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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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Journal of
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Mathematical PhysicsAdvances in
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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
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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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
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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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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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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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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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
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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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 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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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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
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
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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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
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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
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
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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