119898) satisfy (10b) and (10c) and(11a) and (11b) and hence are the solution of 119902-NKdVHSCS((10a) (10b) and (10c)) where 1198791 = 120579(ℎ1)120597119902ℎ

minus1

1= 120597119902 minus 1205721

1205721 = [120597119902ℎ1]ℎ1 and 119882119902[ℎ1 120601119895] and Ω119902[ℎ1 120595119895] are defined asfollows

119882119902 [1206011 120601] =

1003816100381610038161003816100381610038161003816100381610038161003816

ℎ1 120601

[120597119902ℎ1] [120597119902120601]

1003816100381610038161003816100381610038161003816100381610038161003816

Ω (ℎ1 120595119895) = [120597minus1

119902ℎ1120595119895]

(14)

Remark 2 Here it should be pointed out that the formulaholds

119861119896 = 1198791119861119896119879minus1

1+ 1198791120591

119896

119879minus1

1

119861119896 = (119861119899)119896119899

ge0

119861119896 = (119861119899)119896119899

ge0

(15)

where the gauge operator 1198791 is defined above The proof hasbeen given in [7]

Proof (1)We firstly show that119861119899 120601119895120595119895 (119895 = 1 119898) satisfy(10b) and (10c)

Noting that 119861119899 120601119895 120595119895 (119895 = 1 119898) are the solution of(10a) (10b) and (10c) we have

[119861119899120601119895] = 120582119899

119895120601119895

[119861lowast

119899120595119895] = 120583

119899

119895120595119895

(16)

Hence

[119861119899120601119895] minus 120582119899

119895120601119895= [1198791119861119899119879

minus1

1[1198791120601119895]] minus 120582

119899

119895120601119895

= [1198791 [119861119899120601119895]] minus 120582119899

119895120601119895

= 120582119899

119895([1198791120601119895] minus 120601

119895) = 0

[119861lowast

119899120595119895] minus 120583119899

119895120595119895= [(119879

minus1

1)lowast

119861lowast

119899119879lowast

1[(119879minus1

1)lowast

120595119895]]

minus 120583119899

119895120595119895= [(119879

minus1

1)lowast

119861lowast

119899120595119895] minus 120583

119899

119895120595119895

= 0

(17)

(2)Wefinally show that119861119899120601120601119895120595119895 satisfy (11a) and (11b)Since the proof of (11a) is the same as the case (1) we only

need to verify that 119861119899 120601 120601119895 120595119895 satisfy (11b) that is

120601120591119896

minus [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= 0 119861119896 = (119861119899)119896119899

ge0 (18)

Noting that 120601120591119896

= [1198791120601]120591119896

= [1198791120591119896

120601] + [1198791120601120591119896

] and 120601120591119896

=

[(119861119896 + sum119898

119895=1120601119895120597minus1

119902120595119895)120601] we get

120601120591119896

minus [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= [1198791120591119896

120601] + [1198791120601120591119896

]

minus [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)[1198791120601]]

]

= [

[

(1198791120591119896

+ 1198791119861119896

minus 1198611198961198791 +

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898

sum

119895=1

120601119895120597minus1

1199021205951198951198791)120601]

]

(19)

According to Remark 2 we have

1198791120591119896

+ 1198791119861119896 minus 1198611198961198791 = 0 (20)

Next we prove

119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791) = 0 (21)

Since 1198791 = 120597119902 minus 1205721 and 120597119902120601119895 = 120579(120601119895)120597119902 + [120597119902120601119895] then forall119895 weobtain by the tedious computation

1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791

= (120597119902 minus 1205721) 120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

119902120595119895120597119902 + 120601

119895120597minus1

1199021205951198951205721

= 120579 (120601119895) 120595119895 + 120601119895

Ω(ℎ1 120595119895)

ℎ1

(22a)

4 Advances in Mathematical Physics

In addition we also have

120579 (120601119895) 120595119895 + 120601119895

Ω(ℎ1 120595119895)

ℎ1

=[120579 (120601119895) 120597119902Ω(1206011 120595119895)] + [1198791ℎ1]Ω (ℎ1 120595119895)

ℎ1

=[120597119902120601119895Ω(ℎ1 120595119895)] minus [120597119902120601119895]Ω (ℎ1 120595119895) + ([120597119902120601119895] minus 1205721120601119895)Ω (ℎ1 120595119895)

ℎ1

=[(120597119902 minus 1205721) 120601119895Ω(ℎ1 120595119895)]

ℎ1

=

[1198791 [(120601119895120597minus1

119902120595119895) ℎ1]]

ℎ1

(22b)

Substituting (22b) into (22a) leads to119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

=1

ℎ1

[

[

1198791[

[

119898

sum

119895=1

(120601119895120597minus1

119902120595119895) ℎ1

]

]

]

]

(23)

Since ℎ1 is the solution of (11a) and (11b) with 120582 = 120582119899

1 we have

ℎ1120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

997904rArr

[

[

(

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

= ℎ1120591119896

minus [119861119896ℎ1]

(24)

Moreover by the property of determinant we have

[1198791ℎ1] =119882119902 [ℎ1 ℎ1]

ℎ1

= 0 (25)

Differentiating both sides of (25) with respect to 120591119896 yields

[1198791120591119896

1206011] + [11987911206011120591119896

] = 0 997904rArr

[11987911206011120591119896

] = minus [1198791120591119896

1206011]

(26)

From (23) (24) and (26) we have119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

= minus1

1206011

[(1198791120591119896

+ 1198791119861119896) 1206011] = minus1

1206011

[119861119896 [11987911206011]]

= 0

(27)

This completes the proof

Obviously Theorem 1 provides an auto DBT for 119902-NKdVHSCS ((10a) (10b) and (10c)) However thisDBTdoesnot enable us to obtain the new solution of 119902-NKdVHSCS((10a) (10b) and (10c)) So we have to seek for nonauto DBTsbetween the two 119902-NKdVHSCS ((10a) (10b) and (10c)) withdifferent degrees of sources

4 The Nonauto DBTs of 119902-NKdVHSCS

In this section we will construct the nonauto DBTs of 119902-NKdVHSCS ((10a) (10b) and (10c)) which enables us toobtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH

Theorem 3 Given 119861119899 120601119895 120595119895 (119895 = 1 119898) the solution for119902-NKdVHSCS ((10a) (10b) and (10c)) let 1198911 1198921 equiv 120601119898+1 betwo independent eigenfunctions of (11a) and (11b) with 120582 =

120582119899

119898+1 Let 1198871(120591119896) be a function of 120591119896 such that 1198871(120591119896)120591

119896

=

(minus1)119898+1

1205731(120591119896)1205781(120591119896) Denote ℎ1 = 1198911 + 1198871(120591119896)1198921The DBT is defined by

119861119899 = 1198791119861119899119879minus1

1= 120597119899

119902+ V119899minus1120597

119899minus1

119902+ sdot sdot sdot + V1120597119902 + V0 (28a)

120601 = [1198791120601] =119882119902 [ℎ1 120601]

ℎ1

(28b)

120601119895= [1198791120601119895] =

119882119902 [ℎ1 120601119895]

ℎ1

(28c)

120595119895= [(119879

minus1

1)lowast

120595119895] = minus120579 (Ω (ℎ1 120595119895))

120579 (ℎ1)

119895 = 1 119898

(28d)

120601119898+1

= minus1205731 (120591119896) [11987911198921] (28e)

120595119898+1

= (minus1)119898+1

1205781 (120591119896)1

120579 (ℎ1) (28f)

where1198791 = 120597119902minus1205721 1205721 = [120597119902ℎ1]ℎ1 and then 119861119899 120601 120601119895120595119895 (119895 =

1 119898) 120601119898+1

120595119898+1

satisfy (10b) and (11a) and (11b) with 119898

replaced by119898+1 hence 119861119899 120601119895 120595119895 (119895 = 1 119898) 120601119898+1

120595119898+1

are the solution of 119902-NKdVHSCS ((10a) (10b) and (10c)) with119898 replaced by 119898 + 1

Proof (1) We firstly show that 119861119899 120601 120601119895 120595119895 (119895 = 1 119898)120601119898+1

120595119898+1

are the solution of (10b) and (10c)With the same proof as Theorem 1 119861119899 120601 120601

119895 120595119895(119895 =

1 119898) can be shown to be the solution of (10b) and (10c)

Advances in Mathematical Physics 5

Here we only need to show that 120601119898+1

120595119898+1

are also thesolution of (10b) and (10c) Consider

[119861119899120601119898+1] = minus1205731 (120591119896) [1198791119861119899119879minus1

1[11987911198921]]

= minus1205731 (120591119896) [1198791 [1198611198991198921]]

= minus120582119899

119898+11205731 (120591119896) [11987911198921] = 120582

119899

119898+1120601119898+1

(29)

Taking a proper solution 120595119898+1 of (10c) with 120583 = 120583119899

119898+1such

that Ω(ℎ1 120595119898+1) = minus1 then we get

[119861lowast

119899120595119898+1

] = (minus1)119898+1

1205781 (120591119896)

sdot [(119879minus1

1)lowast

119861lowast

119899119879lowast

1(

1

120579 (ℎ1))] = (minus1)

119898+11205781 (120591119896)

sdot [(119879minus1

1)lowast

119861lowast

119899119879lowast

1(minus

Ω (ℎ1 120595119898+1)

120579 (ℎ1))]

(30)

Noting that [(119879minus11

)lowast120595119898+1] = minus(Ω(ℎ1 120595119898+1))120579(ℎ1) we derive

from (30)

[119861lowast

119899120601119898+1

]

= (minus1)119898+1

1205781 (120591119896) [(119879minus1

1)lowast

119861lowast

119899119879lowast

1[(119879minus1

1)lowast

120595119898+1]]

= (minus1)119898+1

1205781 (120591119896) [(119879minus1

1)lowast

[119861lowast

119899120595119898+1]]

= 120583119899

119898+1120595119898+1

(31)

(2) We finally show that 119861119899 120601119895 120595119895 (119895 = 1 119898) 120601119898+1

120595119898+1

are the solution of (11a) and (11b) with119898 replaced by119898+

1 Evidently we only need to prove 119861119899 120601119895 120595119895 (119895 = 1 119898)120601119898+1

120595119898+1

satisfy (11b) that is120601120591119896

minus[(119861119896+sum119898+1

119895=1120601119895120597minus1

119902120595119895)120601] =

0Noting that 120601 = [1198791120601] rArr 120601

120591119896

= [1198791120601]120591119896

= [1198791120591119896

120601] +

[1198791120601120591119896

] we have

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= [

[

(1198791120591119896

+ 1198791119861119896

minus 1198611198961198791 +

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791)120601]

]

(32a)

From (15) a direct computation leads to

1198791120591119896

+ 1198791119861119896 minus 1198611198961198791 +

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= 0 (32b)

Noticing that119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

minus 120601119898+1

120597minus1

119902120595119898+1

1198791

(33a)

then forall119895 = 1 119898 we obtain by the tedious computation

1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791

= 120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1)

(33b)

Substituting (33b) into (33a) we get

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

minus 120601119898+1

120597minus1

119902120595119898+1

120597119902 + 120601119898+1

120597minus1

119902120595119898+1

1205721

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

minus (minus1)119898+1

1205781 (120591119896) 120601119898+1120597minus1

119902(120597119902

1

ℎ1

minus [120597119902

1

ℎ1

])

+ 120601119898+1

120597minus1

119902120595119898+1

1205721

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

+ (minus1)119898+1

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(33c)

In addition since1198911 1198921 are the solutions of (11a) and (11b) wehave

1198911120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)1198911

]

]

1198921120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)1198921

]

]

(34)

hence

ℎ1120591119896

= 1198911120591119896

+ 1198871 (120591119896) 1198921120591119896

+ 1198871 (120591119896)120591119896

1198921

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (120591119896)120591119896

1198921 997904rArr

[

[

119898

sum

119895=1

(120601119895120597minus1

119902120595119895) ℎ1

]

]

= ℎ1120591119896

minus [119861119896ℎ1] minus 1198871 (120591119896)120591119896

1198921

(35)

6 Advances in Mathematical Physics

Noting [1198791ℎ1] = 119882119902[ℎ1 ℎ1]ℎ1 = 0 and differentiating bothsides of this equation with respect to 120591119896 lead to

[1198791120591119896

ℎ1] + [1198791ℎ1120591119896

] = 0 997904rArr

[1198791ℎ1120591119896

] = minus [1198791120591119896

ℎ1]

(36)

Rewriting (33c) leads to

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

sum119898

119895=1[1198791 (120601119895120597

minus1

119902120595119895) ℎ1]

ℎ1

+ (minus1)119898+1

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(37a)

Combining (32a) and (32b) and (35) and (36) we get

sum119898

119895=1[1198791 (120601119895120597

minus1

119902120595119895) ℎ1]

ℎ1

=

1198791 [sum119898

119895=1(120601119895120597minus1

119902120595119895) ℎ1]

ℎ1

=

[1198791ℎ1120591119896

] minus [1198791119861119896ℎ1] minus 1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[1198791120591

119896

ℎ1] + [1198791119861119896ℎ1]

ℎ1

minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[119861119896 [1198791ℎ1]]

ℎ1

+

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[119861119896 [1198791ℎ1]]

ℎ1

= 0

(37b)

Substituting (32b) (37a) and (37b) into (32a) we have

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

+ (minus1)(119898+1)

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(38)

Noting 1198871(120591119896)120591119896

= (minus1)119898+1

1205731(120591119896)1205781(120591119896) we immediately getfrom (38)

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= 0 (39)

This completes the proof

Theorem 4 (the 119873-times repeated nonauto DBT) Given119861119899 1206011 120601119898 1205951 120595119898 are the solution for 119902-NKdVHSCS

((10a) (10b) and (10c)) 1198911 119891119873 1198921 119892119873 are inde-pendent eigenfunctions of (11a) and (11b) with 120582 =

120582119899

119898+1 120582

119899

119898+119873 119887119894(120591119896) 119894 = 1 119873 are functions of 120591119896 such

that 119887119894(120591119896)120591119896

= (minus1)119898+119873

120573119894(120591119896)120578119894(120591119896)Denote ℎ119894 = 119891119894 + 119887119894(120591119896)119892119894 The 119873-times repeated DBT is

defined by

119861(119873)

119899= 119879119873119861119899119879

minus1

119873= 120597119899

119902+ V(119873)119899minus1

120597119899minus1

119902+ sdot sdot sdot + V(119873)

1120597119902

+ V(119873)0

(40a)

120601(119873)

= [119879119873120601] =119882119902 [ℎ1 ℎ2 ℎ119873 120601]

119882119902 [ℎ1 ℎ2 ℎ119873] (40b)

120601(119873)

119895= [119879119873120601119895] =

119882119902 [ℎ1 ℎ2 ℎ119873 120601119895]

119882119902 [ℎ1 ℎ2 ℎ119873] (40c)

120595(119873)

119895= [(119879

minus1

119873)lowast

120595119895] = minus120579 (119866119902 [ℎ1 ℎ2 ℎ119873 120595119895])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

119895 = 1 119898

(40d)

120601(119873)

119898+119894= minus120573119894 (120591119904) [119879119873119892119894] (40e)

120595(119873)

119898+119894= (minus1)

119898+119894120578119894 (120591119896)

sdot120579 (119882119902 [ℎ1 ℎ119894minus1 ℎ119894+1 ℎ119873])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

119894 = 1 119873

(40f)

where

119879119873 =1

119882119902 [ℎ1 ℎ2 ℎ119873]

sdot

[[[[[[[

[

ℎ1 ℎ2 sdot sdot sdot ℎ119873 1

[120597119902ℎ1] [120597119902ℎ2] sdot sdot sdot [120597119902ℎ119873] 120597119902

[120597119873

119902ℎ1] [120597

119873

119902ℎ2] sdot sdot sdot [120597

119873

119902ℎ119873] 120597

119873

119902

]]]]]]]

]

119866119902 [ℎ1 ℎ2 ℎ119873]

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1 ℎ2 sdot sdot sdot ℎ119873

[120597119873minus2

119902ℎ1] [120597

119873minus2

119902ℎ2] sdot sdot sdot [120597

119873minus2

119902ℎ119873]

[120597minus1

119902ℎ1120595119895] [120597

minus1

119902ℎ2120595119895] sdot sdot sdot [120597

minus1

119902ℎ119873120595119895]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879119873 = 119863119873119863119873minus1 sdot sdot sdot 1198631

119863119896 = (120597119902 minus 120572(119896minus1)

119896)

Advances in Mathematical Physics 7

120572(119896)

119894=

[120597119902ℎ(119896)

119894]

ℎ(119896)

119894

ℎ(119896)

119894= [119879119896ℎ119894] 119896 = 0 1 119873 minus 1

(41)

then 119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) satisfy (10b) and (10c)

and (11a) and (11b) with 119898 replaced by 119898 + 119873 hence119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

are the solution of 119902-NKdVHSCS ((10a) (10b) and (10c)) with119898 replaced by 119898 + 119873

Proof With the same method as Theorem 3 we can showthat 120601

(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (10b) (10c) and (11a) Here we only need to show119861(119873)

119899 120601(119873)

120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 =

1 119873) satisfy (11b) Next we will show it by themathematical induction method Theorem 3 indicates119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (11b) in the case of 119873 = 1Provided that119861(119873)

119897 120601(119897)

119895 120595(119897)

119895 120601(119897)

119898+119894 120595(119897)

119898+119894satisfy (11b) for 119897 le

119873 minus 1

120601(119897)

120591119896

= [

[

(119861(119897)

119896+

119898+119897

sum

119895=1

120601(119897)

119895120597minus1

119902120595(119897)

119895)120601(119897)]

]

119861(119897)

119896= (119861(119897)

119899)119896119899

ge0

(42a)

119887119895 (120591119896)120591119896

= (minus1)(119897+119894)

120573119894 (120591119896) 120578119894 (120591119896)

119897 = 1 119873 minus 1

(42b)

Noticing that 120601(119873) = [119863119873120601(119873minus1)

] then when 119897 = 119873 we have

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= [119863119873120601(119873minus1)

]120591119896

minus [

[

(119861(119873)

119896119863119873

+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

= [

[

(119863119873120591119896

+ 119863119873119861(119873minus1)

119896minus 119861(119873)

119896119863119873

+

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895

minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

(43)

simplifying sum119898+119873minus1

119895=1119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

sum119898+119873

119895=1120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873 leads to

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873119863119873

(44a)

From (40f) we obtain

120595(119873)

119898+119873= (minus1)

119898+119873120578119873 (120591119896)

120579 (119882119902 [ℎ1 ℎ2 ℎ119894minus1])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 ([119879119873minus1ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 (ℎ(119873minus1)

119873)

(44b)

Substituting (44b) into (44a) yields

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873

ℎ(119873minus1)

119873

+ (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902[120597119902

1

ℎ(119873minus1)

119873

]

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902

[120597119902ℎ(119873minus1)

119873]

120579 (ℎ(119873minus1)

119873) ℎ(119873minus1)

119873

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

+ (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

(45)

8 Advances in Mathematical Physics

From (37a) for one DBT 119863119873 we have

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

=1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)

sdot ℎ(119873minus1)

119873]

]

]

]

(46)

Note that ℎ(119873minus1)119873

satisfies

ℎ(119873minus1)

119873120591119896

= [

[

(119861(119873minus1)

119896+

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

+ 119887119873 (120591119896)120591119896

119892(119873minus1)

119873997904rArr

[

[

(

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

= ℎ(119873minus1)

119873120591119896

minus [119861(119873minus1)

119896ℎ(119873minus1)

119873] minus 119887119873 (120591119896)120591

119896

119892(119873minus1)

119873

(47a)

and that

[119879119873ℎ119873] = [119863119873 [119879119873minus1ℎ119873]] = [119863119873ℎ(119873minus1)

119873] = 0 (47b)

Differentiating both sides of (47b) with respect to 120591119896 yields

[119863119873120591119896

ℎ(119873minus1)

119873] + [119863119873ℎ

(119873minus1)

119873120591119896

] = 0 997904rArr

[119863119873ℎ(119873minus1)

119873120591119896

] = minus [119863119873120591119896

ℎ(119873minus1)

119873]

(48)

we obtain

1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895) ℎ(119873minus1)

119873]

]

]

]

= minus[119861(119873minus1)

119896[119863119873ℎ

(119873minus1)

119873]]

ℎ(119873minus1)

119873

= 0

(49)

Combining (43) (45) (46) and (49) we get

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

minus

119887119873 (120591119896)120591119896

[119879119873119892119873]

ℎ(119873minus1)

119873

= 0

(50)

This completes the proof

5 Soliton Solution of 119902-KdVHSCS

It is known that KdV equation is the first nontrivial equationof the KdV hierarchy However the first nontrivial equationof 119902-KdVHSCS is not the 119902-KdVESCS but (12a) (12b) (12c)(12d) and (12e) In this section we aim to construct thesoliton solution to (12a) (12b) (12c) (12d) and (12e) In orderto get the soliton solution of (12a) (12b) (12c) (12d) and(12e) the following proposition is firstly presented

Proposition 5 Let 1198911 1198921 be two independent wave functionsof (12e) ℎ1 equiv 1198911 + 1198871(1205911)1198921 under the nonauto DBT and thetransformed coefficients are given by

V1 minus V1 = 119909 (119902 minus 1) (V0 minus V0) (51)

where

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(52)

Proof It was shown in [7] that formula (51) holds for (12a)(12b) (12c) (12d) and (12e) and that

V0 minus V0 = [120597119902 (V1 + 1205721 + 120579 (1205721))] (53)

Noting that ℎ1 = 1198911 + 1198871(1205911)1198921 (1198612)12

ge0= 1198611 = 120597119902 + 1199060 then

we have

ℎ11205911

= [

[

((1198612)12

ge0+

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (1205911)1205911

1198921

= [120597119902ℎ1] + 1199060ℎ1 +

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

(54)

From (54) we get

1199060 =ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)

minus 1198871 (1205911)1205911

1198921

ℎ1

(55)

Noticing that (12c) implies

V1 = 120579 (1199060) + 1199060 (56)

Advances in Mathematical Physics 9

we have

V0 minus V0 = [

[

120597119902(120579(ℎ11205911

minus [120597119902ℎ1]

ℎ1

) +ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

) +[120597119902ℎ1]

ℎ1

+ 120579([120597119902ℎ1]

ℎ1

))]

]

= [

[

120597119902 (120579(ℎ11205911

ℎ1

)) +ℎ11205911

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

minus [

[

120597119902(120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))

+ (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)))]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)] + (119902120579 + 1)

sdot [

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

(57)

Next we consider

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

=

119898

sum

119895=1

(120579(120601119895

ℎ1

) [120597119902Ω(120595119895 ℎ1)]

+ [120597119902

120601119895

ℎ1

]Ω (120595119895 ℎ1)) =

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1

120579 (ℎ1)

+[120597119902120601119895] ℎ1 minus 120601119895 [120597119902ℎ1]

120579 (ℎ1) ℎ1

Ω(120595119895 ℎ1)) =1

120579 (ℎ1)

sdot

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1 + ([120597119902120601119895] minus 1205721120601119895)Ω (120595119895 ℎ1))

=ℎ1

120579 (ℎ1)(

119898

sum

119895=1

120579 (120601119895) 120595119895 + 120601119895

Ω(120595119895 ℎ1)

ℎ1

)

(58)

Noting (37b) we can immediately derive

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

= 0 (59)

Hence we obtain from (57)

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(60)

This completes the proof

Next we will start from the trivial solution to (12a) (12b)(12c) (12d) and (12e) without sources that is V0 = V1 = 0and useTheorem 3 and Proposition 5 to construct one solitonsolution to (12a) (12b) (12c) (12d) and (12e) with 119898 = 1When V0 = V1 = 0 then 1198612 = 120597

2

119902 hence the wave functions

1198911 1198921 of Lax operator 1198612 = 1205972

119902satisfy

[1205972

119902120593] = 120582

2

1120593

1206011205911

= [120597119902120593]

(61)

We take the solution 1198911 1198921 of system (61) as follows

1198911 = 119864119902 (1199011119909) exp (11990111205911)

1198921 = 119864119902 (minus1199011119909) exp (minus11990111205911)

(62)

where 119864119902(119909) denotes the 119902-exponential function satisfying

[120597119902119864119902 (1199011119909)] = 1199011119864119902 (1199011119909) (63)

with an equivalent form

119864119902 (119909) =

infin

sum

119896=0

1

[119896]119902119909119896 (64)

10 Advances in Mathematical Physics

Noting ℎ1 equiv 1198911 + 1198871(1205911)1198921 where 1198911 1198921 are defined by (62)we get from (51) and (52)

V0

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

119909 (119902 minus 1) ℎ11205792 (ℎ1)

(65a)

V1

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

ℎ11205792 (ℎ1)

(65b)

In addition by Theorem 3 we obtain

1206011= minus1205731 (1205911)

ℎ1 [1205971199021198921] minus 1198921 [120597119902ℎ1]

ℎ21

(65c)

1205951= minus

1205781 (1205911)

1205791 (ℎ1) (65d)

where 1198871(1205911) 1205731(1205911) and 1205781(1205911) satisfy

1198871 (1205911)1205911

= minus120573119902 (1205911) 1205781 (1205911) (65e)

Then (65a) (65b) (65c) (65d) and (65e) present one solitonsolution of (12a) (12b) (12c) (12d) and (12e) with 119898 =

1 In particular when 1198871(1205911) = 119888 where 119888 is an arbitraryconstant (65a) (65b) (65c) (65d) and (65e) can be reducedto one soliton solution to the first nontrivial equation of the119902-KdV hierarchy [7] Certainly we also use Theorem 4 andProposition 5 to construct the multisoliton solution to (12a)(12b) (12c) (12d) and (12e) But owing to the complexity ofthe computation we omit it here

6 Summary

As 119899-reduction of the extended 119902-deformed KP hierarchy 119902-NKdVHSCS is explored in this paper Two kinds of DBTsare constructed and the soliton solution to the first nontrivialequation of 119902-KdVHSCS is also obtained We find that oneof the DBTs provides a nonauto Backlund transformation forthe two 119902-NKdVESCSwith different degree which enables usto obtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH Noting that we only investigate DBTand solution of 119902-NKdVESCS other integrable structureswill be studied in our forthcoming paper such as infiniteconservation law tau function and Hamiltonian structure

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant nos 11201178 and 11171175) FujianNational Science Foundation (Grant no 2012J01013) Fujian

Higher College Special Project of Scientfic Research (Grantno JK2012025) Fujian provincial visiting scholar programand the Scientific Research Foundation of Jimei UniversityChina

References

[1] A Klimyk andK Schmudgen ldquoq-calculusrdquo inQuantumGroupsand their Represntaions pp 37ndash52 Springer Berlin Germany1997

[2] Z YWuDH Zhang andQR Zheng ldquoQuantumdeformationof KdV hierarchies and their exact solutions 119902-deformedsolitonsrdquo Journal of Physics A Mathematical and General vol27 no 15 pp 5307ndash5312 1994

[3] E Frenkel and N Reshetikhin ldquoQuantum affine algebras anddeformations of the Virasoro and 119882-algebrasrdquo Communica-tions in Mathematical Physics vol 178 no 1 pp 237ndash264 1996

[4] E Frenkel ldquoDeformations of the KdV hierarchy and relatedsoliton equationsrdquo International Mathematics Research Noticesno 2 pp 55ndash76 1996

[5] L Haine and P Iliev ldquoThe bispectral property of a 119902-deformation of the Schur polynomials and the 119902-KdV hierar-chyrdquo Journal of Physics A Mathematical and General vol 30no 20 pp 7217ndash7227 1997

[6] M Adler E Horozov and P vanMoerbeke ldquoThe solution to the119902-KdV equationrdquo Physics Letters A vol 242 no 3 pp 139ndash1511998

[7] M-H Tu J-C Shaw and C-R Lee ldquoOn DarbouxndashBacklundtransformations for the 119902-deformed Korteweg-de Vries hierar-chyrdquo Letters in Mathematical Physics vol 49 no 1 pp 33ndash451999

[8] M-H Tu and C-R Lee ldquoOn the 119902-deformed modifiedKorteweg-de Vries hierarchyrdquo Physics Letters A vol 266 no2-3 pp 155ndash159 2000

[9] J Mas and M Seco ldquoThe algebra of q-pseudodifferentialsymbols and 119882

(119873)

119870119875-algebrardquo Journal of Mathematical Physics

vol 37 pp 6510ndash6529 1996[10] P Iliev ldquoTau function solutions to a 119902-deformation of the KP

hierarchyrdquo Letters in Mathematical Physics vol 44 no 3 pp187ndash200 1998

[11] P Iliev ldquo119902-KP hierarchy bispectrality and Calogero-Mosersystemsrdquo Journal of Geometry and Physics vol 35 no 2-3 pp157ndash182 2000

[12] M-H Tu ldquo119902-deformedKP hierarchy its additional symmetriesand infinitesimal Backlund transformationsrdquo Letters in Mathe-matical Physics vol 49 no 2 pp 95ndash103 1999

[13] J S He Y H Li and Y Cheng ldquo119902-deformed KP hierarchy and119902-deformed constrained KP hierarchyrdquo Symmetry Integrabilityand Geometry Methods and Applications vol 2 no 60 p 322006

[14] J S He Y H Li and Y Cheng ldquo119902-deformed Gelfand-Dickeyhierarchy and the determinant representation of its gaugetransformationrdquo Chinese Annals of Mathematics A vol 25 no3 pp 373ndash382 2004

[15] K L Tian J S He Y C Su and Y Cheng ldquoString equations ofthe 119902-KP hierarchyrdquo Chinese Annals of Mathematics B vol 32no 6 pp 895ndash904 2011

[16] R L Lin X J Liu and Y B Zeng ldquoA new extended 119902-deformedKP hierarchyrdquo Journal of Nonlinear Mathematical Physics vol15 no 3 pp 333ndash347 2008

Advances in Mathematical Physics 11

[17] R L Lin H Peng and M Manas ldquoThe 119902-deformed mKPhierarchywith self-consistent sourcesWronskian solutions andsolitonsrdquo Journal of Physics A Mathematical and Theoreticalvol 43 Article ID 434022 2010

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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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

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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

2 Advances in Mathematical Physics

The 119902-derivative operator 120597119902 and 119902-shift operator 120579 aredefined by

[120597119902119891 (119909)] =119891 (119902119909) minus 119891 (119909)

119909 (119902 minus 1) (1)

120579 (119891 (119909)) = 119891 (119902119909) (2)

In this paper we introduce twonotations [119875119891] and119875∘119891 = 119875119891in which 119875 is a 119902-pseudo-differential operator (119902-PDO) givenby

119875 =

119899

sum

119894=minusinfin

119901119894120597119894

119902 (3)

[119875119891] denotes 119875 acting on the function 119891 while 119875119891 indicatesthe multiplication of 119875 and 119891 that is 120597119902119891 = 120579(119891)120597119902 + [120597119902119891]

It can be easily shown from (1) that when 119902 rarr 1 120597119902

reduces to the ordinary differential operator 120597119909 and that 120579 and120597119902 do not commute but satisfy

[120597119902120579119896(119891)] = 119902

119896120579119896[120597119902119891] 119896 isin 119885 (4)

Let 120597minus1119902

be the formal inverse of 120597119902 such as 120597119902120597minus1

119902119891 = 120597

minus1

119902120597119902119891 =

119891 In general the 119902-deformed Leibnitz rule holds

120597119899

119902119891 = sum

119896ge0

(119899

119896)

119902

120579119899minus119896

(120597119896

119902119891) 120597119899minus119896

119902 119899 isin 119885 (5)

where 119902-number and 119902-binomial are defined by

(119899)119902 =119902119899minus 1

119902 minus 1

(119899

119896)

119902

=(119899)119902 (119899 minus 1)119902 sdot sdot sdot (119899 minus 119896 + 1)119902

(1)119902 (2)119902 sdot sdot sdot (119896)119902

(119899

0)

119902

= 1

(6)

For a 119902-PDO 119875 = sum119899

119894=minusinfin119901119894120597119894

119902 we separate 119875 into the

differential part 119875+ = sum119899

119894=0119901119894120597119894

119902and the integral part 119875minus =

sum119894leminus1

119901119894120597119894

119902 The conjugate operation 119875

lowast is given by

119875lowast

=

119899

sum

119894=minusinfin

(120597lowast

119902)119894

119901119894 (7)

where 120597lowast

119902= minus120597119902120579

minus1= minus(1119902)1205971119902 (120597

minus1

119902)lowast

= (120597lowast

119902)minus1

= minus120579120597minus1

119902

The 119902-exponential function 119864119902(119909) is defined as

119864119902 (119909) = exp(

infin

sum

119896=1

(1 minus 119902)119896

119896 (1 minus 119902119896)119909119896) (8)

satisfying [120597119896

119902119864119902(119909119911)] = 119911

119896119864119902(119909119911) 119896 isin 119885

The extended 119902-KPH was given by [16]

119871 119905119899

= [119861119899 119871] 119861119899 = (119871119899)ge0

(9a)

119871120591119896

= [

[

119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895 119871

]

]

119861119896 = (119871119896)ge0

119896 = 119899 (9b)

120601119895119905119899

= [119861119899120601119895]

120595119895119905119899

= minus [119861lowast

119899120595119895]

119895 = 1 119898

(9c)

where 119871 = 120597119902 + 1199060 + 1199061120597minus1

119902+ 1199062120597

minus2

119902+ sdot sdot sdot and the coefficients

119906119894 (119894 = 0 1 ) are the functions of 119905 = (119909 1199051 )The commutativity of (9a) (9b) and (9c) leads to the

zero-curvature representation of 119902-KPH ((9a) (9b) and(9c)) As the 119899-reduction of the extended 119902-KPH the 119902-NKdVHSCS is defined as follows [16]

119861119899120591119896

= [

[

119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895 119861119899

]

]

119861119896 = (119861119899)119896119899

ge0(10a)

[119861119899120601119895] = 120582119899

119895120601119895 (10b)

[119861lowast

119899120595119895] = 120583

119899

119895120595119895 119895 = 1 119898 (10c)

Under (10b) and (10c) the Lax representation for (10a) is

[119861119899120593] = 120582120593 (11a)

120593120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)120593]

]

(11b)

We find that when 119898 = 0 119902-NKdVHSCS ((10a) (10b) and(10c) and (11a) and (11b)) can be reduced to the 119902-NKdVHand its related Lax representation respectively In additionwhen 119899 = 2 119896 = 1 ((10a) (10b) and (10c)) becomes the firstnontrivial soliton equation of 119902-KdVHSCS given by

V11205911

+ 1198901 +

119898

sum

119895=1

1198921198951 = 0 (12a)

V01205913

+ 1198900 +

119898

sum

119895=1

1198921198950 = 0 (12b)

120579 (V1) minus V1 + 1199060 minus 1205792(1199060) = 0 (12c)

[(1205972

119902+ V1120597119902 + V0) 120593119895] minus 120582

2

119895120593119895 = 0 (12d)

[(1205972

119902+ V1120597119902 + V0)

lowast

120595119895] minus 1205832

119895120595119895 = 0 119895 = 1 119898 (12e)

3 The Auto Darboux-BaumlcklundTransformation (DBT) for 119902-NKdVHSCS

In this section we will focus on the construction of auto DBTfor 119902-NKdVHSCS

Advances in Mathematical Physics 3

Theorem 1 Assume 119861119899 120601119895 120595119895 (119895 = 1 119898) be the solutionof 119902-NKdVHSCS ((10a) (10b) and (10c)) and ℎ1 satisfies (11a)and (11b) with 120582 = 120582

119899

1 the DBT is defined by

119861119899 = 1198791119861119899119879minus1

1= 120597119899

119902+ V119899minus1120597

119899minus1

119902+ sdot sdot sdot + V1120597119902 + V0 (13a)

120601 = [1198791120601] =119882119902 [ℎ1 120601]

ℎ1

(13b)

120601119895= [1198791120601119895] =

119882119902 [ℎ1 120601119895]

ℎ1

(13c)

120595119895= [(119879

minus1

1)lowast

120595119895] = minus120579 (Ω (ℎ1 120595119895))

120579 (ℎ1)

119895 = 1 119898

(13d)

Then 119861119899 120601 120601119895 120595119895 (119895 = 1 119898) satisfy (10b) and (10c) and(11a) and (11b) and hence are the solution of 119902-NKdVHSCS((10a) (10b) and (10c)) where 1198791 = 120579(ℎ1)120597119902ℎ

minus1

1= 120597119902 minus 1205721

1205721 = [120597119902ℎ1]ℎ1 and 119882119902[ℎ1 120601119895] and Ω119902[ℎ1 120595119895] are defined asfollows

119882119902 [1206011 120601] =

1003816100381610038161003816100381610038161003816100381610038161003816

ℎ1 120601

[120597119902ℎ1] [120597119902120601]

1003816100381610038161003816100381610038161003816100381610038161003816

Ω (ℎ1 120595119895) = [120597minus1

119902ℎ1120595119895]

(14)

Remark 2 Here it should be pointed out that the formulaholds

119861119896 = 1198791119861119896119879minus1

1+ 1198791120591

119896

119879minus1

1

119861119896 = (119861119899)119896119899

ge0

119861119896 = (119861119899)119896119899

ge0

(15)

where the gauge operator 1198791 is defined above The proof hasbeen given in [7]

Proof (1)We firstly show that119861119899 120601119895120595119895 (119895 = 1 119898) satisfy(10b) and (10c)

Noting that 119861119899 120601119895 120595119895 (119895 = 1 119898) are the solution of(10a) (10b) and (10c) we have

[119861119899120601119895] = 120582119899

119895120601119895

[119861lowast

119899120595119895] = 120583

119899

119895120595119895

(16)

Hence

[119861119899120601119895] minus 120582119899

119895120601119895= [1198791119861119899119879

minus1

1[1198791120601119895]] minus 120582

119899

119895120601119895

= [1198791 [119861119899120601119895]] minus 120582119899

119895120601119895

= 120582119899

119895([1198791120601119895] minus 120601

119895) = 0

[119861lowast

119899120595119895] minus 120583119899

119895120595119895= [(119879

minus1

1)lowast

119861lowast

119899119879lowast

1[(119879minus1

1)lowast

120595119895]]

minus 120583119899

119895120595119895= [(119879

minus1

1)lowast

119861lowast

119899120595119895] minus 120583

119899

119895120595119895

= 0

(17)

(2)Wefinally show that119861119899120601120601119895120595119895 satisfy (11a) and (11b)Since the proof of (11a) is the same as the case (1) we only

need to verify that 119861119899 120601 120601119895 120595119895 satisfy (11b) that is

120601120591119896

minus [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= 0 119861119896 = (119861119899)119896119899

ge0 (18)

Noting that 120601120591119896

= [1198791120601]120591119896

= [1198791120591119896

120601] + [1198791120601120591119896

] and 120601120591119896

=

[(119861119896 + sum119898

119895=1120601119895120597minus1

119902120595119895)120601] we get

120601120591119896

minus [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= [1198791120591119896

120601] + [1198791120601120591119896

]

minus [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)[1198791120601]]

]

= [

[

(1198791120591119896

+ 1198791119861119896

minus 1198611198961198791 +

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898

sum

119895=1

120601119895120597minus1

1199021205951198951198791)120601]

]

(19)

According to Remark 2 we have

1198791120591119896

+ 1198791119861119896 minus 1198611198961198791 = 0 (20)

Next we prove

119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791) = 0 (21)

Since 1198791 = 120597119902 minus 1205721 and 120597119902120601119895 = 120579(120601119895)120597119902 + [120597119902120601119895] then forall119895 weobtain by the tedious computation

1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791

= (120597119902 minus 1205721) 120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

119902120595119895120597119902 + 120601

119895120597minus1

1199021205951198951205721

= 120579 (120601119895) 120595119895 + 120601119895

Ω(ℎ1 120595119895)

ℎ1

(22a)

4 Advances in Mathematical Physics

In addition we also have

120579 (120601119895) 120595119895 + 120601119895

Ω(ℎ1 120595119895)

ℎ1

=[120579 (120601119895) 120597119902Ω(1206011 120595119895)] + [1198791ℎ1]Ω (ℎ1 120595119895)

ℎ1

=[120597119902120601119895Ω(ℎ1 120595119895)] minus [120597119902120601119895]Ω (ℎ1 120595119895) + ([120597119902120601119895] minus 1205721120601119895)Ω (ℎ1 120595119895)

ℎ1

=[(120597119902 minus 1205721) 120601119895Ω(ℎ1 120595119895)]

ℎ1

=

[1198791 [(120601119895120597minus1

119902120595119895) ℎ1]]

ℎ1

(22b)

Substituting (22b) into (22a) leads to119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

=1

ℎ1

[

[

1198791[

[

119898

sum

119895=1

(120601119895120597minus1

119902120595119895) ℎ1

]

]

]

]

(23)

Since ℎ1 is the solution of (11a) and (11b) with 120582 = 120582119899

1 we have

ℎ1120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

997904rArr

[

[

(

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

= ℎ1120591119896

minus [119861119896ℎ1]

(24)

Moreover by the property of determinant we have

[1198791ℎ1] =119882119902 [ℎ1 ℎ1]

ℎ1

= 0 (25)

Differentiating both sides of (25) with respect to 120591119896 yields

[1198791120591119896

1206011] + [11987911206011120591119896

] = 0 997904rArr

[11987911206011120591119896

] = minus [1198791120591119896

1206011]

(26)

From (23) (24) and (26) we have119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

= minus1

1206011

[(1198791120591119896

+ 1198791119861119896) 1206011] = minus1

1206011

[119861119896 [11987911206011]]

= 0

(27)

This completes the proof

Obviously Theorem 1 provides an auto DBT for 119902-NKdVHSCS ((10a) (10b) and (10c)) However thisDBTdoesnot enable us to obtain the new solution of 119902-NKdVHSCS((10a) (10b) and (10c)) So we have to seek for nonauto DBTsbetween the two 119902-NKdVHSCS ((10a) (10b) and (10c)) withdifferent degrees of sources

4 The Nonauto DBTs of 119902-NKdVHSCS

In this section we will construct the nonauto DBTs of 119902-NKdVHSCS ((10a) (10b) and (10c)) which enables us toobtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH

Theorem 3 Given 119861119899 120601119895 120595119895 (119895 = 1 119898) the solution for119902-NKdVHSCS ((10a) (10b) and (10c)) let 1198911 1198921 equiv 120601119898+1 betwo independent eigenfunctions of (11a) and (11b) with 120582 =

120582119899

119898+1 Let 1198871(120591119896) be a function of 120591119896 such that 1198871(120591119896)120591

119896

=

(minus1)119898+1

1205731(120591119896)1205781(120591119896) Denote ℎ1 = 1198911 + 1198871(120591119896)1198921The DBT is defined by

119861119899 = 1198791119861119899119879minus1

1= 120597119899

119902+ V119899minus1120597

119899minus1

119902+ sdot sdot sdot + V1120597119902 + V0 (28a)

120601 = [1198791120601] =119882119902 [ℎ1 120601]

ℎ1

(28b)

120601119895= [1198791120601119895] =

119882119902 [ℎ1 120601119895]

ℎ1

(28c)

120595119895= [(119879

minus1

1)lowast

120595119895] = minus120579 (Ω (ℎ1 120595119895))

120579 (ℎ1)

119895 = 1 119898

(28d)

120601119898+1

= minus1205731 (120591119896) [11987911198921] (28e)

120595119898+1

= (minus1)119898+1

1205781 (120591119896)1

120579 (ℎ1) (28f)

where1198791 = 120597119902minus1205721 1205721 = [120597119902ℎ1]ℎ1 and then 119861119899 120601 120601119895120595119895 (119895 =

1 119898) 120601119898+1

120595119898+1

satisfy (10b) and (11a) and (11b) with 119898

replaced by119898+1 hence 119861119899 120601119895 120595119895 (119895 = 1 119898) 120601119898+1

120595119898+1

are the solution of 119902-NKdVHSCS ((10a) (10b) and (10c)) with119898 replaced by 119898 + 1

Proof (1) We firstly show that 119861119899 120601 120601119895 120595119895 (119895 = 1 119898)120601119898+1

120595119898+1

are the solution of (10b) and (10c)With the same proof as Theorem 1 119861119899 120601 120601

119895 120595119895(119895 =

1 119898) can be shown to be the solution of (10b) and (10c)

Advances in Mathematical Physics 5

Here we only need to show that 120601119898+1

120595119898+1

are also thesolution of (10b) and (10c) Consider

[119861119899120601119898+1] = minus1205731 (120591119896) [1198791119861119899119879minus1

1[11987911198921]]

= minus1205731 (120591119896) [1198791 [1198611198991198921]]

= minus120582119899

119898+11205731 (120591119896) [11987911198921] = 120582

119899

119898+1120601119898+1

(29)

Taking a proper solution 120595119898+1 of (10c) with 120583 = 120583119899

119898+1such

that Ω(ℎ1 120595119898+1) = minus1 then we get

[119861lowast

119899120595119898+1

] = (minus1)119898+1

1205781 (120591119896)

sdot [(119879minus1

1)lowast

119861lowast

119899119879lowast

1(

1

120579 (ℎ1))] = (minus1)

119898+11205781 (120591119896)

sdot [(119879minus1

1)lowast

119861lowast

119899119879lowast

1(minus

Ω (ℎ1 120595119898+1)

120579 (ℎ1))]

(30)

Noting that [(119879minus11

)lowast120595119898+1] = minus(Ω(ℎ1 120595119898+1))120579(ℎ1) we derive

from (30)

[119861lowast

119899120601119898+1

]

= (minus1)119898+1

1205781 (120591119896) [(119879minus1

1)lowast

119861lowast

119899119879lowast

1[(119879minus1

1)lowast

120595119898+1]]

= (minus1)119898+1

1205781 (120591119896) [(119879minus1

1)lowast

[119861lowast

119899120595119898+1]]

= 120583119899

119898+1120595119898+1

(31)

(2) We finally show that 119861119899 120601119895 120595119895 (119895 = 1 119898) 120601119898+1

120595119898+1

are the solution of (11a) and (11b) with119898 replaced by119898+

1 Evidently we only need to prove 119861119899 120601119895 120595119895 (119895 = 1 119898)120601119898+1

120595119898+1

satisfy (11b) that is120601120591119896

minus[(119861119896+sum119898+1

119895=1120601119895120597minus1

119902120595119895)120601] =

0Noting that 120601 = [1198791120601] rArr 120601

120591119896

= [1198791120601]120591119896

= [1198791120591119896

120601] +

[1198791120601120591119896

] we have

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= [

[

(1198791120591119896

+ 1198791119861119896

minus 1198611198961198791 +

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791)120601]

]

(32a)

From (15) a direct computation leads to

1198791120591119896

+ 1198791119861119896 minus 1198611198961198791 +

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= 0 (32b)

Noticing that119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

minus 120601119898+1

120597minus1

119902120595119898+1

1198791

(33a)

then forall119895 = 1 119898 we obtain by the tedious computation

1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791

= 120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1)

(33b)

Substituting (33b) into (33a) we get

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

minus 120601119898+1

120597minus1

119902120595119898+1

120597119902 + 120601119898+1

120597minus1

119902120595119898+1

1205721

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

minus (minus1)119898+1

1205781 (120591119896) 120601119898+1120597minus1

119902(120597119902

1

ℎ1

minus [120597119902

1

ℎ1

])

+ 120601119898+1

120597minus1

119902120595119898+1

1205721

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

+ (minus1)119898+1

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(33c)

In addition since1198911 1198921 are the solutions of (11a) and (11b) wehave

1198911120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)1198911

]

]

1198921120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)1198921

]

]

(34)

hence

ℎ1120591119896

= 1198911120591119896

+ 1198871 (120591119896) 1198921120591119896

+ 1198871 (120591119896)120591119896

1198921

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (120591119896)120591119896

1198921 997904rArr

[

[

119898

sum

119895=1

(120601119895120597minus1

119902120595119895) ℎ1

]

]

= ℎ1120591119896

minus [119861119896ℎ1] minus 1198871 (120591119896)120591119896

1198921

(35)

6 Advances in Mathematical Physics

Noting [1198791ℎ1] = 119882119902[ℎ1 ℎ1]ℎ1 = 0 and differentiating bothsides of this equation with respect to 120591119896 lead to

[1198791120591119896

ℎ1] + [1198791ℎ1120591119896

] = 0 997904rArr

[1198791ℎ1120591119896

] = minus [1198791120591119896

ℎ1]

(36)

Rewriting (33c) leads to

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

sum119898

119895=1[1198791 (120601119895120597

minus1

119902120595119895) ℎ1]

ℎ1

+ (minus1)119898+1

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(37a)

Combining (32a) and (32b) and (35) and (36) we get

sum119898

119895=1[1198791 (120601119895120597

minus1

119902120595119895) ℎ1]

ℎ1

=

1198791 [sum119898

119895=1(120601119895120597minus1

119902120595119895) ℎ1]

ℎ1

=

[1198791ℎ1120591119896

] minus [1198791119861119896ℎ1] minus 1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[1198791120591

119896

ℎ1] + [1198791119861119896ℎ1]

ℎ1

minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[119861119896 [1198791ℎ1]]

ℎ1

+

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[119861119896 [1198791ℎ1]]

ℎ1

= 0

(37b)

Substituting (32b) (37a) and (37b) into (32a) we have

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

+ (minus1)(119898+1)

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(38)

Noting 1198871(120591119896)120591119896

= (minus1)119898+1

1205731(120591119896)1205781(120591119896) we immediately getfrom (38)

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= 0 (39)

This completes the proof

Theorem 4 (the 119873-times repeated nonauto DBT) Given119861119899 1206011 120601119898 1205951 120595119898 are the solution for 119902-NKdVHSCS

((10a) (10b) and (10c)) 1198911 119891119873 1198921 119892119873 are inde-pendent eigenfunctions of (11a) and (11b) with 120582 =

120582119899

119898+1 120582

119899

119898+119873 119887119894(120591119896) 119894 = 1 119873 are functions of 120591119896 such

that 119887119894(120591119896)120591119896

= (minus1)119898+119873

120573119894(120591119896)120578119894(120591119896)Denote ℎ119894 = 119891119894 + 119887119894(120591119896)119892119894 The 119873-times repeated DBT is

defined by

119861(119873)

119899= 119879119873119861119899119879

minus1

119873= 120597119899

119902+ V(119873)119899minus1

120597119899minus1

119902+ sdot sdot sdot + V(119873)

1120597119902

+ V(119873)0

(40a)

120601(119873)

= [119879119873120601] =119882119902 [ℎ1 ℎ2 ℎ119873 120601]

119882119902 [ℎ1 ℎ2 ℎ119873] (40b)

120601(119873)

119895= [119879119873120601119895] =

119882119902 [ℎ1 ℎ2 ℎ119873 120601119895]

119882119902 [ℎ1 ℎ2 ℎ119873] (40c)

120595(119873)

119895= [(119879

minus1

119873)lowast

120595119895] = minus120579 (119866119902 [ℎ1 ℎ2 ℎ119873 120595119895])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

119895 = 1 119898

(40d)

120601(119873)

119898+119894= minus120573119894 (120591119904) [119879119873119892119894] (40e)

120595(119873)

119898+119894= (minus1)

119898+119894120578119894 (120591119896)

sdot120579 (119882119902 [ℎ1 ℎ119894minus1 ℎ119894+1 ℎ119873])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

119894 = 1 119873

(40f)

where

119879119873 =1

119882119902 [ℎ1 ℎ2 ℎ119873]

sdot

[[[[[[[

[

ℎ1 ℎ2 sdot sdot sdot ℎ119873 1

[120597119902ℎ1] [120597119902ℎ2] sdot sdot sdot [120597119902ℎ119873] 120597119902

[120597119873

119902ℎ1] [120597

119873

119902ℎ2] sdot sdot sdot [120597

119873

119902ℎ119873] 120597

119873

119902

]]]]]]]

]

119866119902 [ℎ1 ℎ2 ℎ119873]

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1 ℎ2 sdot sdot sdot ℎ119873

[120597119873minus2

119902ℎ1] [120597

119873minus2

119902ℎ2] sdot sdot sdot [120597

119873minus2

119902ℎ119873]

[120597minus1

119902ℎ1120595119895] [120597

minus1

119902ℎ2120595119895] sdot sdot sdot [120597

minus1

119902ℎ119873120595119895]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879119873 = 119863119873119863119873minus1 sdot sdot sdot 1198631

119863119896 = (120597119902 minus 120572(119896minus1)

119896)

Advances in Mathematical Physics 7

120572(119896)

119894=

[120597119902ℎ(119896)

119894]

ℎ(119896)

119894

ℎ(119896)

119894= [119879119896ℎ119894] 119896 = 0 1 119873 minus 1

(41)

then 119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) satisfy (10b) and (10c)

and (11a) and (11b) with 119898 replaced by 119898 + 119873 hence119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

are the solution of 119902-NKdVHSCS ((10a) (10b) and (10c)) with119898 replaced by 119898 + 119873

Proof With the same method as Theorem 3 we can showthat 120601

(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (10b) (10c) and (11a) Here we only need to show119861(119873)

119899 120601(119873)

120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 =

1 119873) satisfy (11b) Next we will show it by themathematical induction method Theorem 3 indicates119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (11b) in the case of 119873 = 1Provided that119861(119873)

119897 120601(119897)

119895 120595(119897)

119895 120601(119897)

119898+119894 120595(119897)

119898+119894satisfy (11b) for 119897 le

119873 minus 1

120601(119897)

120591119896

= [

[

(119861(119897)

119896+

119898+119897

sum

119895=1

120601(119897)

119895120597minus1

119902120595(119897)

119895)120601(119897)]

]

119861(119897)

119896= (119861(119897)

119899)119896119899

ge0

(42a)

119887119895 (120591119896)120591119896

= (minus1)(119897+119894)

120573119894 (120591119896) 120578119894 (120591119896)

119897 = 1 119873 minus 1

(42b)

Noticing that 120601(119873) = [119863119873120601(119873minus1)

] then when 119897 = 119873 we have

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= [119863119873120601(119873minus1)

]120591119896

minus [

[

(119861(119873)

119896119863119873

+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

= [

[

(119863119873120591119896

+ 119863119873119861(119873minus1)

119896minus 119861(119873)

119896119863119873

+

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895

minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

(43)

simplifying sum119898+119873minus1

119895=1119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

sum119898+119873

119895=1120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873 leads to

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873119863119873

(44a)

From (40f) we obtain

120595(119873)

119898+119873= (minus1)

119898+119873120578119873 (120591119896)

120579 (119882119902 [ℎ1 ℎ2 ℎ119894minus1])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 ([119879119873minus1ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 (ℎ(119873minus1)

119873)

(44b)

Substituting (44b) into (44a) yields

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873

ℎ(119873minus1)

119873

+ (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902[120597119902

1

ℎ(119873minus1)

119873

]

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902

[120597119902ℎ(119873minus1)

119873]

120579 (ℎ(119873minus1)

119873) ℎ(119873minus1)

119873

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

+ (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

(45)

8 Advances in Mathematical Physics

From (37a) for one DBT 119863119873 we have

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

=1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)

sdot ℎ(119873minus1)

119873]

]

]

]

(46)

Note that ℎ(119873minus1)119873

satisfies

ℎ(119873minus1)

119873120591119896

= [

[

(119861(119873minus1)

119896+

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

+ 119887119873 (120591119896)120591119896

119892(119873minus1)

119873997904rArr

[

[

(

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

= ℎ(119873minus1)

119873120591119896

minus [119861(119873minus1)

119896ℎ(119873minus1)

119873] minus 119887119873 (120591119896)120591

119896

119892(119873minus1)

119873

(47a)

and that

[119879119873ℎ119873] = [119863119873 [119879119873minus1ℎ119873]] = [119863119873ℎ(119873minus1)

119873] = 0 (47b)

Differentiating both sides of (47b) with respect to 120591119896 yields

[119863119873120591119896

ℎ(119873minus1)

119873] + [119863119873ℎ

(119873minus1)

119873120591119896

] = 0 997904rArr

[119863119873ℎ(119873minus1)

119873120591119896

] = minus [119863119873120591119896

ℎ(119873minus1)

119873]

(48)

we obtain

1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895) ℎ(119873minus1)

119873]

]

]

]

= minus[119861(119873minus1)

119896[119863119873ℎ

(119873minus1)

119873]]

ℎ(119873minus1)

119873

= 0

(49)

Combining (43) (45) (46) and (49) we get

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

minus

119887119873 (120591119896)120591119896

[119879119873119892119873]

ℎ(119873minus1)

119873

= 0

(50)

This completes the proof

5 Soliton Solution of 119902-KdVHSCS

It is known that KdV equation is the first nontrivial equationof the KdV hierarchy However the first nontrivial equationof 119902-KdVHSCS is not the 119902-KdVESCS but (12a) (12b) (12c)(12d) and (12e) In this section we aim to construct thesoliton solution to (12a) (12b) (12c) (12d) and (12e) In orderto get the soliton solution of (12a) (12b) (12c) (12d) and(12e) the following proposition is firstly presented

Proposition 5 Let 1198911 1198921 be two independent wave functionsof (12e) ℎ1 equiv 1198911 + 1198871(1205911)1198921 under the nonauto DBT and thetransformed coefficients are given by

V1 minus V1 = 119909 (119902 minus 1) (V0 minus V0) (51)

where

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(52)

Proof It was shown in [7] that formula (51) holds for (12a)(12b) (12c) (12d) and (12e) and that

V0 minus V0 = [120597119902 (V1 + 1205721 + 120579 (1205721))] (53)

Noting that ℎ1 = 1198911 + 1198871(1205911)1198921 (1198612)12

ge0= 1198611 = 120597119902 + 1199060 then

we have

ℎ11205911

= [

[

((1198612)12

ge0+

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (1205911)1205911

1198921

= [120597119902ℎ1] + 1199060ℎ1 +

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

(54)

From (54) we get

1199060 =ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)

minus 1198871 (1205911)1205911

1198921

ℎ1

(55)

Noticing that (12c) implies

V1 = 120579 (1199060) + 1199060 (56)

Advances in Mathematical Physics 9

we have

V0 minus V0 = [

[

120597119902(120579(ℎ11205911

minus [120597119902ℎ1]

ℎ1

) +ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

) +[120597119902ℎ1]

ℎ1

+ 120579([120597119902ℎ1]

ℎ1

))]

]

= [

[

120597119902 (120579(ℎ11205911

ℎ1

)) +ℎ11205911

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

minus [

[

120597119902(120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))

+ (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)))]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)] + (119902120579 + 1)

sdot [

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

(57)

Next we consider

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

=

119898

sum

119895=1

(120579(120601119895

ℎ1

) [120597119902Ω(120595119895 ℎ1)]

+ [120597119902

120601119895

ℎ1

]Ω (120595119895 ℎ1)) =

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1

120579 (ℎ1)

+[120597119902120601119895] ℎ1 minus 120601119895 [120597119902ℎ1]

120579 (ℎ1) ℎ1

Ω(120595119895 ℎ1)) =1

120579 (ℎ1)

sdot

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1 + ([120597119902120601119895] minus 1205721120601119895)Ω (120595119895 ℎ1))

=ℎ1

120579 (ℎ1)(

119898

sum

119895=1

120579 (120601119895) 120595119895 + 120601119895

Ω(120595119895 ℎ1)

ℎ1

)

(58)

Noting (37b) we can immediately derive

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

= 0 (59)

Hence we obtain from (57)

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(60)

This completes the proof

Next we will start from the trivial solution to (12a) (12b)(12c) (12d) and (12e) without sources that is V0 = V1 = 0and useTheorem 3 and Proposition 5 to construct one solitonsolution to (12a) (12b) (12c) (12d) and (12e) with 119898 = 1When V0 = V1 = 0 then 1198612 = 120597

2

119902 hence the wave functions

1198911 1198921 of Lax operator 1198612 = 1205972

119902satisfy

[1205972

119902120593] = 120582

2

1120593

1206011205911

= [120597119902120593]

(61)

We take the solution 1198911 1198921 of system (61) as follows

1198911 = 119864119902 (1199011119909) exp (11990111205911)

1198921 = 119864119902 (minus1199011119909) exp (minus11990111205911)

(62)

where 119864119902(119909) denotes the 119902-exponential function satisfying

[120597119902119864119902 (1199011119909)] = 1199011119864119902 (1199011119909) (63)

with an equivalent form

119864119902 (119909) =

infin

sum

119896=0

1

[119896]119902119909119896 (64)

10 Advances in Mathematical Physics

Noting ℎ1 equiv 1198911 + 1198871(1205911)1198921 where 1198911 1198921 are defined by (62)we get from (51) and (52)

V0

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

119909 (119902 minus 1) ℎ11205792 (ℎ1)

(65a)

V1

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

ℎ11205792 (ℎ1)

(65b)

In addition by Theorem 3 we obtain

1206011= minus1205731 (1205911)

ℎ1 [1205971199021198921] minus 1198921 [120597119902ℎ1]

ℎ21

(65c)

1205951= minus

1205781 (1205911)

1205791 (ℎ1) (65d)

where 1198871(1205911) 1205731(1205911) and 1205781(1205911) satisfy

1198871 (1205911)1205911

= minus120573119902 (1205911) 1205781 (1205911) (65e)

Then (65a) (65b) (65c) (65d) and (65e) present one solitonsolution of (12a) (12b) (12c) (12d) and (12e) with 119898 =

1 In particular when 1198871(1205911) = 119888 where 119888 is an arbitraryconstant (65a) (65b) (65c) (65d) and (65e) can be reducedto one soliton solution to the first nontrivial equation of the119902-KdV hierarchy [7] Certainly we also use Theorem 4 andProposition 5 to construct the multisoliton solution to (12a)(12b) (12c) (12d) and (12e) But owing to the complexity ofthe computation we omit it here

6 Summary

As 119899-reduction of the extended 119902-deformed KP hierarchy 119902-NKdVHSCS is explored in this paper Two kinds of DBTsare constructed and the soliton solution to the first nontrivialequation of 119902-KdVHSCS is also obtained We find that oneof the DBTs provides a nonauto Backlund transformation forthe two 119902-NKdVESCSwith different degree which enables usto obtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH Noting that we only investigate DBTand solution of 119902-NKdVESCS other integrable structureswill be studied in our forthcoming paper such as infiniteconservation law tau function and Hamiltonian structure

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant nos 11201178 and 11171175) FujianNational Science Foundation (Grant no 2012J01013) Fujian

Higher College Special Project of Scientfic Research (Grantno JK2012025) Fujian provincial visiting scholar programand the Scientific Research Foundation of Jimei UniversityChina

References

[1] A Klimyk andK Schmudgen ldquoq-calculusrdquo inQuantumGroupsand their Represntaions pp 37ndash52 Springer Berlin Germany1997

[2] Z YWuDH Zhang andQR Zheng ldquoQuantumdeformationof KdV hierarchies and their exact solutions 119902-deformedsolitonsrdquo Journal of Physics A Mathematical and General vol27 no 15 pp 5307ndash5312 1994

[3] E Frenkel and N Reshetikhin ldquoQuantum affine algebras anddeformations of the Virasoro and 119882-algebrasrdquo Communica-tions in Mathematical Physics vol 178 no 1 pp 237ndash264 1996

[4] E Frenkel ldquoDeformations of the KdV hierarchy and relatedsoliton equationsrdquo International Mathematics Research Noticesno 2 pp 55ndash76 1996

[5] L Haine and P Iliev ldquoThe bispectral property of a 119902-deformation of the Schur polynomials and the 119902-KdV hierar-chyrdquo Journal of Physics A Mathematical and General vol 30no 20 pp 7217ndash7227 1997

[6] M Adler E Horozov and P vanMoerbeke ldquoThe solution to the119902-KdV equationrdquo Physics Letters A vol 242 no 3 pp 139ndash1511998

[7] M-H Tu J-C Shaw and C-R Lee ldquoOn DarbouxndashBacklundtransformations for the 119902-deformed Korteweg-de Vries hierar-chyrdquo Letters in Mathematical Physics vol 49 no 1 pp 33ndash451999

[8] M-H Tu and C-R Lee ldquoOn the 119902-deformed modifiedKorteweg-de Vries hierarchyrdquo Physics Letters A vol 266 no2-3 pp 155ndash159 2000

[9] J Mas and M Seco ldquoThe algebra of q-pseudodifferentialsymbols and 119882

(119873)

119870119875-algebrardquo Journal of Mathematical Physics

vol 37 pp 6510ndash6529 1996[10] P Iliev ldquoTau function solutions to a 119902-deformation of the KP

hierarchyrdquo Letters in Mathematical Physics vol 44 no 3 pp187ndash200 1998

[11] P Iliev ldquo119902-KP hierarchy bispectrality and Calogero-Mosersystemsrdquo Journal of Geometry and Physics vol 35 no 2-3 pp157ndash182 2000

[12] M-H Tu ldquo119902-deformedKP hierarchy its additional symmetriesand infinitesimal Backlund transformationsrdquo Letters in Mathe-matical Physics vol 49 no 2 pp 95ndash103 1999

[13] J S He Y H Li and Y Cheng ldquo119902-deformed KP hierarchy and119902-deformed constrained KP hierarchyrdquo Symmetry Integrabilityand Geometry Methods and Applications vol 2 no 60 p 322006

[14] J S He Y H Li and Y Cheng ldquo119902-deformed Gelfand-Dickeyhierarchy and the determinant representation of its gaugetransformationrdquo Chinese Annals of Mathematics A vol 25 no3 pp 373ndash382 2004

[15] K L Tian J S He Y C Su and Y Cheng ldquoString equations ofthe 119902-KP hierarchyrdquo Chinese Annals of Mathematics B vol 32no 6 pp 895ndash904 2011

[16] R L Lin X J Liu and Y B Zeng ldquoA new extended 119902-deformedKP hierarchyrdquo Journal of Nonlinear Mathematical Physics vol15 no 3 pp 333ndash347 2008

Advances in Mathematical Physics 11

[17] R L Lin H Peng and M Manas ldquoThe 119902-deformed mKPhierarchywith self-consistent sourcesWronskian solutions andsolitonsrdquo Journal of Physics A Mathematical and Theoreticalvol 43 Article ID 434022 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Advances in Mathematical Physics 3

Theorem 1 Assume 119861119899 120601119895 120595119895 (119895 = 1 119898) be the solutionof 119902-NKdVHSCS ((10a) (10b) and (10c)) and ℎ1 satisfies (11a)and (11b) with 120582 = 120582

119899

1 the DBT is defined by

119861119899 = 1198791119861119899119879minus1

1= 120597119899

119902+ V119899minus1120597

119899minus1

119902+ sdot sdot sdot + V1120597119902 + V0 (13a)

120601 = [1198791120601] =119882119902 [ℎ1 120601]

ℎ1

(13b)

120601119895= [1198791120601119895] =

119882119902 [ℎ1 120601119895]

ℎ1

(13c)

120595119895= [(119879

minus1

1)lowast

120595119895] = minus120579 (Ω (ℎ1 120595119895))

120579 (ℎ1)

119895 = 1 119898

(13d)

Then 119861119899 120601 120601119895 120595119895 (119895 = 1 119898) satisfy (10b) and (10c) and(11a) and (11b) and hence are the solution of 119902-NKdVHSCS((10a) (10b) and (10c)) where 1198791 = 120579(ℎ1)120597119902ℎ

minus1

1= 120597119902 minus 1205721

1205721 = [120597119902ℎ1]ℎ1 and 119882119902[ℎ1 120601119895] and Ω119902[ℎ1 120595119895] are defined asfollows

119882119902 [1206011 120601] =

1003816100381610038161003816100381610038161003816100381610038161003816

ℎ1 120601

[120597119902ℎ1] [120597119902120601]

1003816100381610038161003816100381610038161003816100381610038161003816

Ω (ℎ1 120595119895) = [120597minus1

119902ℎ1120595119895]

(14)

Remark 2 Here it should be pointed out that the formulaholds

119861119896 = 1198791119861119896119879minus1

1+ 1198791120591

119896

119879minus1

1

119861119896 = (119861119899)119896119899

ge0

119861119896 = (119861119899)119896119899

ge0

(15)

where the gauge operator 1198791 is defined above The proof hasbeen given in [7]

Proof (1)We firstly show that119861119899 120601119895120595119895 (119895 = 1 119898) satisfy(10b) and (10c)

Noting that 119861119899 120601119895 120595119895 (119895 = 1 119898) are the solution of(10a) (10b) and (10c) we have

[119861119899120601119895] = 120582119899

119895120601119895

[119861lowast

119899120595119895] = 120583

119899

119895120595119895

(16)

Hence

[119861119899120601119895] minus 120582119899

119895120601119895= [1198791119861119899119879

minus1

1[1198791120601119895]] minus 120582

119899

119895120601119895

= [1198791 [119861119899120601119895]] minus 120582119899

119895120601119895

= 120582119899

119895([1198791120601119895] minus 120601

119895) = 0

[119861lowast

119899120595119895] minus 120583119899

119895120595119895= [(119879

minus1

1)lowast

119861lowast

119899119879lowast

1[(119879minus1

1)lowast

120595119895]]

minus 120583119899

119895120595119895= [(119879

minus1

1)lowast

119861lowast

119899120595119895] minus 120583

119899

119895120595119895

= 0

(17)

(2)Wefinally show that119861119899120601120601119895120595119895 satisfy (11a) and (11b)Since the proof of (11a) is the same as the case (1) we only

need to verify that 119861119899 120601 120601119895 120595119895 satisfy (11b) that is

120601120591119896

minus [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= 0 119861119896 = (119861119899)119896119899

ge0 (18)

Noting that 120601120591119896

= [1198791120601]120591119896

= [1198791120591119896

120601] + [1198791120601120591119896

] and 120601120591119896

=

[(119861119896 + sum119898

119895=1120601119895120597minus1

119902120595119895)120601] we get

120601120591119896

minus [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= [1198791120591119896

120601] + [1198791120601120591119896

]

minus [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)[1198791120601]]

]

= [

[

(1198791120591119896

+ 1198791119861119896

minus 1198611198961198791 +

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898

sum

119895=1

120601119895120597minus1

1199021205951198951198791)120601]

]

(19)

According to Remark 2 we have

1198791120591119896

+ 1198791119861119896 minus 1198611198961198791 = 0 (20)

Next we prove

119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791) = 0 (21)

Since 1198791 = 120597119902 minus 1205721 and 120597119902120601119895 = 120579(120601119895)120597119902 + [120597119902120601119895] then forall119895 weobtain by the tedious computation

1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791

= (120597119902 minus 1205721) 120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

119902120595119895120597119902 + 120601

119895120597minus1

1199021205951198951205721

= 120579 (120601119895) 120595119895 + 120601119895

Ω(ℎ1 120595119895)

ℎ1

(22a)

4 Advances in Mathematical Physics

In addition we also have

120579 (120601119895) 120595119895 + 120601119895

Ω(ℎ1 120595119895)

ℎ1

=[120579 (120601119895) 120597119902Ω(1206011 120595119895)] + [1198791ℎ1]Ω (ℎ1 120595119895)

ℎ1

=[120597119902120601119895Ω(ℎ1 120595119895)] minus [120597119902120601119895]Ω (ℎ1 120595119895) + ([120597119902120601119895] minus 1205721120601119895)Ω (ℎ1 120595119895)

ℎ1

=[(120597119902 minus 1205721) 120601119895Ω(ℎ1 120595119895)]

ℎ1

=

[1198791 [(120601119895120597minus1

119902120595119895) ℎ1]]

ℎ1

(22b)

Substituting (22b) into (22a) leads to119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

=1

ℎ1

[

[

1198791[

[

119898

sum

119895=1

(120601119895120597minus1

119902120595119895) ℎ1

]

]

]

]

(23)

Since ℎ1 is the solution of (11a) and (11b) with 120582 = 120582119899

1 we have

ℎ1120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

997904rArr

[

[

(

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

= ℎ1120591119896

minus [119861119896ℎ1]

(24)

Moreover by the property of determinant we have

[1198791ℎ1] =119882119902 [ℎ1 ℎ1]

ℎ1

= 0 (25)

Differentiating both sides of (25) with respect to 120591119896 yields

[1198791120591119896

1206011] + [11987911206011120591119896

] = 0 997904rArr

[11987911206011120591119896

] = minus [1198791120591119896

1206011]

(26)

From (23) (24) and (26) we have119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

= minus1

1206011

[(1198791120591119896

+ 1198791119861119896) 1206011] = minus1

1206011

[119861119896 [11987911206011]]

= 0

(27)

This completes the proof

Obviously Theorem 1 provides an auto DBT for 119902-NKdVHSCS ((10a) (10b) and (10c)) However thisDBTdoesnot enable us to obtain the new solution of 119902-NKdVHSCS((10a) (10b) and (10c)) So we have to seek for nonauto DBTsbetween the two 119902-NKdVHSCS ((10a) (10b) and (10c)) withdifferent degrees of sources

4 The Nonauto DBTs of 119902-NKdVHSCS

In this section we will construct the nonauto DBTs of 119902-NKdVHSCS ((10a) (10b) and (10c)) which enables us toobtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH

Theorem 3 Given 119861119899 120601119895 120595119895 (119895 = 1 119898) the solution for119902-NKdVHSCS ((10a) (10b) and (10c)) let 1198911 1198921 equiv 120601119898+1 betwo independent eigenfunctions of (11a) and (11b) with 120582 =

120582119899

119898+1 Let 1198871(120591119896) be a function of 120591119896 such that 1198871(120591119896)120591

119896

=

(minus1)119898+1

1205731(120591119896)1205781(120591119896) Denote ℎ1 = 1198911 + 1198871(120591119896)1198921The DBT is defined by

119861119899 = 1198791119861119899119879minus1

1= 120597119899

119902+ V119899minus1120597

119899minus1

119902+ sdot sdot sdot + V1120597119902 + V0 (28a)

120601 = [1198791120601] =119882119902 [ℎ1 120601]

ℎ1

(28b)

120601119895= [1198791120601119895] =

119882119902 [ℎ1 120601119895]

ℎ1

(28c)

120595119895= [(119879

minus1

1)lowast

120595119895] = minus120579 (Ω (ℎ1 120595119895))

120579 (ℎ1)

119895 = 1 119898

(28d)

120601119898+1

= minus1205731 (120591119896) [11987911198921] (28e)

120595119898+1

= (minus1)119898+1

1205781 (120591119896)1

120579 (ℎ1) (28f)

where1198791 = 120597119902minus1205721 1205721 = [120597119902ℎ1]ℎ1 and then 119861119899 120601 120601119895120595119895 (119895 =

1 119898) 120601119898+1

120595119898+1

satisfy (10b) and (11a) and (11b) with 119898

replaced by119898+1 hence 119861119899 120601119895 120595119895 (119895 = 1 119898) 120601119898+1

120595119898+1

are the solution of 119902-NKdVHSCS ((10a) (10b) and (10c)) with119898 replaced by 119898 + 1

Proof (1) We firstly show that 119861119899 120601 120601119895 120595119895 (119895 = 1 119898)120601119898+1

120595119898+1

are the solution of (10b) and (10c)With the same proof as Theorem 1 119861119899 120601 120601

119895 120595119895(119895 =

1 119898) can be shown to be the solution of (10b) and (10c)

Advances in Mathematical Physics 5

Here we only need to show that 120601119898+1

120595119898+1

are also thesolution of (10b) and (10c) Consider

[119861119899120601119898+1] = minus1205731 (120591119896) [1198791119861119899119879minus1

1[11987911198921]]

= minus1205731 (120591119896) [1198791 [1198611198991198921]]

= minus120582119899

119898+11205731 (120591119896) [11987911198921] = 120582

119899

119898+1120601119898+1

(29)

Taking a proper solution 120595119898+1 of (10c) with 120583 = 120583119899

119898+1such

that Ω(ℎ1 120595119898+1) = minus1 then we get

[119861lowast

119899120595119898+1

] = (minus1)119898+1

1205781 (120591119896)

sdot [(119879minus1

1)lowast

119861lowast

119899119879lowast

1(

1

120579 (ℎ1))] = (minus1)

119898+11205781 (120591119896)

sdot [(119879minus1

1)lowast

119861lowast

119899119879lowast

1(minus

Ω (ℎ1 120595119898+1)

120579 (ℎ1))]

(30)

Noting that [(119879minus11

)lowast120595119898+1] = minus(Ω(ℎ1 120595119898+1))120579(ℎ1) we derive

from (30)

[119861lowast

119899120601119898+1

]

= (minus1)119898+1

1205781 (120591119896) [(119879minus1

1)lowast

119861lowast

119899119879lowast

1[(119879minus1

1)lowast

120595119898+1]]

= (minus1)119898+1

1205781 (120591119896) [(119879minus1

1)lowast

[119861lowast

119899120595119898+1]]

= 120583119899

119898+1120595119898+1

(31)

(2) We finally show that 119861119899 120601119895 120595119895 (119895 = 1 119898) 120601119898+1

120595119898+1

are the solution of (11a) and (11b) with119898 replaced by119898+

1 Evidently we only need to prove 119861119899 120601119895 120595119895 (119895 = 1 119898)120601119898+1

120595119898+1

satisfy (11b) that is120601120591119896

minus[(119861119896+sum119898+1

119895=1120601119895120597minus1

119902120595119895)120601] =

0Noting that 120601 = [1198791120601] rArr 120601

120591119896

= [1198791120601]120591119896

= [1198791120591119896

120601] +

[1198791120601120591119896

] we have

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= [

[

(1198791120591119896

+ 1198791119861119896

minus 1198611198961198791 +

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791)120601]

]

(32a)

From (15) a direct computation leads to

1198791120591119896

+ 1198791119861119896 minus 1198611198961198791 +

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= 0 (32b)

Noticing that119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

minus 120601119898+1

120597minus1

119902120595119898+1

1198791

(33a)

then forall119895 = 1 119898 we obtain by the tedious computation

1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791

= 120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1)

(33b)

Substituting (33b) into (33a) we get

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

minus 120601119898+1

120597minus1

119902120595119898+1

120597119902 + 120601119898+1

120597minus1

119902120595119898+1

1205721

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

minus (minus1)119898+1

1205781 (120591119896) 120601119898+1120597minus1

119902(120597119902

1

ℎ1

minus [120597119902

1

ℎ1

])

+ 120601119898+1

120597minus1

119902120595119898+1

1205721

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

+ (minus1)119898+1

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(33c)

In addition since1198911 1198921 are the solutions of (11a) and (11b) wehave

1198911120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)1198911

]

]

1198921120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)1198921

]

]

(34)

hence

ℎ1120591119896

= 1198911120591119896

+ 1198871 (120591119896) 1198921120591119896

+ 1198871 (120591119896)120591119896

1198921

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (120591119896)120591119896

1198921 997904rArr

[

[

119898

sum

119895=1

(120601119895120597minus1

119902120595119895) ℎ1

]

]

= ℎ1120591119896

minus [119861119896ℎ1] minus 1198871 (120591119896)120591119896

1198921

(35)

6 Advances in Mathematical Physics

Noting [1198791ℎ1] = 119882119902[ℎ1 ℎ1]ℎ1 = 0 and differentiating bothsides of this equation with respect to 120591119896 lead to

[1198791120591119896

ℎ1] + [1198791ℎ1120591119896

] = 0 997904rArr

[1198791ℎ1120591119896

] = minus [1198791120591119896

ℎ1]

(36)

Rewriting (33c) leads to

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

sum119898

119895=1[1198791 (120601119895120597

minus1

119902120595119895) ℎ1]

ℎ1

+ (minus1)119898+1

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(37a)

Combining (32a) and (32b) and (35) and (36) we get

sum119898

119895=1[1198791 (120601119895120597

minus1

119902120595119895) ℎ1]

ℎ1

=

1198791 [sum119898

119895=1(120601119895120597minus1

119902120595119895) ℎ1]

ℎ1

=

[1198791ℎ1120591119896

] minus [1198791119861119896ℎ1] minus 1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[1198791120591

119896

ℎ1] + [1198791119861119896ℎ1]

ℎ1

minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[119861119896 [1198791ℎ1]]

ℎ1

+

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[119861119896 [1198791ℎ1]]

ℎ1

= 0

(37b)

Substituting (32b) (37a) and (37b) into (32a) we have

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

+ (minus1)(119898+1)

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(38)

Noting 1198871(120591119896)120591119896

= (minus1)119898+1

1205731(120591119896)1205781(120591119896) we immediately getfrom (38)

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= 0 (39)

This completes the proof

Theorem 4 (the 119873-times repeated nonauto DBT) Given119861119899 1206011 120601119898 1205951 120595119898 are the solution for 119902-NKdVHSCS

((10a) (10b) and (10c)) 1198911 119891119873 1198921 119892119873 are inde-pendent eigenfunctions of (11a) and (11b) with 120582 =

120582119899

119898+1 120582

119899

119898+119873 119887119894(120591119896) 119894 = 1 119873 are functions of 120591119896 such

that 119887119894(120591119896)120591119896

= (minus1)119898+119873

120573119894(120591119896)120578119894(120591119896)Denote ℎ119894 = 119891119894 + 119887119894(120591119896)119892119894 The 119873-times repeated DBT is

defined by

119861(119873)

119899= 119879119873119861119899119879

minus1

119873= 120597119899

119902+ V(119873)119899minus1

120597119899minus1

119902+ sdot sdot sdot + V(119873)

1120597119902

+ V(119873)0

(40a)

120601(119873)

= [119879119873120601] =119882119902 [ℎ1 ℎ2 ℎ119873 120601]

119882119902 [ℎ1 ℎ2 ℎ119873] (40b)

120601(119873)

119895= [119879119873120601119895] =

119882119902 [ℎ1 ℎ2 ℎ119873 120601119895]

119882119902 [ℎ1 ℎ2 ℎ119873] (40c)

120595(119873)

119895= [(119879

minus1

119873)lowast

120595119895] = minus120579 (119866119902 [ℎ1 ℎ2 ℎ119873 120595119895])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

119895 = 1 119898

(40d)

120601(119873)

119898+119894= minus120573119894 (120591119904) [119879119873119892119894] (40e)

120595(119873)

119898+119894= (minus1)

119898+119894120578119894 (120591119896)

sdot120579 (119882119902 [ℎ1 ℎ119894minus1 ℎ119894+1 ℎ119873])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

119894 = 1 119873

(40f)

where

119879119873 =1

119882119902 [ℎ1 ℎ2 ℎ119873]

sdot

[[[[[[[

[

ℎ1 ℎ2 sdot sdot sdot ℎ119873 1

[120597119902ℎ1] [120597119902ℎ2] sdot sdot sdot [120597119902ℎ119873] 120597119902

[120597119873

119902ℎ1] [120597

119873

119902ℎ2] sdot sdot sdot [120597

119873

119902ℎ119873] 120597

119873

119902

]]]]]]]

]

119866119902 [ℎ1 ℎ2 ℎ119873]

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1 ℎ2 sdot sdot sdot ℎ119873

[120597119873minus2

119902ℎ1] [120597

119873minus2

119902ℎ2] sdot sdot sdot [120597

119873minus2

119902ℎ119873]

[120597minus1

119902ℎ1120595119895] [120597

minus1

119902ℎ2120595119895] sdot sdot sdot [120597

minus1

119902ℎ119873120595119895]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879119873 = 119863119873119863119873minus1 sdot sdot sdot 1198631

119863119896 = (120597119902 minus 120572(119896minus1)

119896)

Advances in Mathematical Physics 7

120572(119896)

119894=

[120597119902ℎ(119896)

119894]

ℎ(119896)

119894

ℎ(119896)

119894= [119879119896ℎ119894] 119896 = 0 1 119873 minus 1

(41)

then 119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) satisfy (10b) and (10c)

and (11a) and (11b) with 119898 replaced by 119898 + 119873 hence119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

are the solution of 119902-NKdVHSCS ((10a) (10b) and (10c)) with119898 replaced by 119898 + 119873

Proof With the same method as Theorem 3 we can showthat 120601

(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (10b) (10c) and (11a) Here we only need to show119861(119873)

119899 120601(119873)

120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 =

1 119873) satisfy (11b) Next we will show it by themathematical induction method Theorem 3 indicates119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (11b) in the case of 119873 = 1Provided that119861(119873)

119897 120601(119897)

119895 120595(119897)

119895 120601(119897)

119898+119894 120595(119897)

119898+119894satisfy (11b) for 119897 le

119873 minus 1

120601(119897)

120591119896

= [

[

(119861(119897)

119896+

119898+119897

sum

119895=1

120601(119897)

119895120597minus1

119902120595(119897)

119895)120601(119897)]

]

119861(119897)

119896= (119861(119897)

119899)119896119899

ge0

(42a)

119887119895 (120591119896)120591119896

= (minus1)(119897+119894)

120573119894 (120591119896) 120578119894 (120591119896)

119897 = 1 119873 minus 1

(42b)

Noticing that 120601(119873) = [119863119873120601(119873minus1)

] then when 119897 = 119873 we have

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= [119863119873120601(119873minus1)

]120591119896

minus [

[

(119861(119873)

119896119863119873

+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

= [

[

(119863119873120591119896

+ 119863119873119861(119873minus1)

119896minus 119861(119873)

119896119863119873

+

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895

minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

(43)

simplifying sum119898+119873minus1

119895=1119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

sum119898+119873

119895=1120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873 leads to

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873119863119873

(44a)

From (40f) we obtain

120595(119873)

119898+119873= (minus1)

119898+119873120578119873 (120591119896)

120579 (119882119902 [ℎ1 ℎ2 ℎ119894minus1])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 ([119879119873minus1ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 (ℎ(119873minus1)

119873)

(44b)

Substituting (44b) into (44a) yields

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873

ℎ(119873minus1)

119873

+ (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902[120597119902

1

ℎ(119873minus1)

119873

]

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902

[120597119902ℎ(119873minus1)

119873]

120579 (ℎ(119873minus1)

119873) ℎ(119873minus1)

119873

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

+ (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

(45)

8 Advances in Mathematical Physics

From (37a) for one DBT 119863119873 we have

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

=1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)

sdot ℎ(119873minus1)

119873]

]

]

]

(46)

Note that ℎ(119873minus1)119873

satisfies

ℎ(119873minus1)

119873120591119896

= [

[

(119861(119873minus1)

119896+

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

+ 119887119873 (120591119896)120591119896

119892(119873minus1)

119873997904rArr

[

[

(

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

= ℎ(119873minus1)

119873120591119896

minus [119861(119873minus1)

119896ℎ(119873minus1)

119873] minus 119887119873 (120591119896)120591

119896

119892(119873minus1)

119873

(47a)

and that

[119879119873ℎ119873] = [119863119873 [119879119873minus1ℎ119873]] = [119863119873ℎ(119873minus1)

119873] = 0 (47b)

Differentiating both sides of (47b) with respect to 120591119896 yields

[119863119873120591119896

ℎ(119873minus1)

119873] + [119863119873ℎ

(119873minus1)

119873120591119896

] = 0 997904rArr

[119863119873ℎ(119873minus1)

119873120591119896

] = minus [119863119873120591119896

ℎ(119873minus1)

119873]

(48)

we obtain

1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895) ℎ(119873minus1)

119873]

]

]

]

= minus[119861(119873minus1)

119896[119863119873ℎ

(119873minus1)

119873]]

ℎ(119873minus1)

119873

= 0

(49)

Combining (43) (45) (46) and (49) we get

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

minus

119887119873 (120591119896)120591119896

[119879119873119892119873]

ℎ(119873minus1)

119873

= 0

(50)

This completes the proof

5 Soliton Solution of 119902-KdVHSCS

It is known that KdV equation is the first nontrivial equationof the KdV hierarchy However the first nontrivial equationof 119902-KdVHSCS is not the 119902-KdVESCS but (12a) (12b) (12c)(12d) and (12e) In this section we aim to construct thesoliton solution to (12a) (12b) (12c) (12d) and (12e) In orderto get the soliton solution of (12a) (12b) (12c) (12d) and(12e) the following proposition is firstly presented

Proposition 5 Let 1198911 1198921 be two independent wave functionsof (12e) ℎ1 equiv 1198911 + 1198871(1205911)1198921 under the nonauto DBT and thetransformed coefficients are given by

V1 minus V1 = 119909 (119902 minus 1) (V0 minus V0) (51)

where

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(52)

Proof It was shown in [7] that formula (51) holds for (12a)(12b) (12c) (12d) and (12e) and that

V0 minus V0 = [120597119902 (V1 + 1205721 + 120579 (1205721))] (53)

Noting that ℎ1 = 1198911 + 1198871(1205911)1198921 (1198612)12

ge0= 1198611 = 120597119902 + 1199060 then

we have

ℎ11205911

= [

[

((1198612)12

ge0+

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (1205911)1205911

1198921

= [120597119902ℎ1] + 1199060ℎ1 +

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

(54)

From (54) we get

1199060 =ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)

minus 1198871 (1205911)1205911

1198921

ℎ1

(55)

Noticing that (12c) implies

V1 = 120579 (1199060) + 1199060 (56)

Advances in Mathematical Physics 9

we have

V0 minus V0 = [

[

120597119902(120579(ℎ11205911

minus [120597119902ℎ1]

ℎ1

) +ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

) +[120597119902ℎ1]

ℎ1

+ 120579([120597119902ℎ1]

ℎ1

))]

]

= [

[

120597119902 (120579(ℎ11205911

ℎ1

)) +ℎ11205911

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

minus [

[

120597119902(120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))

+ (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)))]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)] + (119902120579 + 1)

sdot [

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

(57)

Next we consider

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

=

119898

sum

119895=1

(120579(120601119895

ℎ1

) [120597119902Ω(120595119895 ℎ1)]

+ [120597119902

120601119895

ℎ1

]Ω (120595119895 ℎ1)) =

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1

120579 (ℎ1)

+[120597119902120601119895] ℎ1 minus 120601119895 [120597119902ℎ1]

120579 (ℎ1) ℎ1

Ω(120595119895 ℎ1)) =1

120579 (ℎ1)

sdot

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1 + ([120597119902120601119895] minus 1205721120601119895)Ω (120595119895 ℎ1))

=ℎ1

120579 (ℎ1)(

119898

sum

119895=1

120579 (120601119895) 120595119895 + 120601119895

Ω(120595119895 ℎ1)

ℎ1

)

(58)

Noting (37b) we can immediately derive

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

= 0 (59)

Hence we obtain from (57)

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(60)

This completes the proof

Next we will start from the trivial solution to (12a) (12b)(12c) (12d) and (12e) without sources that is V0 = V1 = 0and useTheorem 3 and Proposition 5 to construct one solitonsolution to (12a) (12b) (12c) (12d) and (12e) with 119898 = 1When V0 = V1 = 0 then 1198612 = 120597

2

119902 hence the wave functions

1198911 1198921 of Lax operator 1198612 = 1205972

119902satisfy

[1205972

119902120593] = 120582

2

1120593

1206011205911

= [120597119902120593]

(61)

We take the solution 1198911 1198921 of system (61) as follows

1198911 = 119864119902 (1199011119909) exp (11990111205911)

1198921 = 119864119902 (minus1199011119909) exp (minus11990111205911)

(62)

where 119864119902(119909) denotes the 119902-exponential function satisfying

[120597119902119864119902 (1199011119909)] = 1199011119864119902 (1199011119909) (63)

with an equivalent form

119864119902 (119909) =

infin

sum

119896=0

1

[119896]119902119909119896 (64)

10 Advances in Mathematical Physics

Noting ℎ1 equiv 1198911 + 1198871(1205911)1198921 where 1198911 1198921 are defined by (62)we get from (51) and (52)

V0

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

119909 (119902 minus 1) ℎ11205792 (ℎ1)

(65a)

V1

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

ℎ11205792 (ℎ1)

(65b)

In addition by Theorem 3 we obtain

1206011= minus1205731 (1205911)

ℎ1 [1205971199021198921] minus 1198921 [120597119902ℎ1]

ℎ21

(65c)

1205951= minus

1205781 (1205911)

1205791 (ℎ1) (65d)

where 1198871(1205911) 1205731(1205911) and 1205781(1205911) satisfy

1198871 (1205911)1205911

= minus120573119902 (1205911) 1205781 (1205911) (65e)

Then (65a) (65b) (65c) (65d) and (65e) present one solitonsolution of (12a) (12b) (12c) (12d) and (12e) with 119898 =

1 In particular when 1198871(1205911) = 119888 where 119888 is an arbitraryconstant (65a) (65b) (65c) (65d) and (65e) can be reducedto one soliton solution to the first nontrivial equation of the119902-KdV hierarchy [7] Certainly we also use Theorem 4 andProposition 5 to construct the multisoliton solution to (12a)(12b) (12c) (12d) and (12e) But owing to the complexity ofthe computation we omit it here

6 Summary

As 119899-reduction of the extended 119902-deformed KP hierarchy 119902-NKdVHSCS is explored in this paper Two kinds of DBTsare constructed and the soliton solution to the first nontrivialequation of 119902-KdVHSCS is also obtained We find that oneof the DBTs provides a nonauto Backlund transformation forthe two 119902-NKdVESCSwith different degree which enables usto obtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH Noting that we only investigate DBTand solution of 119902-NKdVESCS other integrable structureswill be studied in our forthcoming paper such as infiniteconservation law tau function and Hamiltonian structure

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant nos 11201178 and 11171175) FujianNational Science Foundation (Grant no 2012J01013) Fujian

Higher College Special Project of Scientfic Research (Grantno JK2012025) Fujian provincial visiting scholar programand the Scientific Research Foundation of Jimei UniversityChina

References

[1] A Klimyk andK Schmudgen ldquoq-calculusrdquo inQuantumGroupsand their Represntaions pp 37ndash52 Springer Berlin Germany1997

[2] Z YWuDH Zhang andQR Zheng ldquoQuantumdeformationof KdV hierarchies and their exact solutions 119902-deformedsolitonsrdquo Journal of Physics A Mathematical and General vol27 no 15 pp 5307ndash5312 1994

[3] E Frenkel and N Reshetikhin ldquoQuantum affine algebras anddeformations of the Virasoro and 119882-algebrasrdquo Communica-tions in Mathematical Physics vol 178 no 1 pp 237ndash264 1996

[4] E Frenkel ldquoDeformations of the KdV hierarchy and relatedsoliton equationsrdquo International Mathematics Research Noticesno 2 pp 55ndash76 1996

[5] L Haine and P Iliev ldquoThe bispectral property of a 119902-deformation of the Schur polynomials and the 119902-KdV hierar-chyrdquo Journal of Physics A Mathematical and General vol 30no 20 pp 7217ndash7227 1997

[6] M Adler E Horozov and P vanMoerbeke ldquoThe solution to the119902-KdV equationrdquo Physics Letters A vol 242 no 3 pp 139ndash1511998

[7] M-H Tu J-C Shaw and C-R Lee ldquoOn DarbouxndashBacklundtransformations for the 119902-deformed Korteweg-de Vries hierar-chyrdquo Letters in Mathematical Physics vol 49 no 1 pp 33ndash451999

[8] M-H Tu and C-R Lee ldquoOn the 119902-deformed modifiedKorteweg-de Vries hierarchyrdquo Physics Letters A vol 266 no2-3 pp 155ndash159 2000

[9] J Mas and M Seco ldquoThe algebra of q-pseudodifferentialsymbols and 119882

(119873)

119870119875-algebrardquo Journal of Mathematical Physics

vol 37 pp 6510ndash6529 1996[10] P Iliev ldquoTau function solutions to a 119902-deformation of the KP

hierarchyrdquo Letters in Mathematical Physics vol 44 no 3 pp187ndash200 1998

[11] P Iliev ldquo119902-KP hierarchy bispectrality and Calogero-Mosersystemsrdquo Journal of Geometry and Physics vol 35 no 2-3 pp157ndash182 2000

[12] M-H Tu ldquo119902-deformedKP hierarchy its additional symmetriesand infinitesimal Backlund transformationsrdquo Letters in Mathe-matical Physics vol 49 no 2 pp 95ndash103 1999

[13] J S He Y H Li and Y Cheng ldquo119902-deformed KP hierarchy and119902-deformed constrained KP hierarchyrdquo Symmetry Integrabilityand Geometry Methods and Applications vol 2 no 60 p 322006

[14] J S He Y H Li and Y Cheng ldquo119902-deformed Gelfand-Dickeyhierarchy and the determinant representation of its gaugetransformationrdquo Chinese Annals of Mathematics A vol 25 no3 pp 373ndash382 2004

[15] K L Tian J S He Y C Su and Y Cheng ldquoString equations ofthe 119902-KP hierarchyrdquo Chinese Annals of Mathematics B vol 32no 6 pp 895ndash904 2011

[16] R L Lin X J Liu and Y B Zeng ldquoA new extended 119902-deformedKP hierarchyrdquo Journal of Nonlinear Mathematical Physics vol15 no 3 pp 333ndash347 2008

Advances in Mathematical Physics 11

[17] R L Lin H Peng and M Manas ldquoThe 119902-deformed mKPhierarchywith self-consistent sourcesWronskian solutions andsolitonsrdquo Journal of Physics A Mathematical and Theoreticalvol 43 Article ID 434022 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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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

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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

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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

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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

In addition we also have

120579 (120601119895) 120595119895 + 120601119895

Ω(ℎ1 120595119895)

ℎ1

=[120579 (120601119895) 120597119902Ω(1206011 120595119895)] + [1198791ℎ1]Ω (ℎ1 120595119895)

ℎ1

=[120597119902120601119895Ω(ℎ1 120595119895)] minus [120597119902120601119895]Ω (ℎ1 120595119895) + ([120597119902120601119895] minus 1205721120601119895)Ω (ℎ1 120595119895)

ℎ1

=[(120597119902 minus 1205721) 120601119895Ω(ℎ1 120595119895)]

ℎ1

=

[1198791 [(120601119895120597minus1

119902120595119895) ℎ1]]

ℎ1

(22b)

Substituting (22b) into (22a) leads to119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

=1

ℎ1

[

[

1198791[

[

119898

sum

119895=1

(120601119895120597minus1

119902120595119895) ℎ1

]

]

]

]

(23)

Since ℎ1 is the solution of (11a) and (11b) with 120582 = 120582119899

1 we have

ℎ1120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

997904rArr

[

[

(

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

= ℎ1120591119896

minus [119861119896ℎ1]

(24)

Moreover by the property of determinant we have

[1198791ℎ1] =119882119902 [ℎ1 ℎ1]

ℎ1

= 0 (25)

Differentiating both sides of (25) with respect to 120591119896 yields

[1198791120591119896

1206011] + [11987911206011120591119896

] = 0 997904rArr

[11987911206011120591119896

] = minus [1198791120591119896

1206011]

(26)

From (23) (24) and (26) we have119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

= minus1

1206011

[(1198791120591119896

+ 1198791119861119896) 1206011] = minus1

1206011

[119861119896 [11987911206011]]

= 0

(27)

This completes the proof

Obviously Theorem 1 provides an auto DBT for 119902-NKdVHSCS ((10a) (10b) and (10c)) However thisDBTdoesnot enable us to obtain the new solution of 119902-NKdVHSCS((10a) (10b) and (10c)) So we have to seek for nonauto DBTsbetween the two 119902-NKdVHSCS ((10a) (10b) and (10c)) withdifferent degrees of sources

4 The Nonauto DBTs of 119902-NKdVHSCS

In this section we will construct the nonauto DBTs of 119902-NKdVHSCS ((10a) (10b) and (10c)) which enables us toobtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH

Theorem 3 Given 119861119899 120601119895 120595119895 (119895 = 1 119898) the solution for119902-NKdVHSCS ((10a) (10b) and (10c)) let 1198911 1198921 equiv 120601119898+1 betwo independent eigenfunctions of (11a) and (11b) with 120582 =

120582119899

119898+1 Let 1198871(120591119896) be a function of 120591119896 such that 1198871(120591119896)120591

119896

=

(minus1)119898+1

1205731(120591119896)1205781(120591119896) Denote ℎ1 = 1198911 + 1198871(120591119896)1198921The DBT is defined by

119861119899 = 1198791119861119899119879minus1

1= 120597119899

119902+ V119899minus1120597

119899minus1

119902+ sdot sdot sdot + V1120597119902 + V0 (28a)

120601 = [1198791120601] =119882119902 [ℎ1 120601]

ℎ1

(28b)

120601119895= [1198791120601119895] =

119882119902 [ℎ1 120601119895]

ℎ1

(28c)

120595119895= [(119879

minus1

1)lowast

120595119895] = minus120579 (Ω (ℎ1 120595119895))

120579 (ℎ1)

119895 = 1 119898

(28d)

120601119898+1

= minus1205731 (120591119896) [11987911198921] (28e)

120595119898+1

= (minus1)119898+1

1205781 (120591119896)1

120579 (ℎ1) (28f)

where1198791 = 120597119902minus1205721 1205721 = [120597119902ℎ1]ℎ1 and then 119861119899 120601 120601119895120595119895 (119895 =

1 119898) 120601119898+1

120595119898+1

satisfy (10b) and (11a) and (11b) with 119898

replaced by119898+1 hence 119861119899 120601119895 120595119895 (119895 = 1 119898) 120601119898+1

120595119898+1

are the solution of 119902-NKdVHSCS ((10a) (10b) and (10c)) with119898 replaced by 119898 + 1

Proof (1) We firstly show that 119861119899 120601 120601119895 120595119895 (119895 = 1 119898)120601119898+1

120595119898+1

are the solution of (10b) and (10c)With the same proof as Theorem 1 119861119899 120601 120601

119895 120595119895(119895 =

1 119898) can be shown to be the solution of (10b) and (10c)

Advances in Mathematical Physics 5

Here we only need to show that 120601119898+1

120595119898+1

are also thesolution of (10b) and (10c) Consider

[119861119899120601119898+1] = minus1205731 (120591119896) [1198791119861119899119879minus1

1[11987911198921]]

= minus1205731 (120591119896) [1198791 [1198611198991198921]]

= minus120582119899

119898+11205731 (120591119896) [11987911198921] = 120582

119899

119898+1120601119898+1

(29)

Taking a proper solution 120595119898+1 of (10c) with 120583 = 120583119899

119898+1such

that Ω(ℎ1 120595119898+1) = minus1 then we get

[119861lowast

119899120595119898+1

] = (minus1)119898+1

1205781 (120591119896)

sdot [(119879minus1

1)lowast

119861lowast

119899119879lowast

1(

1

120579 (ℎ1))] = (minus1)

119898+11205781 (120591119896)

sdot [(119879minus1

1)lowast

119861lowast

119899119879lowast

1(minus

Ω (ℎ1 120595119898+1)

120579 (ℎ1))]

(30)

Noting that [(119879minus11

)lowast120595119898+1] = minus(Ω(ℎ1 120595119898+1))120579(ℎ1) we derive

from (30)

[119861lowast

119899120601119898+1

]

= (minus1)119898+1

1205781 (120591119896) [(119879minus1

1)lowast

119861lowast

119899119879lowast

1[(119879minus1

1)lowast

120595119898+1]]

= (minus1)119898+1

1205781 (120591119896) [(119879minus1

1)lowast

[119861lowast

119899120595119898+1]]

= 120583119899

119898+1120595119898+1

(31)

(2) We finally show that 119861119899 120601119895 120595119895 (119895 = 1 119898) 120601119898+1

120595119898+1

are the solution of (11a) and (11b) with119898 replaced by119898+

1 Evidently we only need to prove 119861119899 120601119895 120595119895 (119895 = 1 119898)120601119898+1

120595119898+1

satisfy (11b) that is120601120591119896

minus[(119861119896+sum119898+1

119895=1120601119895120597minus1

119902120595119895)120601] =

0Noting that 120601 = [1198791120601] rArr 120601

120591119896

= [1198791120601]120591119896

= [1198791120591119896

120601] +

[1198791120601120591119896

] we have

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= [

[

(1198791120591119896

+ 1198791119861119896

minus 1198611198961198791 +

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791)120601]

]

(32a)

From (15) a direct computation leads to

1198791120591119896

+ 1198791119861119896 minus 1198611198961198791 +

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= 0 (32b)

Noticing that119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

minus 120601119898+1

120597minus1

119902120595119898+1

1198791

(33a)

then forall119895 = 1 119898 we obtain by the tedious computation

1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791

= 120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1)

(33b)

Substituting (33b) into (33a) we get

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

minus 120601119898+1

120597minus1

119902120595119898+1

120597119902 + 120601119898+1

120597minus1

119902120595119898+1

1205721

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

minus (minus1)119898+1

1205781 (120591119896) 120601119898+1120597minus1

119902(120597119902

1

ℎ1

minus [120597119902

1

ℎ1

])

+ 120601119898+1

120597minus1

119902120595119898+1

1205721

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

+ (minus1)119898+1

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(33c)

In addition since1198911 1198921 are the solutions of (11a) and (11b) wehave

1198911120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)1198911

]

]

1198921120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)1198921

]

]

(34)

hence

ℎ1120591119896

= 1198911120591119896

+ 1198871 (120591119896) 1198921120591119896

+ 1198871 (120591119896)120591119896

1198921

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (120591119896)120591119896

1198921 997904rArr

[

[

119898

sum

119895=1

(120601119895120597minus1

119902120595119895) ℎ1

]

]

= ℎ1120591119896

minus [119861119896ℎ1] minus 1198871 (120591119896)120591119896

1198921

(35)

6 Advances in Mathematical Physics

Noting [1198791ℎ1] = 119882119902[ℎ1 ℎ1]ℎ1 = 0 and differentiating bothsides of this equation with respect to 120591119896 lead to

[1198791120591119896

ℎ1] + [1198791ℎ1120591119896

] = 0 997904rArr

[1198791ℎ1120591119896

] = minus [1198791120591119896

ℎ1]

(36)

Rewriting (33c) leads to

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

sum119898

119895=1[1198791 (120601119895120597

minus1

119902120595119895) ℎ1]

ℎ1

+ (minus1)119898+1

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(37a)

Combining (32a) and (32b) and (35) and (36) we get

sum119898

119895=1[1198791 (120601119895120597

minus1

119902120595119895) ℎ1]

ℎ1

=

1198791 [sum119898

119895=1(120601119895120597minus1

119902120595119895) ℎ1]

ℎ1

=

[1198791ℎ1120591119896

] minus [1198791119861119896ℎ1] minus 1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[1198791120591

119896

ℎ1] + [1198791119861119896ℎ1]

ℎ1

minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[119861119896 [1198791ℎ1]]

ℎ1

+

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[119861119896 [1198791ℎ1]]

ℎ1

= 0

(37b)

Substituting (32b) (37a) and (37b) into (32a) we have

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

+ (minus1)(119898+1)

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(38)

Noting 1198871(120591119896)120591119896

= (minus1)119898+1

1205731(120591119896)1205781(120591119896) we immediately getfrom (38)

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= 0 (39)

This completes the proof

Theorem 4 (the 119873-times repeated nonauto DBT) Given119861119899 1206011 120601119898 1205951 120595119898 are the solution for 119902-NKdVHSCS

((10a) (10b) and (10c)) 1198911 119891119873 1198921 119892119873 are inde-pendent eigenfunctions of (11a) and (11b) with 120582 =

120582119899

119898+1 120582

119899

119898+119873 119887119894(120591119896) 119894 = 1 119873 are functions of 120591119896 such

that 119887119894(120591119896)120591119896

= (minus1)119898+119873

120573119894(120591119896)120578119894(120591119896)Denote ℎ119894 = 119891119894 + 119887119894(120591119896)119892119894 The 119873-times repeated DBT is

defined by

119861(119873)

119899= 119879119873119861119899119879

minus1

119873= 120597119899

119902+ V(119873)119899minus1

120597119899minus1

119902+ sdot sdot sdot + V(119873)

1120597119902

+ V(119873)0

(40a)

120601(119873)

= [119879119873120601] =119882119902 [ℎ1 ℎ2 ℎ119873 120601]

119882119902 [ℎ1 ℎ2 ℎ119873] (40b)

120601(119873)

119895= [119879119873120601119895] =

119882119902 [ℎ1 ℎ2 ℎ119873 120601119895]

119882119902 [ℎ1 ℎ2 ℎ119873] (40c)

120595(119873)

119895= [(119879

minus1

119873)lowast

120595119895] = minus120579 (119866119902 [ℎ1 ℎ2 ℎ119873 120595119895])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

119895 = 1 119898

(40d)

120601(119873)

119898+119894= minus120573119894 (120591119904) [119879119873119892119894] (40e)

120595(119873)

119898+119894= (minus1)

119898+119894120578119894 (120591119896)

sdot120579 (119882119902 [ℎ1 ℎ119894minus1 ℎ119894+1 ℎ119873])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

119894 = 1 119873

(40f)

where

119879119873 =1

119882119902 [ℎ1 ℎ2 ℎ119873]

sdot

[[[[[[[

[

ℎ1 ℎ2 sdot sdot sdot ℎ119873 1

[120597119902ℎ1] [120597119902ℎ2] sdot sdot sdot [120597119902ℎ119873] 120597119902

[120597119873

119902ℎ1] [120597

119873

119902ℎ2] sdot sdot sdot [120597

119873

119902ℎ119873] 120597

119873

119902

]]]]]]]

]

119866119902 [ℎ1 ℎ2 ℎ119873]

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1 ℎ2 sdot sdot sdot ℎ119873

[120597119873minus2

119902ℎ1] [120597

119873minus2

119902ℎ2] sdot sdot sdot [120597

119873minus2

119902ℎ119873]

[120597minus1

119902ℎ1120595119895] [120597

minus1

119902ℎ2120595119895] sdot sdot sdot [120597

minus1

119902ℎ119873120595119895]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879119873 = 119863119873119863119873minus1 sdot sdot sdot 1198631

119863119896 = (120597119902 minus 120572(119896minus1)

119896)

Advances in Mathematical Physics 7

120572(119896)

119894=

[120597119902ℎ(119896)

119894]

ℎ(119896)

119894

ℎ(119896)

119894= [119879119896ℎ119894] 119896 = 0 1 119873 minus 1

(41)

then 119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) satisfy (10b) and (10c)

and (11a) and (11b) with 119898 replaced by 119898 + 119873 hence119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

are the solution of 119902-NKdVHSCS ((10a) (10b) and (10c)) with119898 replaced by 119898 + 119873

Proof With the same method as Theorem 3 we can showthat 120601

(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (10b) (10c) and (11a) Here we only need to show119861(119873)

119899 120601(119873)

120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 =

1 119873) satisfy (11b) Next we will show it by themathematical induction method Theorem 3 indicates119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (11b) in the case of 119873 = 1Provided that119861(119873)

119897 120601(119897)

119895 120595(119897)

119895 120601(119897)

119898+119894 120595(119897)

119898+119894satisfy (11b) for 119897 le

119873 minus 1

120601(119897)

120591119896

= [

[

(119861(119897)

119896+

119898+119897

sum

119895=1

120601(119897)

119895120597minus1

119902120595(119897)

119895)120601(119897)]

]

119861(119897)

119896= (119861(119897)

119899)119896119899

ge0

(42a)

119887119895 (120591119896)120591119896

= (minus1)(119897+119894)

120573119894 (120591119896) 120578119894 (120591119896)

119897 = 1 119873 minus 1

(42b)

Noticing that 120601(119873) = [119863119873120601(119873minus1)

] then when 119897 = 119873 we have

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= [119863119873120601(119873minus1)

]120591119896

minus [

[

(119861(119873)

119896119863119873

+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

= [

[

(119863119873120591119896

+ 119863119873119861(119873minus1)

119896minus 119861(119873)

119896119863119873

+

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895

minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

(43)

simplifying sum119898+119873minus1

119895=1119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

sum119898+119873

119895=1120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873 leads to

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873119863119873

(44a)

From (40f) we obtain

120595(119873)

119898+119873= (minus1)

119898+119873120578119873 (120591119896)

120579 (119882119902 [ℎ1 ℎ2 ℎ119894minus1])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 ([119879119873minus1ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 (ℎ(119873minus1)

119873)

(44b)

Substituting (44b) into (44a) yields

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873

ℎ(119873minus1)

119873

+ (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902[120597119902

1

ℎ(119873minus1)

119873

]

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902

[120597119902ℎ(119873minus1)

119873]

120579 (ℎ(119873minus1)

119873) ℎ(119873minus1)

119873

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

+ (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

(45)

8 Advances in Mathematical Physics

From (37a) for one DBT 119863119873 we have

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

=1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)

sdot ℎ(119873minus1)

119873]

]

]

]

(46)

Note that ℎ(119873minus1)119873

satisfies

ℎ(119873minus1)

119873120591119896

= [

[

(119861(119873minus1)

119896+

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

+ 119887119873 (120591119896)120591119896

119892(119873minus1)

119873997904rArr

[

[

(

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

= ℎ(119873minus1)

119873120591119896

minus [119861(119873minus1)

119896ℎ(119873minus1)

119873] minus 119887119873 (120591119896)120591

119896

119892(119873minus1)

119873

(47a)

and that

[119879119873ℎ119873] = [119863119873 [119879119873minus1ℎ119873]] = [119863119873ℎ(119873minus1)

119873] = 0 (47b)

Differentiating both sides of (47b) with respect to 120591119896 yields

[119863119873120591119896

ℎ(119873minus1)

119873] + [119863119873ℎ

(119873minus1)

119873120591119896

] = 0 997904rArr

[119863119873ℎ(119873minus1)

119873120591119896

] = minus [119863119873120591119896

ℎ(119873minus1)

119873]

(48)

we obtain

1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895) ℎ(119873minus1)

119873]

]

]

]

= minus[119861(119873minus1)

119896[119863119873ℎ

(119873minus1)

119873]]

ℎ(119873minus1)

119873

= 0

(49)

Combining (43) (45) (46) and (49) we get

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

minus

119887119873 (120591119896)120591119896

[119879119873119892119873]

ℎ(119873minus1)

119873

= 0

(50)

This completes the proof

5 Soliton Solution of 119902-KdVHSCS

It is known that KdV equation is the first nontrivial equationof the KdV hierarchy However the first nontrivial equationof 119902-KdVHSCS is not the 119902-KdVESCS but (12a) (12b) (12c)(12d) and (12e) In this section we aim to construct thesoliton solution to (12a) (12b) (12c) (12d) and (12e) In orderto get the soliton solution of (12a) (12b) (12c) (12d) and(12e) the following proposition is firstly presented

Proposition 5 Let 1198911 1198921 be two independent wave functionsof (12e) ℎ1 equiv 1198911 + 1198871(1205911)1198921 under the nonauto DBT and thetransformed coefficients are given by

V1 minus V1 = 119909 (119902 minus 1) (V0 minus V0) (51)

where

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(52)

Proof It was shown in [7] that formula (51) holds for (12a)(12b) (12c) (12d) and (12e) and that

V0 minus V0 = [120597119902 (V1 + 1205721 + 120579 (1205721))] (53)

Noting that ℎ1 = 1198911 + 1198871(1205911)1198921 (1198612)12

ge0= 1198611 = 120597119902 + 1199060 then

we have

ℎ11205911

= [

[

((1198612)12

ge0+

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (1205911)1205911

1198921

= [120597119902ℎ1] + 1199060ℎ1 +

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

(54)

From (54) we get

1199060 =ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)

minus 1198871 (1205911)1205911

1198921

ℎ1

(55)

Noticing that (12c) implies

V1 = 120579 (1199060) + 1199060 (56)

Advances in Mathematical Physics 9

we have

V0 minus V0 = [

[

120597119902(120579(ℎ11205911

minus [120597119902ℎ1]

ℎ1

) +ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

) +[120597119902ℎ1]

ℎ1

+ 120579([120597119902ℎ1]

ℎ1

))]

]

= [

[

120597119902 (120579(ℎ11205911

ℎ1

)) +ℎ11205911

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

minus [

[

120597119902(120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))

+ (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)))]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)] + (119902120579 + 1)

sdot [

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

(57)

Next we consider

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

=

119898

sum

119895=1

(120579(120601119895

ℎ1

) [120597119902Ω(120595119895 ℎ1)]

+ [120597119902

120601119895

ℎ1

]Ω (120595119895 ℎ1)) =

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1

120579 (ℎ1)

+[120597119902120601119895] ℎ1 minus 120601119895 [120597119902ℎ1]

120579 (ℎ1) ℎ1

Ω(120595119895 ℎ1)) =1

120579 (ℎ1)

sdot

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1 + ([120597119902120601119895] minus 1205721120601119895)Ω (120595119895 ℎ1))

=ℎ1

120579 (ℎ1)(

119898

sum

119895=1

120579 (120601119895) 120595119895 + 120601119895

Ω(120595119895 ℎ1)

ℎ1

)

(58)

Noting (37b) we can immediately derive

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

= 0 (59)

Hence we obtain from (57)

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(60)

This completes the proof

Next we will start from the trivial solution to (12a) (12b)(12c) (12d) and (12e) without sources that is V0 = V1 = 0and useTheorem 3 and Proposition 5 to construct one solitonsolution to (12a) (12b) (12c) (12d) and (12e) with 119898 = 1When V0 = V1 = 0 then 1198612 = 120597

2

119902 hence the wave functions

1198911 1198921 of Lax operator 1198612 = 1205972

119902satisfy

[1205972

119902120593] = 120582

2

1120593

1206011205911

= [120597119902120593]

(61)

We take the solution 1198911 1198921 of system (61) as follows

1198911 = 119864119902 (1199011119909) exp (11990111205911)

1198921 = 119864119902 (minus1199011119909) exp (minus11990111205911)

(62)

where 119864119902(119909) denotes the 119902-exponential function satisfying

[120597119902119864119902 (1199011119909)] = 1199011119864119902 (1199011119909) (63)

with an equivalent form

119864119902 (119909) =

infin

sum

119896=0

1

[119896]119902119909119896 (64)

10 Advances in Mathematical Physics

Noting ℎ1 equiv 1198911 + 1198871(1205911)1198921 where 1198911 1198921 are defined by (62)we get from (51) and (52)

V0

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

119909 (119902 minus 1) ℎ11205792 (ℎ1)

(65a)

V1

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

ℎ11205792 (ℎ1)

(65b)

In addition by Theorem 3 we obtain

1206011= minus1205731 (1205911)

ℎ1 [1205971199021198921] minus 1198921 [120597119902ℎ1]

ℎ21

(65c)

1205951= minus

1205781 (1205911)

1205791 (ℎ1) (65d)

where 1198871(1205911) 1205731(1205911) and 1205781(1205911) satisfy

1198871 (1205911)1205911

= minus120573119902 (1205911) 1205781 (1205911) (65e)

Then (65a) (65b) (65c) (65d) and (65e) present one solitonsolution of (12a) (12b) (12c) (12d) and (12e) with 119898 =

1 In particular when 1198871(1205911) = 119888 where 119888 is an arbitraryconstant (65a) (65b) (65c) (65d) and (65e) can be reducedto one soliton solution to the first nontrivial equation of the119902-KdV hierarchy [7] Certainly we also use Theorem 4 andProposition 5 to construct the multisoliton solution to (12a)(12b) (12c) (12d) and (12e) But owing to the complexity ofthe computation we omit it here

6 Summary

As 119899-reduction of the extended 119902-deformed KP hierarchy 119902-NKdVHSCS is explored in this paper Two kinds of DBTsare constructed and the soliton solution to the first nontrivialequation of 119902-KdVHSCS is also obtained We find that oneof the DBTs provides a nonauto Backlund transformation forthe two 119902-NKdVESCSwith different degree which enables usto obtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH Noting that we only investigate DBTand solution of 119902-NKdVESCS other integrable structureswill be studied in our forthcoming paper such as infiniteconservation law tau function and Hamiltonian structure

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant nos 11201178 and 11171175) FujianNational Science Foundation (Grant no 2012J01013) Fujian

Higher College Special Project of Scientfic Research (Grantno JK2012025) Fujian provincial visiting scholar programand the Scientific Research Foundation of Jimei UniversityChina

References

[1] A Klimyk andK Schmudgen ldquoq-calculusrdquo inQuantumGroupsand their Represntaions pp 37ndash52 Springer Berlin Germany1997

[2] Z YWuDH Zhang andQR Zheng ldquoQuantumdeformationof KdV hierarchies and their exact solutions 119902-deformedsolitonsrdquo Journal of Physics A Mathematical and General vol27 no 15 pp 5307ndash5312 1994

[3] E Frenkel and N Reshetikhin ldquoQuantum affine algebras anddeformations of the Virasoro and 119882-algebrasrdquo Communica-tions in Mathematical Physics vol 178 no 1 pp 237ndash264 1996

[4] E Frenkel ldquoDeformations of the KdV hierarchy and relatedsoliton equationsrdquo International Mathematics Research Noticesno 2 pp 55ndash76 1996

[5] L Haine and P Iliev ldquoThe bispectral property of a 119902-deformation of the Schur polynomials and the 119902-KdV hierar-chyrdquo Journal of Physics A Mathematical and General vol 30no 20 pp 7217ndash7227 1997

[6] M Adler E Horozov and P vanMoerbeke ldquoThe solution to the119902-KdV equationrdquo Physics Letters A vol 242 no 3 pp 139ndash1511998

[7] M-H Tu J-C Shaw and C-R Lee ldquoOn DarbouxndashBacklundtransformations for the 119902-deformed Korteweg-de Vries hierar-chyrdquo Letters in Mathematical Physics vol 49 no 1 pp 33ndash451999

[8] M-H Tu and C-R Lee ldquoOn the 119902-deformed modifiedKorteweg-de Vries hierarchyrdquo Physics Letters A vol 266 no2-3 pp 155ndash159 2000

[9] J Mas and M Seco ldquoThe algebra of q-pseudodifferentialsymbols and 119882

(119873)

119870119875-algebrardquo Journal of Mathematical Physics

vol 37 pp 6510ndash6529 1996[10] P Iliev ldquoTau function solutions to a 119902-deformation of the KP

hierarchyrdquo Letters in Mathematical Physics vol 44 no 3 pp187ndash200 1998

[11] P Iliev ldquo119902-KP hierarchy bispectrality and Calogero-Mosersystemsrdquo Journal of Geometry and Physics vol 35 no 2-3 pp157ndash182 2000

[12] M-H Tu ldquo119902-deformedKP hierarchy its additional symmetriesand infinitesimal Backlund transformationsrdquo Letters in Mathe-matical Physics vol 49 no 2 pp 95ndash103 1999

[13] J S He Y H Li and Y Cheng ldquo119902-deformed KP hierarchy and119902-deformed constrained KP hierarchyrdquo Symmetry Integrabilityand Geometry Methods and Applications vol 2 no 60 p 322006

[14] J S He Y H Li and Y Cheng ldquo119902-deformed Gelfand-Dickeyhierarchy and the determinant representation of its gaugetransformationrdquo Chinese Annals of Mathematics A vol 25 no3 pp 373ndash382 2004

[15] K L Tian J S He Y C Su and Y Cheng ldquoString equations ofthe 119902-KP hierarchyrdquo Chinese Annals of Mathematics B vol 32no 6 pp 895ndash904 2011

[16] R L Lin X J Liu and Y B Zeng ldquoA new extended 119902-deformedKP hierarchyrdquo Journal of Nonlinear Mathematical Physics vol15 no 3 pp 333ndash347 2008

Advances in Mathematical Physics 11

[17] R L Lin H Peng and M Manas ldquoThe 119902-deformed mKPhierarchywith self-consistent sourcesWronskian solutions andsolitonsrdquo Journal of Physics A Mathematical and Theoreticalvol 43 Article ID 434022 2010

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

Advances in Mathematical Physics 5

Here we only need to show that 120601119898+1

120595119898+1

are also thesolution of (10b) and (10c) Consider

[119861119899120601119898+1] = minus1205731 (120591119896) [1198791119861119899119879minus1

1[11987911198921]]

= minus1205731 (120591119896) [1198791 [1198611198991198921]]

= minus120582119899

119898+11205731 (120591119896) [11987911198921] = 120582

119899

119898+1120601119898+1

(29)

Taking a proper solution 120595119898+1 of (10c) with 120583 = 120583119899

119898+1such

that Ω(ℎ1 120595119898+1) = minus1 then we get

[119861lowast

119899120595119898+1

] = (minus1)119898+1

1205781 (120591119896)

sdot [(119879minus1

1)lowast

119861lowast

119899119879lowast

1(

1

120579 (ℎ1))] = (minus1)

119898+11205781 (120591119896)

sdot [(119879minus1

1)lowast

119861lowast

119899119879lowast

1(minus

Ω (ℎ1 120595119898+1)

120579 (ℎ1))]

(30)

Noting that [(119879minus11

)lowast120595119898+1] = minus(Ω(ℎ1 120595119898+1))120579(ℎ1) we derive

from (30)

[119861lowast

119899120601119898+1

]

= (minus1)119898+1

1205781 (120591119896) [(119879minus1

1)lowast

119861lowast

119899119879lowast

1[(119879minus1

1)lowast

120595119898+1]]

= (minus1)119898+1

1205781 (120591119896) [(119879minus1

1)lowast

[119861lowast

119899120595119898+1]]

= 120583119899

119898+1120595119898+1

(31)

(2) We finally show that 119861119899 120601119895 120595119895 (119895 = 1 119898) 120601119898+1

120595119898+1

are the solution of (11a) and (11b) with119898 replaced by119898+

1 Evidently we only need to prove 119861119899 120601119895 120595119895 (119895 = 1 119898)120601119898+1

120595119898+1

satisfy (11b) that is120601120591119896

minus[(119861119896+sum119898+1

119895=1120601119895120597minus1

119902120595119895)120601] =

0Noting that 120601 = [1198791120601] rArr 120601

120591119896

= [1198791120601]120591119896

= [1198791120591119896

120601] +

[1198791120601120591119896

] we have

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= [

[

(1198791120591119896

+ 1198791119861119896

minus 1198611198961198791 +

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791)120601]

]

(32a)

From (15) a direct computation leads to

1198791120591119896

+ 1198791119861119896 minus 1198611198961198791 +

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= 0 (32b)

Noticing that119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

119898

sum

119895=1

(1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791)

minus 120601119898+1

120597minus1

119902120595119898+1

1198791

(33a)

then forall119895 = 1 119898 we obtain by the tedious computation

1198791120601119895120597minus1

119902120595119895 minus 120601

119895120597minus1

1199021205951198951198791

= 120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1)

(33b)

Substituting (33b) into (33a) we get

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

minus 120601119898+1

120597minus1

119902120595119898+1

120597119902 + 120601119898+1

120597minus1

119902120595119898+1

1205721

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

minus (minus1)119898+1

1205781 (120591119896) 120601119898+1120597minus1

119902(120597119902

1

ℎ1

minus [120597119902

1

ℎ1

])

+ 120601119898+1

120597minus1

119902120595119898+1

1205721

=

119898

sum

119895=1

(120579 (120601119895) 120595119895 + 120601119895

120579 (Ω (ℎ1 120595119895))

120579 (ℎ1))

+ (minus1)119898+1

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(33c)

In addition since1198911 1198921 are the solutions of (11a) and (11b) wehave

1198911120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)1198911

]

]

1198921120591119896

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)1198921

]

]

(34)

hence

ℎ1120591119896

= 1198911120591119896

+ 1198871 (120591119896) 1198921120591119896

+ 1198871 (120591119896)120591119896

1198921

= [

[

(119861119896 +

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (120591119896)120591119896

1198921 997904rArr

[

[

119898

sum

119895=1

(120601119895120597minus1

119902120595119895) ℎ1

]

]

= ℎ1120591119896

minus [119861119896ℎ1] minus 1198871 (120591119896)120591119896

1198921

(35)

6 Advances in Mathematical Physics

Noting [1198791ℎ1] = 119882119902[ℎ1 ℎ1]ℎ1 = 0 and differentiating bothsides of this equation with respect to 120591119896 lead to

[1198791120591119896

ℎ1] + [1198791ℎ1120591119896

] = 0 997904rArr

[1198791ℎ1120591119896

] = minus [1198791120591119896

ℎ1]

(36)

Rewriting (33c) leads to

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

sum119898

119895=1[1198791 (120601119895120597

minus1

119902120595119895) ℎ1]

ℎ1

+ (minus1)119898+1

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(37a)

Combining (32a) and (32b) and (35) and (36) we get

sum119898

119895=1[1198791 (120601119895120597

minus1

119902120595119895) ℎ1]

ℎ1

=

1198791 [sum119898

119895=1(120601119895120597minus1

119902120595119895) ℎ1]

ℎ1

=

[1198791ℎ1120591119896

] minus [1198791119861119896ℎ1] minus 1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[1198791120591

119896

ℎ1] + [1198791119861119896ℎ1]

ℎ1

minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[119861119896 [1198791ℎ1]]

ℎ1

+

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[119861119896 [1198791ℎ1]]

ℎ1

= 0

(37b)

Substituting (32b) (37a) and (37b) into (32a) we have

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

+ (minus1)(119898+1)

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(38)

Noting 1198871(120591119896)120591119896

= (minus1)119898+1

1205731(120591119896)1205781(120591119896) we immediately getfrom (38)

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= 0 (39)

This completes the proof

Theorem 4 (the 119873-times repeated nonauto DBT) Given119861119899 1206011 120601119898 1205951 120595119898 are the solution for 119902-NKdVHSCS

((10a) (10b) and (10c)) 1198911 119891119873 1198921 119892119873 are inde-pendent eigenfunctions of (11a) and (11b) with 120582 =

120582119899

119898+1 120582

119899

119898+119873 119887119894(120591119896) 119894 = 1 119873 are functions of 120591119896 such

that 119887119894(120591119896)120591119896

= (minus1)119898+119873

120573119894(120591119896)120578119894(120591119896)Denote ℎ119894 = 119891119894 + 119887119894(120591119896)119892119894 The 119873-times repeated DBT is

defined by

119861(119873)

119899= 119879119873119861119899119879

minus1

119873= 120597119899

119902+ V(119873)119899minus1

120597119899minus1

119902+ sdot sdot sdot + V(119873)

1120597119902

+ V(119873)0

(40a)

120601(119873)

= [119879119873120601] =119882119902 [ℎ1 ℎ2 ℎ119873 120601]

119882119902 [ℎ1 ℎ2 ℎ119873] (40b)

120601(119873)

119895= [119879119873120601119895] =

119882119902 [ℎ1 ℎ2 ℎ119873 120601119895]

119882119902 [ℎ1 ℎ2 ℎ119873] (40c)

120595(119873)

119895= [(119879

minus1

119873)lowast

120595119895] = minus120579 (119866119902 [ℎ1 ℎ2 ℎ119873 120595119895])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

119895 = 1 119898

(40d)

120601(119873)

119898+119894= minus120573119894 (120591119904) [119879119873119892119894] (40e)

120595(119873)

119898+119894= (minus1)

119898+119894120578119894 (120591119896)

sdot120579 (119882119902 [ℎ1 ℎ119894minus1 ℎ119894+1 ℎ119873])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

119894 = 1 119873

(40f)

where

119879119873 =1

119882119902 [ℎ1 ℎ2 ℎ119873]

sdot

[[[[[[[

[

ℎ1 ℎ2 sdot sdot sdot ℎ119873 1

[120597119902ℎ1] [120597119902ℎ2] sdot sdot sdot [120597119902ℎ119873] 120597119902

[120597119873

119902ℎ1] [120597

119873

119902ℎ2] sdot sdot sdot [120597

119873

119902ℎ119873] 120597

119873

119902

]]]]]]]

]

119866119902 [ℎ1 ℎ2 ℎ119873]

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1 ℎ2 sdot sdot sdot ℎ119873

[120597119873minus2

119902ℎ1] [120597

119873minus2

119902ℎ2] sdot sdot sdot [120597

119873minus2

119902ℎ119873]

[120597minus1

119902ℎ1120595119895] [120597

minus1

119902ℎ2120595119895] sdot sdot sdot [120597

minus1

119902ℎ119873120595119895]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879119873 = 119863119873119863119873minus1 sdot sdot sdot 1198631

119863119896 = (120597119902 minus 120572(119896minus1)

119896)

Advances in Mathematical Physics 7

120572(119896)

119894=

[120597119902ℎ(119896)

119894]

ℎ(119896)

119894

ℎ(119896)

119894= [119879119896ℎ119894] 119896 = 0 1 119873 minus 1

(41)

then 119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) satisfy (10b) and (10c)

and (11a) and (11b) with 119898 replaced by 119898 + 119873 hence119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

are the solution of 119902-NKdVHSCS ((10a) (10b) and (10c)) with119898 replaced by 119898 + 119873

Proof With the same method as Theorem 3 we can showthat 120601

(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (10b) (10c) and (11a) Here we only need to show119861(119873)

119899 120601(119873)

120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 =

1 119873) satisfy (11b) Next we will show it by themathematical induction method Theorem 3 indicates119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (11b) in the case of 119873 = 1Provided that119861(119873)

119897 120601(119897)

119895 120595(119897)

119895 120601(119897)

119898+119894 120595(119897)

119898+119894satisfy (11b) for 119897 le

119873 minus 1

120601(119897)

120591119896

= [

[

(119861(119897)

119896+

119898+119897

sum

119895=1

120601(119897)

119895120597minus1

119902120595(119897)

119895)120601(119897)]

]

119861(119897)

119896= (119861(119897)

119899)119896119899

ge0

(42a)

119887119895 (120591119896)120591119896

= (minus1)(119897+119894)

120573119894 (120591119896) 120578119894 (120591119896)

119897 = 1 119873 minus 1

(42b)

Noticing that 120601(119873) = [119863119873120601(119873minus1)

] then when 119897 = 119873 we have

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= [119863119873120601(119873minus1)

]120591119896

minus [

[

(119861(119873)

119896119863119873

+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

= [

[

(119863119873120591119896

+ 119863119873119861(119873minus1)

119896minus 119861(119873)

119896119863119873

+

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895

minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

(43)

simplifying sum119898+119873minus1

119895=1119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

sum119898+119873

119895=1120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873 leads to

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873119863119873

(44a)

From (40f) we obtain

120595(119873)

119898+119873= (minus1)

119898+119873120578119873 (120591119896)

120579 (119882119902 [ℎ1 ℎ2 ℎ119894minus1])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 ([119879119873minus1ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 (ℎ(119873minus1)

119873)

(44b)

Substituting (44b) into (44a) yields

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873

ℎ(119873minus1)

119873

+ (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902[120597119902

1

ℎ(119873minus1)

119873

]

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902

[120597119902ℎ(119873minus1)

119873]

120579 (ℎ(119873minus1)

119873) ℎ(119873minus1)

119873

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

+ (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

(45)

8 Advances in Mathematical Physics

From (37a) for one DBT 119863119873 we have

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

=1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)

sdot ℎ(119873minus1)

119873]

]

]

]

(46)

Note that ℎ(119873minus1)119873

satisfies

ℎ(119873minus1)

119873120591119896

= [

[

(119861(119873minus1)

119896+

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

+ 119887119873 (120591119896)120591119896

119892(119873minus1)

119873997904rArr

[

[

(

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

= ℎ(119873minus1)

119873120591119896

minus [119861(119873minus1)

119896ℎ(119873minus1)

119873] minus 119887119873 (120591119896)120591

119896

119892(119873minus1)

119873

(47a)

and that

[119879119873ℎ119873] = [119863119873 [119879119873minus1ℎ119873]] = [119863119873ℎ(119873minus1)

119873] = 0 (47b)

Differentiating both sides of (47b) with respect to 120591119896 yields

[119863119873120591119896

ℎ(119873minus1)

119873] + [119863119873ℎ

(119873minus1)

119873120591119896

] = 0 997904rArr

[119863119873ℎ(119873minus1)

119873120591119896

] = minus [119863119873120591119896

ℎ(119873minus1)

119873]

(48)

we obtain

1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895) ℎ(119873minus1)

119873]

]

]

]

= minus[119861(119873minus1)

119896[119863119873ℎ

(119873minus1)

119873]]

ℎ(119873minus1)

119873

= 0

(49)

Combining (43) (45) (46) and (49) we get

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

minus

119887119873 (120591119896)120591119896

[119879119873119892119873]

ℎ(119873minus1)

119873

= 0

(50)

This completes the proof

5 Soliton Solution of 119902-KdVHSCS

It is known that KdV equation is the first nontrivial equationof the KdV hierarchy However the first nontrivial equationof 119902-KdVHSCS is not the 119902-KdVESCS but (12a) (12b) (12c)(12d) and (12e) In this section we aim to construct thesoliton solution to (12a) (12b) (12c) (12d) and (12e) In orderto get the soliton solution of (12a) (12b) (12c) (12d) and(12e) the following proposition is firstly presented

Proposition 5 Let 1198911 1198921 be two independent wave functionsof (12e) ℎ1 equiv 1198911 + 1198871(1205911)1198921 under the nonauto DBT and thetransformed coefficients are given by

V1 minus V1 = 119909 (119902 minus 1) (V0 minus V0) (51)

where

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(52)

Proof It was shown in [7] that formula (51) holds for (12a)(12b) (12c) (12d) and (12e) and that

V0 minus V0 = [120597119902 (V1 + 1205721 + 120579 (1205721))] (53)

Noting that ℎ1 = 1198911 + 1198871(1205911)1198921 (1198612)12

ge0= 1198611 = 120597119902 + 1199060 then

we have

ℎ11205911

= [

[

((1198612)12

ge0+

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (1205911)1205911

1198921

= [120597119902ℎ1] + 1199060ℎ1 +

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

(54)

From (54) we get

1199060 =ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)

minus 1198871 (1205911)1205911

1198921

ℎ1

(55)

Noticing that (12c) implies

V1 = 120579 (1199060) + 1199060 (56)

Advances in Mathematical Physics 9

we have

V0 minus V0 = [

[

120597119902(120579(ℎ11205911

minus [120597119902ℎ1]

ℎ1

) +ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

) +[120597119902ℎ1]

ℎ1

+ 120579([120597119902ℎ1]

ℎ1

))]

]

= [

[

120597119902 (120579(ℎ11205911

ℎ1

)) +ℎ11205911

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

minus [

[

120597119902(120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))

+ (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)))]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)] + (119902120579 + 1)

sdot [

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

(57)

Next we consider

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

=

119898

sum

119895=1

(120579(120601119895

ℎ1

) [120597119902Ω(120595119895 ℎ1)]

+ [120597119902

120601119895

ℎ1

]Ω (120595119895 ℎ1)) =

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1

120579 (ℎ1)

+[120597119902120601119895] ℎ1 minus 120601119895 [120597119902ℎ1]

120579 (ℎ1) ℎ1

Ω(120595119895 ℎ1)) =1

120579 (ℎ1)

sdot

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1 + ([120597119902120601119895] minus 1205721120601119895)Ω (120595119895 ℎ1))

=ℎ1

120579 (ℎ1)(

119898

sum

119895=1

120579 (120601119895) 120595119895 + 120601119895

Ω(120595119895 ℎ1)

ℎ1

)

(58)

Noting (37b) we can immediately derive

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

= 0 (59)

Hence we obtain from (57)

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(60)

This completes the proof

Next we will start from the trivial solution to (12a) (12b)(12c) (12d) and (12e) without sources that is V0 = V1 = 0and useTheorem 3 and Proposition 5 to construct one solitonsolution to (12a) (12b) (12c) (12d) and (12e) with 119898 = 1When V0 = V1 = 0 then 1198612 = 120597

2

119902 hence the wave functions

1198911 1198921 of Lax operator 1198612 = 1205972

119902satisfy

[1205972

119902120593] = 120582

2

1120593

1206011205911

= [120597119902120593]

(61)

We take the solution 1198911 1198921 of system (61) as follows

1198911 = 119864119902 (1199011119909) exp (11990111205911)

1198921 = 119864119902 (minus1199011119909) exp (minus11990111205911)

(62)

where 119864119902(119909) denotes the 119902-exponential function satisfying

[120597119902119864119902 (1199011119909)] = 1199011119864119902 (1199011119909) (63)

with an equivalent form

119864119902 (119909) =

infin

sum

119896=0

1

[119896]119902119909119896 (64)

10 Advances in Mathematical Physics

Noting ℎ1 equiv 1198911 + 1198871(1205911)1198921 where 1198911 1198921 are defined by (62)we get from (51) and (52)

V0

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

119909 (119902 minus 1) ℎ11205792 (ℎ1)

(65a)

V1

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

ℎ11205792 (ℎ1)

(65b)

In addition by Theorem 3 we obtain

1206011= minus1205731 (1205911)

ℎ1 [1205971199021198921] minus 1198921 [120597119902ℎ1]

ℎ21

(65c)

1205951= minus

1205781 (1205911)

1205791 (ℎ1) (65d)

where 1198871(1205911) 1205731(1205911) and 1205781(1205911) satisfy

1198871 (1205911)1205911

= minus120573119902 (1205911) 1205781 (1205911) (65e)

Then (65a) (65b) (65c) (65d) and (65e) present one solitonsolution of (12a) (12b) (12c) (12d) and (12e) with 119898 =

1 In particular when 1198871(1205911) = 119888 where 119888 is an arbitraryconstant (65a) (65b) (65c) (65d) and (65e) can be reducedto one soliton solution to the first nontrivial equation of the119902-KdV hierarchy [7] Certainly we also use Theorem 4 andProposition 5 to construct the multisoliton solution to (12a)(12b) (12c) (12d) and (12e) But owing to the complexity ofthe computation we omit it here

6 Summary

As 119899-reduction of the extended 119902-deformed KP hierarchy 119902-NKdVHSCS is explored in this paper Two kinds of DBTsare constructed and the soliton solution to the first nontrivialequation of 119902-KdVHSCS is also obtained We find that oneof the DBTs provides a nonauto Backlund transformation forthe two 119902-NKdVESCSwith different degree which enables usto obtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH Noting that we only investigate DBTand solution of 119902-NKdVESCS other integrable structureswill be studied in our forthcoming paper such as infiniteconservation law tau function and Hamiltonian structure

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant nos 11201178 and 11171175) FujianNational Science Foundation (Grant no 2012J01013) Fujian

Higher College Special Project of Scientfic Research (Grantno JK2012025) Fujian provincial visiting scholar programand the Scientific Research Foundation of Jimei UniversityChina

References

[1] A Klimyk andK Schmudgen ldquoq-calculusrdquo inQuantumGroupsand their Represntaions pp 37ndash52 Springer Berlin Germany1997

[2] Z YWuDH Zhang andQR Zheng ldquoQuantumdeformationof KdV hierarchies and their exact solutions 119902-deformedsolitonsrdquo Journal of Physics A Mathematical and General vol27 no 15 pp 5307ndash5312 1994

[3] E Frenkel and N Reshetikhin ldquoQuantum affine algebras anddeformations of the Virasoro and 119882-algebrasrdquo Communica-tions in Mathematical Physics vol 178 no 1 pp 237ndash264 1996

[4] E Frenkel ldquoDeformations of the KdV hierarchy and relatedsoliton equationsrdquo International Mathematics Research Noticesno 2 pp 55ndash76 1996

[5] L Haine and P Iliev ldquoThe bispectral property of a 119902-deformation of the Schur polynomials and the 119902-KdV hierar-chyrdquo Journal of Physics A Mathematical and General vol 30no 20 pp 7217ndash7227 1997

[6] M Adler E Horozov and P vanMoerbeke ldquoThe solution to the119902-KdV equationrdquo Physics Letters A vol 242 no 3 pp 139ndash1511998

[7] M-H Tu J-C Shaw and C-R Lee ldquoOn DarbouxndashBacklundtransformations for the 119902-deformed Korteweg-de Vries hierar-chyrdquo Letters in Mathematical Physics vol 49 no 1 pp 33ndash451999

[8] M-H Tu and C-R Lee ldquoOn the 119902-deformed modifiedKorteweg-de Vries hierarchyrdquo Physics Letters A vol 266 no2-3 pp 155ndash159 2000

[9] J Mas and M Seco ldquoThe algebra of q-pseudodifferentialsymbols and 119882

(119873)

119870119875-algebrardquo Journal of Mathematical Physics

vol 37 pp 6510ndash6529 1996[10] P Iliev ldquoTau function solutions to a 119902-deformation of the KP

hierarchyrdquo Letters in Mathematical Physics vol 44 no 3 pp187ndash200 1998

[11] P Iliev ldquo119902-KP hierarchy bispectrality and Calogero-Mosersystemsrdquo Journal of Geometry and Physics vol 35 no 2-3 pp157ndash182 2000

[12] M-H Tu ldquo119902-deformedKP hierarchy its additional symmetriesand infinitesimal Backlund transformationsrdquo Letters in Mathe-matical Physics vol 49 no 2 pp 95ndash103 1999

[13] J S He Y H Li and Y Cheng ldquo119902-deformed KP hierarchy and119902-deformed constrained KP hierarchyrdquo Symmetry Integrabilityand Geometry Methods and Applications vol 2 no 60 p 322006

[14] J S He Y H Li and Y Cheng ldquo119902-deformed Gelfand-Dickeyhierarchy and the determinant representation of its gaugetransformationrdquo Chinese Annals of Mathematics A vol 25 no3 pp 373ndash382 2004

[15] K L Tian J S He Y C Su and Y Cheng ldquoString equations ofthe 119902-KP hierarchyrdquo Chinese Annals of Mathematics B vol 32no 6 pp 895ndash904 2011

[16] R L Lin X J Liu and Y B Zeng ldquoA new extended 119902-deformedKP hierarchyrdquo Journal of Nonlinear Mathematical Physics vol15 no 3 pp 333ndash347 2008

Advances in Mathematical Physics 11

[17] R L Lin H Peng and M Manas ldquoThe 119902-deformed mKPhierarchywith self-consistent sourcesWronskian solutions andsolitonsrdquo Journal of Physics A Mathematical and Theoreticalvol 43 Article ID 434022 2010

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Mathematical Physics

Noting [1198791ℎ1] = 119882119902[ℎ1 ℎ1]ℎ1 = 0 and differentiating bothsides of this equation with respect to 120591119896 lead to

[1198791120591119896

ℎ1] + [1198791ℎ1120591119896

] = 0 997904rArr

[1198791ℎ1120591119896

] = minus [1198791120591119896

ℎ1]

(36)

Rewriting (33c) leads to

119898

sum

119895=1

1198791120601119895120597minus1

119902120595119895 minus

119898+1

sum

119895=1

120601119895120597minus1

1199021205951198951198791

=

sum119898

119895=1[1198791 (120601119895120597

minus1

119902120595119895) ℎ1]

ℎ1

+ (minus1)119898+1

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(37a)

Combining (32a) and (32b) and (35) and (36) we get

sum119898

119895=1[1198791 (120601119895120597

minus1

119902120595119895) ℎ1]

ℎ1

=

1198791 [sum119898

119895=1(120601119895120597minus1

119902120595119895) ℎ1]

ℎ1

=

[1198791ℎ1120591119896

] minus [1198791119861119896ℎ1] minus 1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[1198791120591

119896

ℎ1] + [1198791119861119896ℎ1]

ℎ1

minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[119861119896 [1198791ℎ1]]

ℎ1

+

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

= minus[119861119896 [1198791ℎ1]]

ℎ1

= 0

(37b)

Substituting (32b) (37a) and (37b) into (32a) we have

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= minus

1198871 (120591119896)120591119896

[11987911198921]

ℎ1

+ (minus1)(119898+1)

1205731 (120591119896) 1205781 (120591119896) [11987911198921]

ℎ1

(38)

Noting 1198871(120591119896)120591119896

= (minus1)119898+1

1205731(120591119896)1205781(120591119896) we immediately getfrom (38)

120601120591119896

minus [

[

(119861119896 +

119898+1

sum

119895=1

120601119895120597minus1

119902120595119895)120601]

]

= 0 (39)

This completes the proof

Theorem 4 (the 119873-times repeated nonauto DBT) Given119861119899 1206011 120601119898 1205951 120595119898 are the solution for 119902-NKdVHSCS

((10a) (10b) and (10c)) 1198911 119891119873 1198921 119892119873 are inde-pendent eigenfunctions of (11a) and (11b) with 120582 =

120582119899

119898+1 120582

119899

119898+119873 119887119894(120591119896) 119894 = 1 119873 are functions of 120591119896 such

that 119887119894(120591119896)120591119896

= (minus1)119898+119873

120573119894(120591119896)120578119894(120591119896)Denote ℎ119894 = 119891119894 + 119887119894(120591119896)119892119894 The 119873-times repeated DBT is

defined by

119861(119873)

119899= 119879119873119861119899119879

minus1

119873= 120597119899

119902+ V(119873)119899minus1

120597119899minus1

119902+ sdot sdot sdot + V(119873)

1120597119902

+ V(119873)0

(40a)

120601(119873)

= [119879119873120601] =119882119902 [ℎ1 ℎ2 ℎ119873 120601]

119882119902 [ℎ1 ℎ2 ℎ119873] (40b)

120601(119873)

119895= [119879119873120601119895] =

119882119902 [ℎ1 ℎ2 ℎ119873 120601119895]

119882119902 [ℎ1 ℎ2 ℎ119873] (40c)

120595(119873)

119895= [(119879

minus1

119873)lowast

120595119895] = minus120579 (119866119902 [ℎ1 ℎ2 ℎ119873 120595119895])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

119895 = 1 119898

(40d)

120601(119873)

119898+119894= minus120573119894 (120591119904) [119879119873119892119894] (40e)

120595(119873)

119898+119894= (minus1)

119898+119894120578119894 (120591119896)

sdot120579 (119882119902 [ℎ1 ℎ119894minus1 ℎ119894+1 ℎ119873])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

119894 = 1 119873

(40f)

where

119879119873 =1

119882119902 [ℎ1 ℎ2 ℎ119873]

sdot

[[[[[[[

[

ℎ1 ℎ2 sdot sdot sdot ℎ119873 1

[120597119902ℎ1] [120597119902ℎ2] sdot sdot sdot [120597119902ℎ119873] 120597119902

[120597119873

119902ℎ1] [120597

119873

119902ℎ2] sdot sdot sdot [120597

119873

119902ℎ119873] 120597

119873

119902

]]]]]]]

]

119866119902 [ℎ1 ℎ2 ℎ119873]

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1 ℎ2 sdot sdot sdot ℎ119873

[120597119873minus2

119902ℎ1] [120597

119873minus2

119902ℎ2] sdot sdot sdot [120597

119873minus2

119902ℎ119873]

[120597minus1

119902ℎ1120595119895] [120597

minus1

119902ℎ2120595119895] sdot sdot sdot [120597

minus1

119902ℎ119873120595119895]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879119873 = 119863119873119863119873minus1 sdot sdot sdot 1198631

119863119896 = (120597119902 minus 120572(119896minus1)

119896)

Advances in Mathematical Physics 7

120572(119896)

119894=

[120597119902ℎ(119896)

119894]

ℎ(119896)

119894

ℎ(119896)

119894= [119879119896ℎ119894] 119896 = 0 1 119873 minus 1

(41)

then 119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) satisfy (10b) and (10c)

and (11a) and (11b) with 119898 replaced by 119898 + 119873 hence119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

are the solution of 119902-NKdVHSCS ((10a) (10b) and (10c)) with119898 replaced by 119898 + 119873

Proof With the same method as Theorem 3 we can showthat 120601

(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (10b) (10c) and (11a) Here we only need to show119861(119873)

119899 120601(119873)

120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 =

1 119873) satisfy (11b) Next we will show it by themathematical induction method Theorem 3 indicates119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (11b) in the case of 119873 = 1Provided that119861(119873)

119897 120601(119897)

119895 120595(119897)

119895 120601(119897)

119898+119894 120595(119897)

119898+119894satisfy (11b) for 119897 le

119873 minus 1

120601(119897)

120591119896

= [

[

(119861(119897)

119896+

119898+119897

sum

119895=1

120601(119897)

119895120597minus1

119902120595(119897)

119895)120601(119897)]

]

119861(119897)

119896= (119861(119897)

119899)119896119899

ge0

(42a)

119887119895 (120591119896)120591119896

= (minus1)(119897+119894)

120573119894 (120591119896) 120578119894 (120591119896)

119897 = 1 119873 minus 1

(42b)

Noticing that 120601(119873) = [119863119873120601(119873minus1)

] then when 119897 = 119873 we have

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= [119863119873120601(119873minus1)

]120591119896

minus [

[

(119861(119873)

119896119863119873

+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

= [

[

(119863119873120591119896

+ 119863119873119861(119873minus1)

119896minus 119861(119873)

119896119863119873

+

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895

minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

(43)

simplifying sum119898+119873minus1

119895=1119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

sum119898+119873

119895=1120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873 leads to

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873119863119873

(44a)

From (40f) we obtain

120595(119873)

119898+119873= (minus1)

119898+119873120578119873 (120591119896)

120579 (119882119902 [ℎ1 ℎ2 ℎ119894minus1])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 ([119879119873minus1ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 (ℎ(119873minus1)

119873)

(44b)

Substituting (44b) into (44a) yields

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873

ℎ(119873minus1)

119873

+ (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902[120597119902

1

ℎ(119873minus1)

119873

]

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902

[120597119902ℎ(119873minus1)

119873]

120579 (ℎ(119873minus1)

119873) ℎ(119873minus1)

119873

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

+ (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

(45)

8 Advances in Mathematical Physics

From (37a) for one DBT 119863119873 we have

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

=1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)

sdot ℎ(119873minus1)

119873]

]

]

]

(46)

Note that ℎ(119873minus1)119873

satisfies

ℎ(119873minus1)

119873120591119896

= [

[

(119861(119873minus1)

119896+

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

+ 119887119873 (120591119896)120591119896

119892(119873minus1)

119873997904rArr

[

[

(

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

= ℎ(119873minus1)

119873120591119896

minus [119861(119873minus1)

119896ℎ(119873minus1)

119873] minus 119887119873 (120591119896)120591

119896

119892(119873minus1)

119873

(47a)

and that

[119879119873ℎ119873] = [119863119873 [119879119873minus1ℎ119873]] = [119863119873ℎ(119873minus1)

119873] = 0 (47b)

Differentiating both sides of (47b) with respect to 120591119896 yields

[119863119873120591119896

ℎ(119873minus1)

119873] + [119863119873ℎ

(119873minus1)

119873120591119896

] = 0 997904rArr

[119863119873ℎ(119873minus1)

119873120591119896

] = minus [119863119873120591119896

ℎ(119873minus1)

119873]

(48)

we obtain

1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895) ℎ(119873minus1)

119873]

]

]

]

= minus[119861(119873minus1)

119896[119863119873ℎ

(119873minus1)

119873]]

ℎ(119873minus1)

119873

= 0

(49)

Combining (43) (45) (46) and (49) we get

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

minus

119887119873 (120591119896)120591119896

[119879119873119892119873]

ℎ(119873minus1)

119873

= 0

(50)

This completes the proof

5 Soliton Solution of 119902-KdVHSCS

It is known that KdV equation is the first nontrivial equationof the KdV hierarchy However the first nontrivial equationof 119902-KdVHSCS is not the 119902-KdVESCS but (12a) (12b) (12c)(12d) and (12e) In this section we aim to construct thesoliton solution to (12a) (12b) (12c) (12d) and (12e) In orderto get the soliton solution of (12a) (12b) (12c) (12d) and(12e) the following proposition is firstly presented

Proposition 5 Let 1198911 1198921 be two independent wave functionsof (12e) ℎ1 equiv 1198911 + 1198871(1205911)1198921 under the nonauto DBT and thetransformed coefficients are given by

V1 minus V1 = 119909 (119902 minus 1) (V0 minus V0) (51)

where

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(52)

Proof It was shown in [7] that formula (51) holds for (12a)(12b) (12c) (12d) and (12e) and that

V0 minus V0 = [120597119902 (V1 + 1205721 + 120579 (1205721))] (53)

Noting that ℎ1 = 1198911 + 1198871(1205911)1198921 (1198612)12

ge0= 1198611 = 120597119902 + 1199060 then

we have

ℎ11205911

= [

[

((1198612)12

ge0+

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (1205911)1205911

1198921

= [120597119902ℎ1] + 1199060ℎ1 +

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

(54)

From (54) we get

1199060 =ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)

minus 1198871 (1205911)1205911

1198921

ℎ1

(55)

Noticing that (12c) implies

V1 = 120579 (1199060) + 1199060 (56)

Advances in Mathematical Physics 9

we have

V0 minus V0 = [

[

120597119902(120579(ℎ11205911

minus [120597119902ℎ1]

ℎ1

) +ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

) +[120597119902ℎ1]

ℎ1

+ 120579([120597119902ℎ1]

ℎ1

))]

]

= [

[

120597119902 (120579(ℎ11205911

ℎ1

)) +ℎ11205911

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

minus [

[

120597119902(120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))

+ (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)))]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)] + (119902120579 + 1)

sdot [

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

(57)

Next we consider

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

=

119898

sum

119895=1

(120579(120601119895

ℎ1

) [120597119902Ω(120595119895 ℎ1)]

+ [120597119902

120601119895

ℎ1

]Ω (120595119895 ℎ1)) =

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1

120579 (ℎ1)

+[120597119902120601119895] ℎ1 minus 120601119895 [120597119902ℎ1]

120579 (ℎ1) ℎ1

Ω(120595119895 ℎ1)) =1

120579 (ℎ1)

sdot

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1 + ([120597119902120601119895] minus 1205721120601119895)Ω (120595119895 ℎ1))

=ℎ1

120579 (ℎ1)(

119898

sum

119895=1

120579 (120601119895) 120595119895 + 120601119895

Ω(120595119895 ℎ1)

ℎ1

)

(58)

Noting (37b) we can immediately derive

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

= 0 (59)

Hence we obtain from (57)

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(60)

This completes the proof

Next we will start from the trivial solution to (12a) (12b)(12c) (12d) and (12e) without sources that is V0 = V1 = 0and useTheorem 3 and Proposition 5 to construct one solitonsolution to (12a) (12b) (12c) (12d) and (12e) with 119898 = 1When V0 = V1 = 0 then 1198612 = 120597

2

119902 hence the wave functions

1198911 1198921 of Lax operator 1198612 = 1205972

119902satisfy

[1205972

119902120593] = 120582

2

1120593

1206011205911

= [120597119902120593]

(61)

We take the solution 1198911 1198921 of system (61) as follows

1198911 = 119864119902 (1199011119909) exp (11990111205911)

1198921 = 119864119902 (minus1199011119909) exp (minus11990111205911)

(62)

where 119864119902(119909) denotes the 119902-exponential function satisfying

[120597119902119864119902 (1199011119909)] = 1199011119864119902 (1199011119909) (63)

with an equivalent form

119864119902 (119909) =

infin

sum

119896=0

1

[119896]119902119909119896 (64)

10 Advances in Mathematical Physics

Noting ℎ1 equiv 1198911 + 1198871(1205911)1198921 where 1198911 1198921 are defined by (62)we get from (51) and (52)

V0

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

119909 (119902 minus 1) ℎ11205792 (ℎ1)

(65a)

V1

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

ℎ11205792 (ℎ1)

(65b)

In addition by Theorem 3 we obtain

1206011= minus1205731 (1205911)

ℎ1 [1205971199021198921] minus 1198921 [120597119902ℎ1]

ℎ21

(65c)

1205951= minus

1205781 (1205911)

1205791 (ℎ1) (65d)

where 1198871(1205911) 1205731(1205911) and 1205781(1205911) satisfy

1198871 (1205911)1205911

= minus120573119902 (1205911) 1205781 (1205911) (65e)

Then (65a) (65b) (65c) (65d) and (65e) present one solitonsolution of (12a) (12b) (12c) (12d) and (12e) with 119898 =

1 In particular when 1198871(1205911) = 119888 where 119888 is an arbitraryconstant (65a) (65b) (65c) (65d) and (65e) can be reducedto one soliton solution to the first nontrivial equation of the119902-KdV hierarchy [7] Certainly we also use Theorem 4 andProposition 5 to construct the multisoliton solution to (12a)(12b) (12c) (12d) and (12e) But owing to the complexity ofthe computation we omit it here

6 Summary

As 119899-reduction of the extended 119902-deformed KP hierarchy 119902-NKdVHSCS is explored in this paper Two kinds of DBTsare constructed and the soliton solution to the first nontrivialequation of 119902-KdVHSCS is also obtained We find that oneof the DBTs provides a nonauto Backlund transformation forthe two 119902-NKdVESCSwith different degree which enables usto obtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH Noting that we only investigate DBTand solution of 119902-NKdVESCS other integrable structureswill be studied in our forthcoming paper such as infiniteconservation law tau function and Hamiltonian structure

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant nos 11201178 and 11171175) FujianNational Science Foundation (Grant no 2012J01013) Fujian

Higher College Special Project of Scientfic Research (Grantno JK2012025) Fujian provincial visiting scholar programand the Scientific Research Foundation of Jimei UniversityChina

References

[1] A Klimyk andK Schmudgen ldquoq-calculusrdquo inQuantumGroupsand their Represntaions pp 37ndash52 Springer Berlin Germany1997

[2] Z YWuDH Zhang andQR Zheng ldquoQuantumdeformationof KdV hierarchies and their exact solutions 119902-deformedsolitonsrdquo Journal of Physics A Mathematical and General vol27 no 15 pp 5307ndash5312 1994

[3] E Frenkel and N Reshetikhin ldquoQuantum affine algebras anddeformations of the Virasoro and 119882-algebrasrdquo Communica-tions in Mathematical Physics vol 178 no 1 pp 237ndash264 1996

[4] E Frenkel ldquoDeformations of the KdV hierarchy and relatedsoliton equationsrdquo International Mathematics Research Noticesno 2 pp 55ndash76 1996

[5] L Haine and P Iliev ldquoThe bispectral property of a 119902-deformation of the Schur polynomials and the 119902-KdV hierar-chyrdquo Journal of Physics A Mathematical and General vol 30no 20 pp 7217ndash7227 1997

[6] M Adler E Horozov and P vanMoerbeke ldquoThe solution to the119902-KdV equationrdquo Physics Letters A vol 242 no 3 pp 139ndash1511998

[7] M-H Tu J-C Shaw and C-R Lee ldquoOn DarbouxndashBacklundtransformations for the 119902-deformed Korteweg-de Vries hierar-chyrdquo Letters in Mathematical Physics vol 49 no 1 pp 33ndash451999

[8] M-H Tu and C-R Lee ldquoOn the 119902-deformed modifiedKorteweg-de Vries hierarchyrdquo Physics Letters A vol 266 no2-3 pp 155ndash159 2000

[9] J Mas and M Seco ldquoThe algebra of q-pseudodifferentialsymbols and 119882

(119873)

119870119875-algebrardquo Journal of Mathematical Physics

vol 37 pp 6510ndash6529 1996[10] P Iliev ldquoTau function solutions to a 119902-deformation of the KP

hierarchyrdquo Letters in Mathematical Physics vol 44 no 3 pp187ndash200 1998

[11] P Iliev ldquo119902-KP hierarchy bispectrality and Calogero-Mosersystemsrdquo Journal of Geometry and Physics vol 35 no 2-3 pp157ndash182 2000

[12] M-H Tu ldquo119902-deformedKP hierarchy its additional symmetriesand infinitesimal Backlund transformationsrdquo Letters in Mathe-matical Physics vol 49 no 2 pp 95ndash103 1999

[13] J S He Y H Li and Y Cheng ldquo119902-deformed KP hierarchy and119902-deformed constrained KP hierarchyrdquo Symmetry Integrabilityand Geometry Methods and Applications vol 2 no 60 p 322006

[14] J S He Y H Li and Y Cheng ldquo119902-deformed Gelfand-Dickeyhierarchy and the determinant representation of its gaugetransformationrdquo Chinese Annals of Mathematics A vol 25 no3 pp 373ndash382 2004

[15] K L Tian J S He Y C Su and Y Cheng ldquoString equations ofthe 119902-KP hierarchyrdquo Chinese Annals of Mathematics B vol 32no 6 pp 895ndash904 2011

[16] R L Lin X J Liu and Y B Zeng ldquoA new extended 119902-deformedKP hierarchyrdquo Journal of Nonlinear Mathematical Physics vol15 no 3 pp 333ndash347 2008

Advances in Mathematical Physics 11

[17] R L Lin H Peng and M Manas ldquoThe 119902-deformed mKPhierarchywith self-consistent sourcesWronskian solutions andsolitonsrdquo Journal of Physics A Mathematical and Theoreticalvol 43 Article ID 434022 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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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

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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

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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 7

120572(119896)

119894=

[120597119902ℎ(119896)

119894]

ℎ(119896)

119894

ℎ(119896)

119894= [119879119896ℎ119894] 119896 = 0 1 119873 minus 1

(41)

then 119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) satisfy (10b) and (10c)

and (11a) and (11b) with 119898 replaced by 119898 + 119873 hence119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

are the solution of 119902-NKdVHSCS ((10a) (10b) and (10c)) with119898 replaced by 119898 + 119873

Proof With the same method as Theorem 3 we can showthat 120601

(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (10b) (10c) and (11a) Here we only need to show119861(119873)

119899 120601(119873)

120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 =

1 119873) satisfy (11b) Next we will show it by themathematical induction method Theorem 3 indicates119861(119873)

119899 120601(119873)

119895 120595(119873)

119895 (119895 = 1 119898) 120601

(119873)

119898+119894 120595(119873)

119898+119894 (119894 = 1 119873)

satisfy (11b) in the case of 119873 = 1Provided that119861(119873)

119897 120601(119897)

119895 120595(119897)

119895 120601(119897)

119898+119894 120595(119897)

119898+119894satisfy (11b) for 119897 le

119873 minus 1

120601(119897)

120591119896

= [

[

(119861(119897)

119896+

119898+119897

sum

119895=1

120601(119897)

119895120597minus1

119902120595(119897)

119895)120601(119897)]

]

119861(119897)

119896= (119861(119897)

119899)119896119899

ge0

(42a)

119887119895 (120591119896)120591119896

= (minus1)(119897+119894)

120573119894 (120591119896) 120578119894 (120591119896)

119897 = 1 119873 minus 1

(42b)

Noticing that 120601(119873) = [119863119873120601(119873minus1)

] then when 119897 = 119873 we have

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= [119863119873120601(119873minus1)

]120591119896

minus [

[

(119861(119873)

119896119863119873

+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

= [

[

(119863119873120591119896

+ 119863119873119861(119873minus1)

119896minus 119861(119873)

119896119863119873

+

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895

minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)120601

(119873minus1)]

]

(43)

simplifying sum119898+119873minus1

119895=1119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

sum119898+119873

119895=1120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873 leads to

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873119863119873

(44a)

From (40f) we obtain

120595(119873)

119898+119873= (minus1)

119898+119873120578119873 (120591119896)

120579 (119882119902 [ℎ1 ℎ2 ℎ119894minus1])

120579 (119882119902 [ℎ1 ℎ2 ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 ([119879119873minus1ℎ119873])

= (minus1)119898+119873

120578119873 (120591119896)

120579 (ℎ(119873minus1)

119873)

(44b)

Substituting (44b) into (44a) yields

119898+119873minus1

sum

119895=1

119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873

ℎ(119873minus1)

119873

+ (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902[120597119902

1

ℎ(119873minus1)

119873

]

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

minus (minus1)119898+119873

120578119873 (120591119896) 120601(119873)

119898+119873120597minus1

119902

[120597119902ℎ(119873minus1)

119873]

120579 (ℎ(119873minus1)

119873) ℎ(119873minus1)

119873

+ 120601(119873)

119898+119873120597minus1

119902120595(119873)

119898+119873120572119873minus1

119873

=

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus 120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

+ (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

(45)

8 Advances in Mathematical Physics

From (37a) for one DBT 119863119873 we have

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

=1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)

sdot ℎ(119873minus1)

119873]

]

]

]

(46)

Note that ℎ(119873minus1)119873

satisfies

ℎ(119873minus1)

119873120591119896

= [

[

(119861(119873minus1)

119896+

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

+ 119887119873 (120591119896)120591119896

119892(119873minus1)

119873997904rArr

[

[

(

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

= ℎ(119873minus1)

119873120591119896

minus [119861(119873minus1)

119896ℎ(119873minus1)

119873] minus 119887119873 (120591119896)120591

119896

119892(119873minus1)

119873

(47a)

and that

[119879119873ℎ119873] = [119863119873 [119879119873minus1ℎ119873]] = [119863119873ℎ(119873minus1)

119873] = 0 (47b)

Differentiating both sides of (47b) with respect to 120591119896 yields

[119863119873120591119896

ℎ(119873minus1)

119873] + [119863119873ℎ

(119873minus1)

119873120591119896

] = 0 997904rArr

[119863119873ℎ(119873minus1)

119873120591119896

] = minus [119863119873120591119896

ℎ(119873minus1)

119873]

(48)

we obtain

1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895) ℎ(119873minus1)

119873]

]

]

]

= minus[119861(119873minus1)

119896[119863119873ℎ

(119873minus1)

119873]]

ℎ(119873minus1)

119873

= 0

(49)

Combining (43) (45) (46) and (49) we get

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

minus

119887119873 (120591119896)120591119896

[119879119873119892119873]

ℎ(119873minus1)

119873

= 0

(50)

This completes the proof

5 Soliton Solution of 119902-KdVHSCS

It is known that KdV equation is the first nontrivial equationof the KdV hierarchy However the first nontrivial equationof 119902-KdVHSCS is not the 119902-KdVESCS but (12a) (12b) (12c)(12d) and (12e) In this section we aim to construct thesoliton solution to (12a) (12b) (12c) (12d) and (12e) In orderto get the soliton solution of (12a) (12b) (12c) (12d) and(12e) the following proposition is firstly presented

Proposition 5 Let 1198911 1198921 be two independent wave functionsof (12e) ℎ1 equiv 1198911 + 1198871(1205911)1198921 under the nonauto DBT and thetransformed coefficients are given by

V1 minus V1 = 119909 (119902 minus 1) (V0 minus V0) (51)

where

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(52)

Proof It was shown in [7] that formula (51) holds for (12a)(12b) (12c) (12d) and (12e) and that

V0 minus V0 = [120597119902 (V1 + 1205721 + 120579 (1205721))] (53)

Noting that ℎ1 = 1198911 + 1198871(1205911)1198921 (1198612)12

ge0= 1198611 = 120597119902 + 1199060 then

we have

ℎ11205911

= [

[

((1198612)12

ge0+

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (1205911)1205911

1198921

= [120597119902ℎ1] + 1199060ℎ1 +

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

(54)

From (54) we get

1199060 =ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)

minus 1198871 (1205911)1205911

1198921

ℎ1

(55)

Noticing that (12c) implies

V1 = 120579 (1199060) + 1199060 (56)

Advances in Mathematical Physics 9

we have

V0 minus V0 = [

[

120597119902(120579(ℎ11205911

minus [120597119902ℎ1]

ℎ1

) +ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

) +[120597119902ℎ1]

ℎ1

+ 120579([120597119902ℎ1]

ℎ1

))]

]

= [

[

120597119902 (120579(ℎ11205911

ℎ1

)) +ℎ11205911

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

minus [

[

120597119902(120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))

+ (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)))]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)] + (119902120579 + 1)

sdot [

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

(57)

Next we consider

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

=

119898

sum

119895=1

(120579(120601119895

ℎ1

) [120597119902Ω(120595119895 ℎ1)]

+ [120597119902

120601119895

ℎ1

]Ω (120595119895 ℎ1)) =

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1

120579 (ℎ1)

+[120597119902120601119895] ℎ1 minus 120601119895 [120597119902ℎ1]

120579 (ℎ1) ℎ1

Ω(120595119895 ℎ1)) =1

120579 (ℎ1)

sdot

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1 + ([120597119902120601119895] minus 1205721120601119895)Ω (120595119895 ℎ1))

=ℎ1

120579 (ℎ1)(

119898

sum

119895=1

120579 (120601119895) 120595119895 + 120601119895

Ω(120595119895 ℎ1)

ℎ1

)

(58)

Noting (37b) we can immediately derive

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

= 0 (59)

Hence we obtain from (57)

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(60)

This completes the proof

Next we will start from the trivial solution to (12a) (12b)(12c) (12d) and (12e) without sources that is V0 = V1 = 0and useTheorem 3 and Proposition 5 to construct one solitonsolution to (12a) (12b) (12c) (12d) and (12e) with 119898 = 1When V0 = V1 = 0 then 1198612 = 120597

2

119902 hence the wave functions

1198911 1198921 of Lax operator 1198612 = 1205972

119902satisfy

[1205972

119902120593] = 120582

2

1120593

1206011205911

= [120597119902120593]

(61)

We take the solution 1198911 1198921 of system (61) as follows

1198911 = 119864119902 (1199011119909) exp (11990111205911)

1198921 = 119864119902 (minus1199011119909) exp (minus11990111205911)

(62)

where 119864119902(119909) denotes the 119902-exponential function satisfying

[120597119902119864119902 (1199011119909)] = 1199011119864119902 (1199011119909) (63)

with an equivalent form

119864119902 (119909) =

infin

sum

119896=0

1

[119896]119902119909119896 (64)

10 Advances in Mathematical Physics

Noting ℎ1 equiv 1198911 + 1198871(1205911)1198921 where 1198911 1198921 are defined by (62)we get from (51) and (52)

V0

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

119909 (119902 minus 1) ℎ11205792 (ℎ1)

(65a)

V1

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

ℎ11205792 (ℎ1)

(65b)

In addition by Theorem 3 we obtain

1206011= minus1205731 (1205911)

ℎ1 [1205971199021198921] minus 1198921 [120597119902ℎ1]

ℎ21

(65c)

1205951= minus

1205781 (1205911)

1205791 (ℎ1) (65d)

where 1198871(1205911) 1205731(1205911) and 1205781(1205911) satisfy

1198871 (1205911)1205911

= minus120573119902 (1205911) 1205781 (1205911) (65e)

Then (65a) (65b) (65c) (65d) and (65e) present one solitonsolution of (12a) (12b) (12c) (12d) and (12e) with 119898 =

1 In particular when 1198871(1205911) = 119888 where 119888 is an arbitraryconstant (65a) (65b) (65c) (65d) and (65e) can be reducedto one soliton solution to the first nontrivial equation of the119902-KdV hierarchy [7] Certainly we also use Theorem 4 andProposition 5 to construct the multisoliton solution to (12a)(12b) (12c) (12d) and (12e) But owing to the complexity ofthe computation we omit it here

6 Summary

As 119899-reduction of the extended 119902-deformed KP hierarchy 119902-NKdVHSCS is explored in this paper Two kinds of DBTsare constructed and the soliton solution to the first nontrivialequation of 119902-KdVHSCS is also obtained We find that oneof the DBTs provides a nonauto Backlund transformation forthe two 119902-NKdVESCSwith different degree which enables usto obtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH Noting that we only investigate DBTand solution of 119902-NKdVESCS other integrable structureswill be studied in our forthcoming paper such as infiniteconservation law tau function and Hamiltonian structure

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant nos 11201178 and 11171175) FujianNational Science Foundation (Grant no 2012J01013) Fujian

Higher College Special Project of Scientfic Research (Grantno JK2012025) Fujian provincial visiting scholar programand the Scientific Research Foundation of Jimei UniversityChina

References

[1] A Klimyk andK Schmudgen ldquoq-calculusrdquo inQuantumGroupsand their Represntaions pp 37ndash52 Springer Berlin Germany1997

[2] Z YWuDH Zhang andQR Zheng ldquoQuantumdeformationof KdV hierarchies and their exact solutions 119902-deformedsolitonsrdquo Journal of Physics A Mathematical and General vol27 no 15 pp 5307ndash5312 1994

[3] E Frenkel and N Reshetikhin ldquoQuantum affine algebras anddeformations of the Virasoro and 119882-algebrasrdquo Communica-tions in Mathematical Physics vol 178 no 1 pp 237ndash264 1996

[4] E Frenkel ldquoDeformations of the KdV hierarchy and relatedsoliton equationsrdquo International Mathematics Research Noticesno 2 pp 55ndash76 1996

[5] L Haine and P Iliev ldquoThe bispectral property of a 119902-deformation of the Schur polynomials and the 119902-KdV hierar-chyrdquo Journal of Physics A Mathematical and General vol 30no 20 pp 7217ndash7227 1997

[6] M Adler E Horozov and P vanMoerbeke ldquoThe solution to the119902-KdV equationrdquo Physics Letters A vol 242 no 3 pp 139ndash1511998

[7] M-H Tu J-C Shaw and C-R Lee ldquoOn DarbouxndashBacklundtransformations for the 119902-deformed Korteweg-de Vries hierar-chyrdquo Letters in Mathematical Physics vol 49 no 1 pp 33ndash451999

[8] M-H Tu and C-R Lee ldquoOn the 119902-deformed modifiedKorteweg-de Vries hierarchyrdquo Physics Letters A vol 266 no2-3 pp 155ndash159 2000

[9] J Mas and M Seco ldquoThe algebra of q-pseudodifferentialsymbols and 119882

(119873)

119870119875-algebrardquo Journal of Mathematical Physics

vol 37 pp 6510ndash6529 1996[10] P Iliev ldquoTau function solutions to a 119902-deformation of the KP

hierarchyrdquo Letters in Mathematical Physics vol 44 no 3 pp187ndash200 1998

[11] P Iliev ldquo119902-KP hierarchy bispectrality and Calogero-Mosersystemsrdquo Journal of Geometry and Physics vol 35 no 2-3 pp157ndash182 2000

[12] M-H Tu ldquo119902-deformedKP hierarchy its additional symmetriesand infinitesimal Backlund transformationsrdquo Letters in Mathe-matical Physics vol 49 no 2 pp 95ndash103 1999

[13] J S He Y H Li and Y Cheng ldquo119902-deformed KP hierarchy and119902-deformed constrained KP hierarchyrdquo Symmetry Integrabilityand Geometry Methods and Applications vol 2 no 60 p 322006

[14] J S He Y H Li and Y Cheng ldquo119902-deformed Gelfand-Dickeyhierarchy and the determinant representation of its gaugetransformationrdquo Chinese Annals of Mathematics A vol 25 no3 pp 373ndash382 2004

[15] K L Tian J S He Y C Su and Y Cheng ldquoString equations ofthe 119902-KP hierarchyrdquo Chinese Annals of Mathematics B vol 32no 6 pp 895ndash904 2011

[16] R L Lin X J Liu and Y B Zeng ldquoA new extended 119902-deformedKP hierarchyrdquo Journal of Nonlinear Mathematical Physics vol15 no 3 pp 333ndash347 2008

Advances in Mathematical Physics 11

[17] R L Lin H Peng and M Manas ldquoThe 119902-deformed mKPhierarchywith self-consistent sourcesWronskian solutions andsolitonsrdquo Journal of Physics A Mathematical and Theoreticalvol 43 Article ID 434022 2010

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

8 Advances in Mathematical Physics

From (37a) for one DBT 119863119873 we have

119898+119873minus1

sum

119895=1

(119863119873120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895minus

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895119863119873)

=1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)

sdot ℎ(119873minus1)

119873]

]

]

]

(46)

Note that ℎ(119873minus1)119873

satisfies

ℎ(119873minus1)

119873120591119896

= [

[

(119861(119873minus1)

119896+

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

+ 119887119873 (120591119896)120591119896

119892(119873minus1)

119873997904rArr

[

[

(

119898+119873minus1

sum

119895=1

120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895)ℎ(119873minus1)

119873]

]

= ℎ(119873minus1)

119873120591119896

minus [119861(119873minus1)

119896ℎ(119873minus1)

119873] minus 119887119873 (120591119896)120591

119896

119892(119873minus1)

119873

(47a)

and that

[119879119873ℎ119873] = [119863119873 [119879119873minus1ℎ119873]] = [119863119873ℎ(119873minus1)

119873] = 0 (47b)

Differentiating both sides of (47b) with respect to 120591119896 yields

[119863119873120591119896

ℎ(119873minus1)

119873] + [119863119873ℎ

(119873minus1)

119873120591119896

] = 0 997904rArr

[119863119873ℎ(119873minus1)

119873120591119896

] = minus [119863119873120591119896

ℎ(119873minus1)

119873]

(48)

we obtain

1

ℎ(119873minus1)

119873

[

[

119863119873[

[

119898+119873minus1

sum

119895=1

(120601(119873minus1)

119895120597minus1

119902120595(119873minus1)

119895) ℎ(119873minus1)

119873]

]

]

]

= minus[119861(119873minus1)

119896[119863119873ℎ

(119873minus1)

119873]]

ℎ(119873minus1)

119873

= 0

(49)

Combining (43) (45) (46) and (49) we get

120601(119873)

120591119896

minus [

[

(119861(119873)

119896+

119898+119873

sum

119895=1

120601(119873)

119895120597minus1

119902120595(119873)

119895)120601(119873)]

]

= (minus1)119898+119873

120573119873 (120591119896) 120578119873 (120591119896) [119879119873119892119873]

ℎ(119873minus1)

119873

minus

119887119873 (120591119896)120591119896

[119879119873119892119873]

ℎ(119873minus1)

119873

= 0

(50)

This completes the proof

5 Soliton Solution of 119902-KdVHSCS

It is known that KdV equation is the first nontrivial equationof the KdV hierarchy However the first nontrivial equationof 119902-KdVHSCS is not the 119902-KdVESCS but (12a) (12b) (12c)(12d) and (12e) In this section we aim to construct thesoliton solution to (12a) (12b) (12c) (12d) and (12e) In orderto get the soliton solution of (12a) (12b) (12c) (12d) and(12e) the following proposition is firstly presented

Proposition 5 Let 1198911 1198921 be two independent wave functionsof (12e) ℎ1 equiv 1198911 + 1198871(1205911)1198921 under the nonauto DBT and thetransformed coefficients are given by

V1 minus V1 = 119909 (119902 minus 1) (V0 minus V0) (51)

where

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(52)

Proof It was shown in [7] that formula (51) holds for (12a)(12b) (12c) (12d) and (12e) and that

V0 minus V0 = [120597119902 (V1 + 1205721 + 120579 (1205721))] (53)

Noting that ℎ1 = 1198911 + 1198871(1205911)1198921 (1198612)12

ge0= 1198611 = 120597119902 + 1199060 then

we have

ℎ11205911

= [

[

((1198612)12

ge0+

119898

sum

119895=1

120601119895120597minus1

119902120595119895)ℎ1

]

]

+ 1198871 (1205911)1205911

1198921

= [120597119902ℎ1] + 1199060ℎ1 +

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

(54)

From (54) we get

1199060 =ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)

minus 1198871 (1205911)1205911

1198921

ℎ1

(55)

Noticing that (12c) implies

V1 = 120579 (1199060) + 1199060 (56)

Advances in Mathematical Physics 9

we have

V0 minus V0 = [

[

120597119902(120579(ℎ11205911

minus [120597119902ℎ1]

ℎ1

) +ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

) +[120597119902ℎ1]

ℎ1

+ 120579([120597119902ℎ1]

ℎ1

))]

]

= [

[

120597119902 (120579(ℎ11205911

ℎ1

)) +ℎ11205911

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

minus [

[

120597119902(120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))

+ (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)))]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)] + (119902120579 + 1)

sdot [

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

(57)

Next we consider

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

=

119898

sum

119895=1

(120579(120601119895

ℎ1

) [120597119902Ω(120595119895 ℎ1)]

+ [120597119902

120601119895

ℎ1

]Ω (120595119895 ℎ1)) =

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1

120579 (ℎ1)

+[120597119902120601119895] ℎ1 minus 120601119895 [120597119902ℎ1]

120579 (ℎ1) ℎ1

Ω(120595119895 ℎ1)) =1

120579 (ℎ1)

sdot

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1 + ([120597119902120601119895] minus 1205721120601119895)Ω (120595119895 ℎ1))

=ℎ1

120579 (ℎ1)(

119898

sum

119895=1

120579 (120601119895) 120595119895 + 120601119895

Ω(120595119895 ℎ1)

ℎ1

)

(58)

Noting (37b) we can immediately derive

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

= 0 (59)

Hence we obtain from (57)

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(60)

This completes the proof

Next we will start from the trivial solution to (12a) (12b)(12c) (12d) and (12e) without sources that is V0 = V1 = 0and useTheorem 3 and Proposition 5 to construct one solitonsolution to (12a) (12b) (12c) (12d) and (12e) with 119898 = 1When V0 = V1 = 0 then 1198612 = 120597

2

119902 hence the wave functions

1198911 1198921 of Lax operator 1198612 = 1205972

119902satisfy

[1205972

119902120593] = 120582

2

1120593

1206011205911

= [120597119902120593]

(61)

We take the solution 1198911 1198921 of system (61) as follows

1198911 = 119864119902 (1199011119909) exp (11990111205911)

1198921 = 119864119902 (minus1199011119909) exp (minus11990111205911)

(62)

where 119864119902(119909) denotes the 119902-exponential function satisfying

[120597119902119864119902 (1199011119909)] = 1199011119864119902 (1199011119909) (63)

with an equivalent form

119864119902 (119909) =

infin

sum

119896=0

1

[119896]119902119909119896 (64)

10 Advances in Mathematical Physics

Noting ℎ1 equiv 1198911 + 1198871(1205911)1198921 where 1198911 1198921 are defined by (62)we get from (51) and (52)

V0

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

119909 (119902 minus 1) ℎ11205792 (ℎ1)

(65a)

V1

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

ℎ11205792 (ℎ1)

(65b)

In addition by Theorem 3 we obtain

1206011= minus1205731 (1205911)

ℎ1 [1205971199021198921] minus 1198921 [120597119902ℎ1]

ℎ21

(65c)

1205951= minus

1205781 (1205911)

1205791 (ℎ1) (65d)

where 1198871(1205911) 1205731(1205911) and 1205781(1205911) satisfy

1198871 (1205911)1205911

= minus120573119902 (1205911) 1205781 (1205911) (65e)

Then (65a) (65b) (65c) (65d) and (65e) present one solitonsolution of (12a) (12b) (12c) (12d) and (12e) with 119898 =

1 In particular when 1198871(1205911) = 119888 where 119888 is an arbitraryconstant (65a) (65b) (65c) (65d) and (65e) can be reducedto one soliton solution to the first nontrivial equation of the119902-KdV hierarchy [7] Certainly we also use Theorem 4 andProposition 5 to construct the multisoliton solution to (12a)(12b) (12c) (12d) and (12e) But owing to the complexity ofthe computation we omit it here

6 Summary

As 119899-reduction of the extended 119902-deformed KP hierarchy 119902-NKdVHSCS is explored in this paper Two kinds of DBTsare constructed and the soliton solution to the first nontrivialequation of 119902-KdVHSCS is also obtained We find that oneof the DBTs provides a nonauto Backlund transformation forthe two 119902-NKdVESCSwith different degree which enables usto obtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH Noting that we only investigate DBTand solution of 119902-NKdVESCS other integrable structureswill be studied in our forthcoming paper such as infiniteconservation law tau function and Hamiltonian structure

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant nos 11201178 and 11171175) FujianNational Science Foundation (Grant no 2012J01013) Fujian

Higher College Special Project of Scientfic Research (Grantno JK2012025) Fujian provincial visiting scholar programand the Scientific Research Foundation of Jimei UniversityChina

References

[1] A Klimyk andK Schmudgen ldquoq-calculusrdquo inQuantumGroupsand their Represntaions pp 37ndash52 Springer Berlin Germany1997

[2] Z YWuDH Zhang andQR Zheng ldquoQuantumdeformationof KdV hierarchies and their exact solutions 119902-deformedsolitonsrdquo Journal of Physics A Mathematical and General vol27 no 15 pp 5307ndash5312 1994

[3] E Frenkel and N Reshetikhin ldquoQuantum affine algebras anddeformations of the Virasoro and 119882-algebrasrdquo Communica-tions in Mathematical Physics vol 178 no 1 pp 237ndash264 1996

[4] E Frenkel ldquoDeformations of the KdV hierarchy and relatedsoliton equationsrdquo International Mathematics Research Noticesno 2 pp 55ndash76 1996

[5] L Haine and P Iliev ldquoThe bispectral property of a 119902-deformation of the Schur polynomials and the 119902-KdV hierar-chyrdquo Journal of Physics A Mathematical and General vol 30no 20 pp 7217ndash7227 1997

[6] M Adler E Horozov and P vanMoerbeke ldquoThe solution to the119902-KdV equationrdquo Physics Letters A vol 242 no 3 pp 139ndash1511998

[7] M-H Tu J-C Shaw and C-R Lee ldquoOn DarbouxndashBacklundtransformations for the 119902-deformed Korteweg-de Vries hierar-chyrdquo Letters in Mathematical Physics vol 49 no 1 pp 33ndash451999

[8] M-H Tu and C-R Lee ldquoOn the 119902-deformed modifiedKorteweg-de Vries hierarchyrdquo Physics Letters A vol 266 no2-3 pp 155ndash159 2000

[9] J Mas and M Seco ldquoThe algebra of q-pseudodifferentialsymbols and 119882

(119873)

119870119875-algebrardquo Journal of Mathematical Physics

vol 37 pp 6510ndash6529 1996[10] P Iliev ldquoTau function solutions to a 119902-deformation of the KP

hierarchyrdquo Letters in Mathematical Physics vol 44 no 3 pp187ndash200 1998

[11] P Iliev ldquo119902-KP hierarchy bispectrality and Calogero-Mosersystemsrdquo Journal of Geometry and Physics vol 35 no 2-3 pp157ndash182 2000

[12] M-H Tu ldquo119902-deformedKP hierarchy its additional symmetriesand infinitesimal Backlund transformationsrdquo Letters in Mathe-matical Physics vol 49 no 2 pp 95ndash103 1999

[13] J S He Y H Li and Y Cheng ldquo119902-deformed KP hierarchy and119902-deformed constrained KP hierarchyrdquo Symmetry Integrabilityand Geometry Methods and Applications vol 2 no 60 p 322006

[14] J S He Y H Li and Y Cheng ldquo119902-deformed Gelfand-Dickeyhierarchy and the determinant representation of its gaugetransformationrdquo Chinese Annals of Mathematics A vol 25 no3 pp 373ndash382 2004

[15] K L Tian J S He Y C Su and Y Cheng ldquoString equations ofthe 119902-KP hierarchyrdquo Chinese Annals of Mathematics B vol 32no 6 pp 895ndash904 2011

[16] R L Lin X J Liu and Y B Zeng ldquoA new extended 119902-deformedKP hierarchyrdquo Journal of Nonlinear Mathematical Physics vol15 no 3 pp 333ndash347 2008

Advances in Mathematical Physics 11

[17] R L Lin H Peng and M Manas ldquoThe 119902-deformed mKPhierarchywith self-consistent sourcesWronskian solutions andsolitonsrdquo Journal of Physics A Mathematical and Theoreticalvol 43 Article ID 434022 2010

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

Advances in Mathematical Physics 9

we have

V0 minus V0 = [

[

120597119902(120579(ℎ11205911

minus [120597119902ℎ1]

ℎ1

) +ℎ11205911

minus [120597119902ℎ1]

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

) +[120597119902ℎ1]

ℎ1

+ 120579([120597119902ℎ1]

ℎ1

))]

]

= [

[

120597119902 (120579(ℎ11205911

ℎ1

)) +ℎ11205911

ℎ1

minus 120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)

minus (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1) + 1198871 (1205911)1205911

1198921

ℎ1

)]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

minus [

[

120597119902(120579(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))

+ (1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1)))]

]

= [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)] + (119902120579 + 1)

sdot [

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

(57)

Next we consider

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

=

119898

sum

119895=1

(120579(120601119895

ℎ1

) [120597119902Ω(120595119895 ℎ1)]

+ [120597119902

120601119895

ℎ1

]Ω (120595119895 ℎ1)) =

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1

120579 (ℎ1)

+[120597119902120601119895] ℎ1 minus 120601119895 [120597119902ℎ1]

120579 (ℎ1) ℎ1

Ω(120595119895 ℎ1)) =1

120579 (ℎ1)

sdot

119898

sum

119895=1

(120579 (120601119895) 120595119895ℎ1 + ([120597119902120601119895] minus 1205721120601119895)Ω (120595119895 ℎ1))

=ℎ1

120579 (ℎ1)(

119898

sum

119895=1

120579 (120601119895) 120595119895 + 120601119895

Ω(120595119895 ℎ1)

ℎ1

)

(58)

Noting (37b) we can immediately derive

[

[

120597119902(1

ℎ1

119898

sum

119895=1

120601119895Ω(120595119895 ℎ1))]

]

= 0 (59)

Hence we obtain from (57)

V0 = V0 + [120597119902 (120579(11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)

+11989111205911

+ 1198871 (1205911) 11989211205911

ℎ1

)]

(60)

This completes the proof

Next we will start from the trivial solution to (12a) (12b)(12c) (12d) and (12e) without sources that is V0 = V1 = 0and useTheorem 3 and Proposition 5 to construct one solitonsolution to (12a) (12b) (12c) (12d) and (12e) with 119898 = 1When V0 = V1 = 0 then 1198612 = 120597

2

119902 hence the wave functions

1198911 1198921 of Lax operator 1198612 = 1205972

119902satisfy

[1205972

119902120593] = 120582

2

1120593

1206011205911

= [120597119902120593]

(61)

We take the solution 1198911 1198921 of system (61) as follows

1198911 = 119864119902 (1199011119909) exp (11990111205911)

1198921 = 119864119902 (minus1199011119909) exp (minus11990111205911)

(62)

where 119864119902(119909) denotes the 119902-exponential function satisfying

[120597119902119864119902 (1199011119909)] = 1199011119864119902 (1199011119909) (63)

with an equivalent form

119864119902 (119909) =

infin

sum

119896=0

1

[119896]119902119909119896 (64)

10 Advances in Mathematical Physics

Noting ℎ1 equiv 1198911 + 1198871(1205911)1198921 where 1198911 1198921 are defined by (62)we get from (51) and (52)

V0

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

119909 (119902 minus 1) ℎ11205792 (ℎ1)

(65a)

V1

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

ℎ11205792 (ℎ1)

(65b)

In addition by Theorem 3 we obtain

1206011= minus1205731 (1205911)

ℎ1 [1205971199021198921] minus 1198921 [120597119902ℎ1]

ℎ21

(65c)

1205951= minus

1205781 (1205911)

1205791 (ℎ1) (65d)

where 1198871(1205911) 1205731(1205911) and 1205781(1205911) satisfy

1198871 (1205911)1205911

= minus120573119902 (1205911) 1205781 (1205911) (65e)

Then (65a) (65b) (65c) (65d) and (65e) present one solitonsolution of (12a) (12b) (12c) (12d) and (12e) with 119898 =

1 In particular when 1198871(1205911) = 119888 where 119888 is an arbitraryconstant (65a) (65b) (65c) (65d) and (65e) can be reducedto one soliton solution to the first nontrivial equation of the119902-KdV hierarchy [7] Certainly we also use Theorem 4 andProposition 5 to construct the multisoliton solution to (12a)(12b) (12c) (12d) and (12e) But owing to the complexity ofthe computation we omit it here

6 Summary

As 119899-reduction of the extended 119902-deformed KP hierarchy 119902-NKdVHSCS is explored in this paper Two kinds of DBTsare constructed and the soliton solution to the first nontrivialequation of 119902-KdVHSCS is also obtained We find that oneof the DBTs provides a nonauto Backlund transformation forthe two 119902-NKdVESCSwith different degree which enables usto obtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH Noting that we only investigate DBTand solution of 119902-NKdVESCS other integrable structureswill be studied in our forthcoming paper such as infiniteconservation law tau function and Hamiltonian structure

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant nos 11201178 and 11171175) FujianNational Science Foundation (Grant no 2012J01013) Fujian

Higher College Special Project of Scientfic Research (Grantno JK2012025) Fujian provincial visiting scholar programand the Scientific Research Foundation of Jimei UniversityChina

References

[1] A Klimyk andK Schmudgen ldquoq-calculusrdquo inQuantumGroupsand their Represntaions pp 37ndash52 Springer Berlin Germany1997

[2] Z YWuDH Zhang andQR Zheng ldquoQuantumdeformationof KdV hierarchies and their exact solutions 119902-deformedsolitonsrdquo Journal of Physics A Mathematical and General vol27 no 15 pp 5307ndash5312 1994

[3] E Frenkel and N Reshetikhin ldquoQuantum affine algebras anddeformations of the Virasoro and 119882-algebrasrdquo Communica-tions in Mathematical Physics vol 178 no 1 pp 237ndash264 1996

[4] E Frenkel ldquoDeformations of the KdV hierarchy and relatedsoliton equationsrdquo International Mathematics Research Noticesno 2 pp 55ndash76 1996

[5] L Haine and P Iliev ldquoThe bispectral property of a 119902-deformation of the Schur polynomials and the 119902-KdV hierar-chyrdquo Journal of Physics A Mathematical and General vol 30no 20 pp 7217ndash7227 1997

[6] M Adler E Horozov and P vanMoerbeke ldquoThe solution to the119902-KdV equationrdquo Physics Letters A vol 242 no 3 pp 139ndash1511998

[7] M-H Tu J-C Shaw and C-R Lee ldquoOn DarbouxndashBacklundtransformations for the 119902-deformed Korteweg-de Vries hierar-chyrdquo Letters in Mathematical Physics vol 49 no 1 pp 33ndash451999

[8] M-H Tu and C-R Lee ldquoOn the 119902-deformed modifiedKorteweg-de Vries hierarchyrdquo Physics Letters A vol 266 no2-3 pp 155ndash159 2000

[9] J Mas and M Seco ldquoThe algebra of q-pseudodifferentialsymbols and 119882

(119873)

119870119875-algebrardquo Journal of Mathematical Physics

vol 37 pp 6510ndash6529 1996[10] P Iliev ldquoTau function solutions to a 119902-deformation of the KP

hierarchyrdquo Letters in Mathematical Physics vol 44 no 3 pp187ndash200 1998

[11] P Iliev ldquo119902-KP hierarchy bispectrality and Calogero-Mosersystemsrdquo Journal of Geometry and Physics vol 35 no 2-3 pp157ndash182 2000

[12] M-H Tu ldquo119902-deformedKP hierarchy its additional symmetriesand infinitesimal Backlund transformationsrdquo Letters in Mathe-matical Physics vol 49 no 2 pp 95ndash103 1999

[13] J S He Y H Li and Y Cheng ldquo119902-deformed KP hierarchy and119902-deformed constrained KP hierarchyrdquo Symmetry Integrabilityand Geometry Methods and Applications vol 2 no 60 p 322006

[14] J S He Y H Li and Y Cheng ldquo119902-deformed Gelfand-Dickeyhierarchy and the determinant representation of its gaugetransformationrdquo Chinese Annals of Mathematics A vol 25 no3 pp 373ndash382 2004

[15] K L Tian J S He Y C Su and Y Cheng ldquoString equations ofthe 119902-KP hierarchyrdquo Chinese Annals of Mathematics B vol 32no 6 pp 895ndash904 2011

[16] R L Lin X J Liu and Y B Zeng ldquoA new extended 119902-deformedKP hierarchyrdquo Journal of Nonlinear Mathematical Physics vol15 no 3 pp 333ndash347 2008

Advances in Mathematical Physics 11

[17] R L Lin H Peng and M Manas ldquoThe 119902-deformed mKPhierarchywith self-consistent sourcesWronskian solutions andsolitonsrdquo Journal of Physics A Mathematical and Theoreticalvol 43 Article ID 434022 2010

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

Noting ℎ1 equiv 1198911 + 1198871(1205911)1198921 where 1198911 1198921 are defined by (62)we get from (51) and (52)

V0

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

119909 (119902 minus 1) ℎ11205792 (ℎ1)

(65a)

V1

=1199011 (ℎ1120579

2(1198911 minus 1198871 (1205911) 1198921) minus (1198911 minus 1198871 (1205911) 1198921) 120579

2(ℎ1))

ℎ11205792 (ℎ1)

(65b)

In addition by Theorem 3 we obtain

1206011= minus1205731 (1205911)

ℎ1 [1205971199021198921] minus 1198921 [120597119902ℎ1]

ℎ21

(65c)

1205951= minus

1205781 (1205911)

1205791 (ℎ1) (65d)

where 1198871(1205911) 1205731(1205911) and 1205781(1205911) satisfy

1198871 (1205911)1205911

= minus120573119902 (1205911) 1205781 (1205911) (65e)

Then (65a) (65b) (65c) (65d) and (65e) present one solitonsolution of (12a) (12b) (12c) (12d) and (12e) with 119898 =

1 In particular when 1198871(1205911) = 119888 where 119888 is an arbitraryconstant (65a) (65b) (65c) (65d) and (65e) can be reducedto one soliton solution to the first nontrivial equation of the119902-KdV hierarchy [7] Certainly we also use Theorem 4 andProposition 5 to construct the multisoliton solution to (12a)(12b) (12c) (12d) and (12e) But owing to the complexity ofthe computation we omit it here

6 Summary

As 119899-reduction of the extended 119902-deformed KP hierarchy 119902-NKdVHSCS is explored in this paper Two kinds of DBTsare constructed and the soliton solution to the first nontrivialequation of 119902-KdVHSCS is also obtained We find that oneof the DBTs provides a nonauto Backlund transformation forthe two 119902-NKdVESCSwith different degree which enables usto obtain the new solution of 119902-NKdVHSCS from the knownsolution of 119902-NKdVH Noting that we only investigate DBTand solution of 119902-NKdVESCS other integrable structureswill be studied in our forthcoming paper such as infiniteconservation law tau function and Hamiltonian structure

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant nos 11201178 and 11171175) FujianNational Science Foundation (Grant no 2012J01013) Fujian

Higher College Special Project of Scientfic Research (Grantno JK2012025) Fujian provincial visiting scholar programand the Scientific Research Foundation of Jimei UniversityChina

References

[1] A Klimyk andK Schmudgen ldquoq-calculusrdquo inQuantumGroupsand their Represntaions pp 37ndash52 Springer Berlin Germany1997

[2] Z YWuDH Zhang andQR Zheng ldquoQuantumdeformationof KdV hierarchies and their exact solutions 119902-deformedsolitonsrdquo Journal of Physics A Mathematical and General vol27 no 15 pp 5307ndash5312 1994

[3] E Frenkel and N Reshetikhin ldquoQuantum affine algebras anddeformations of the Virasoro and 119882-algebrasrdquo Communica-tions in Mathematical Physics vol 178 no 1 pp 237ndash264 1996

[4] E Frenkel ldquoDeformations of the KdV hierarchy and relatedsoliton equationsrdquo International Mathematics Research Noticesno 2 pp 55ndash76 1996

[5] L Haine and P Iliev ldquoThe bispectral property of a 119902-deformation of the Schur polynomials and the 119902-KdV hierar-chyrdquo Journal of Physics A Mathematical and General vol 30no 20 pp 7217ndash7227 1997

[6] M Adler E Horozov and P vanMoerbeke ldquoThe solution to the119902-KdV equationrdquo Physics Letters A vol 242 no 3 pp 139ndash1511998

[7] M-H Tu J-C Shaw and C-R Lee ldquoOn DarbouxndashBacklundtransformations for the 119902-deformed Korteweg-de Vries hierar-chyrdquo Letters in Mathematical Physics vol 49 no 1 pp 33ndash451999

[8] M-H Tu and C-R Lee ldquoOn the 119902-deformed modifiedKorteweg-de Vries hierarchyrdquo Physics Letters A vol 266 no2-3 pp 155ndash159 2000

[9] J Mas and M Seco ldquoThe algebra of q-pseudodifferentialsymbols and 119882

(119873)

119870119875-algebrardquo Journal of Mathematical Physics

vol 37 pp 6510ndash6529 1996[10] P Iliev ldquoTau function solutions to a 119902-deformation of the KP

hierarchyrdquo Letters in Mathematical Physics vol 44 no 3 pp187ndash200 1998

[11] P Iliev ldquo119902-KP hierarchy bispectrality and Calogero-Mosersystemsrdquo Journal of Geometry and Physics vol 35 no 2-3 pp157ndash182 2000

[12] M-H Tu ldquo119902-deformedKP hierarchy its additional symmetriesand infinitesimal Backlund transformationsrdquo Letters in Mathe-matical Physics vol 49 no 2 pp 95ndash103 1999

[13] J S He Y H Li and Y Cheng ldquo119902-deformed KP hierarchy and119902-deformed constrained KP hierarchyrdquo Symmetry Integrabilityand Geometry Methods and Applications vol 2 no 60 p 322006

[14] J S He Y H Li and Y Cheng ldquo119902-deformed Gelfand-Dickeyhierarchy and the determinant representation of its gaugetransformationrdquo Chinese Annals of Mathematics A vol 25 no3 pp 373ndash382 2004

[15] K L Tian J S He Y C Su and Y Cheng ldquoString equations ofthe 119902-KP hierarchyrdquo Chinese Annals of Mathematics B vol 32no 6 pp 895ndash904 2011

[16] R L Lin X J Liu and Y B Zeng ldquoA new extended 119902-deformedKP hierarchyrdquo Journal of Nonlinear Mathematical Physics vol15 no 3 pp 333ndash347 2008

Advances in Mathematical Physics 11

[17] R L Lin H Peng and M Manas ldquoThe 119902-deformed mKPhierarchywith self-consistent sourcesWronskian solutions andsolitonsrdquo Journal of Physics A Mathematical and Theoreticalvol 43 Article ID 434022 2010

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

Advances in Mathematical Physics 11

[17] R L Lin H Peng and M Manas ldquoThe 119902-deformed mKPhierarchywith self-consistent sourcesWronskian solutions andsolitonsrdquo Journal of Physics A Mathematical and Theoreticalvol 43 Article ID 434022 2010

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