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Advances in Mathematical Physics
Volume 2016, Article ID 8153752, 11 pages
http://dx.doi.org/10.1155/2016/8153752
Research Article

Two Kinds of Darboux-Bäcklund Transformations for the -Deformed KdV Hierarchy with Self-Consistent Sources

1Department of Mathematics, School of Sciences, Jimei University, Xiamen 361021, China
2Department of Mathematics, The University of Texas Rio Grande Valley, Edinburg, TX 78539, USA
3Department of Mathematics, Tsinghua University, Beijing 100084, China

Received 19 April 2016; Accepted 23 July 2016

Academic Editor: Pavel Kurasov

Copyright © 2016 Hongxia Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

1. Introduction

The -deformed integrable systems are regarded as the -deformation of the related classical ones. The -deformation is performed by using the -derivative to take the place of usual derivative and it reduces to a classical integrable system as . In recent years, some -deformed integrable systems, especially the -deformed th KdV hierarchy (-NKdVH) and the -deformed KP hierarchy (-KPH), have attracted much interest both in mathematics and in physics [117]. It was shown that -NKdVH inherited some integrable structures from the classical th KdV hierarchy, such as infinite conservation law [2], bi-Hamiltonian structure [3, 4], tau function [5, 6], Darboux-Bäcklund transformation [7], and -Miura transformation [8]. In 1999, some elementary DBTs of the -NKdVH (also called -deformed Gelfand-Dickey hierarchy) were constructed by using the -deformed pseudodifferential operators. The formula for the -times repeated DBTs was also presented, which produces the new soliton solutions to the -NKdVH [7]. For -KPH, its bi-Hamiltonian structure, tau function, additional symmetries, -effect in -soliton, Virasoro constraints of tau function and integrable extension, and -Bäcklund transformation were also explored in [917]. In 2008, based on the symmetry constraint for -KPH, the new extension of this hierarchy was considered [16]. Two kinds of reductions of this new extended -KP hierarchy were also studied, which give many dimensional -deformed soliton equations with self-consistent sources [16]. For example, the -reduction gives -NKdVHSCS. However, to our knowledge, the DBTs and the soliton solution for -NKdVHSCS still remain unexplored. It is known that the DBT is an important property to characterize the integrability of the hierarchy. Thus, it is necessary for us to explore the DBT for -NKdVHSCS. We think our research results will deepen our understanding on soliton solutions of this hierarchy.

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

2. The -Deformed th KdV Hierarchy with Self-Consistent Sources (-NKdVHSCS)

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

The -derivative operator and -shift operator are defined by In this paper, we introduce two notations: [] and , in which is a -pseudo-differential operator (-PDO) given by denotes acting on the function , while indicates the multiplication of and ; that is, .

It can be easily shown from (1) that when , reduces to the ordinary differential operator and that and do not commute but satisfyLet be the formal inverse of such as . In general, the -deformed Leibnitz rule holdswhere -number and -binomial are defined byFor a -PDO , we separate into the differential part and the integral part . The conjugate operation is given by where .

The -exponential function is defined assatisfying .

The extended -KPH was given by [16] where and the coefficients are the functions of .

The commutativity of (9a), (9b), and (9c) leads to the zero-curvature representation of -KPH ((9a), (9b), and (9c)). As the -reduction of the extended -KPH, the -NKdVHSCS is defined as follows [16]: Under (10b) and (10c), the Lax representation for (10a) is We find that when , -NKdVHSCS ((10a), (10b), and (10c) and (11a) and (11b)) can be reduced to the -NKdVH and its related Lax representation, respectively. In addition, when , ((10a), (10b), and (10c)) becomes the first nontrivial soliton equation of -KdVHSCS given by

3. The Auto Darboux-Bäcklund Transformation (DBT) for -NKdVHSCS

In this section, we will focus on the construction of auto DBT for -NKdVHSCS.

Theorem 1. Assume , , be the solution of -NKdVHSCS ((10a), (10b), and (10c)), and satisfies (11a) and (11b) with ; the DBT is defined by Then , , , satisfy (10b) and (10c) and (11a) and (11b) and hence are the solution of -NKdVHSCS ((10a), (10b), and (10c)), where , , and and are defined as follows:

Remark 2. Here it should be pointed out that the formula holds where the gauge operator is defined above. The proof has been given in [7].

Proof. (1) We firstly show that , , satisfy (10b) and (10c).
Noting that , , are the solution of (10a), (10b), and (10c), we have Hence, (2) We finally show that , , , satisfy (11a) and (11b).
Since the proof of (11a) is the same as the case (1), we only need to verify that , , , satisfy (11b); that is, Noting that and , we get According to Remark 2, we have Next, we prove Since and , then , we obtain by the tedious computation In addition, we also have Substituting (22b) into (22a) leads to Since is the solution of (11a) and (11b) with , we have Moreover, by the property of determinant, we haveDifferentiating both sides of (25) with respect to yieldsFrom (23), (24), and (26), we haveThis completes the proof.

Obviously, Theorem 1 provides an auto DBT for -NKdVHSCS ((10a), (10b), and (10c)). However, this DBT does not enable us to obtain the new solution of -NKdVHSCS ((10a), (10b), and (10c)). So we have to seek for nonauto DBTs between the two -NKdVHSCS ((10a), (10b), and (10c)) with different degrees of sources.

4. The Nonauto DBTs of -NKdVHSCS

In this section, we will construct the nonauto DBTs of -NKdVHSCS ((10a), (10b), and (10c)), which enables us to obtain the new solution of -NKdVHSCS from the known solution of -NKdVH.

Theorem 3. Given , , the solution for -NKdVHSCS ((10a), (10b), and (10c)), let be two independent eigenfunctions of (11a) and (11b) with . Let be a function of such that . Denote .
The DBT is defined by where , , and then , , , , , satisfy (10b) and (11a) and (11b) with replaced by ; hence , , , , are the solution of -NKdVHSCS ((10a), (10b), and (10c)) with replaced by .

Proof. (1) We firstly show that , , , , , are the solution of (10b) and (10c).
With the same proof as Theorem 1, , , , can be shown to be the solution of (10b) and (10c). Here we only need to show that , are also the solution of (10b) and (10c). ConsiderTaking a proper solution of (10c) with such that , then we get Noting that , we derive from (30) (2) We finally show that , , , , are the solution of (11a) and (11b) with replaced by . Evidently we only need to prove , , , , satisfy (11b); that is, .
Noting that , we haveFrom (15), a direct computation leads to Noticing that then , we obtain by the tedious computationSubstituting (33b) into (33a), we getIn addition, since are the solutions of (11a) and (11b), we have hence Noting and differentiating both sides of this equation with respect to lead to Rewriting (33c) leads to Combining (32a) and (32b) and (35) and (36), we get Substituting (32b), (37a), and (37b) into (32a), we haveNoting , we immediately get from (38) This completes the proof.

Theorem 4 (the -times repeated nonauto DBT). Given are the solution for -NKdVHSCS ((10a), (10b), and (10c)), are independent eigenfunctions of (11a) and (11b) with . , are functions of such that .
Denote . The -times repeated DBT is defined by wherethen satisfy (10b) and (10c) and (11a) and (11b) with replaced by ; hence are the solution of -NKdVHSCS ((10a), (10b), and (10c)) with replaced by .

Proof. With the same method as Theorem 3, we can show that , satisfy (10b), (10c), and (11a). Here we only need to show satisfy (11b). Next we will show it by the mathematical induction method. Theorem 3 indicates satisfy (11b) in the case of .
Provided that satisfy (11b) for , Noticing that , then when , we have simplifying leads toFrom (40f), we obtain Substituting (44b) into (44a) yields From (37a) for one DBT , we have Note that satisfies and that Differentiating both sides of (47b) with respect to yields we obtain Combining (43), (45), (46), and (49), we get This completes the proof.

5. Soliton Solution of -KdVHSCS

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

Proposition 5. Let be two independent wave functions of (12e), , under the nonauto DBT, and the transformed coefficients are given by where

Proof. It was shown in [7] that formula (51) holds for (12a), (12b), (12c), (12d), and (12e) and that Noting that , then we have From (54), we get Noticing that (12c) implies we have Next we considerNoting (37b), we can immediately derive Hence we obtain from (57) This completes the proof.

Next we will start from the trivial solution to (12a), (12b), (12c), (12d), and (12e) without sources, that is, , and use Theorem 3 and Proposition 5 to construct one soliton solution to (12a), (12b), (12c), (12d), and (12e) with . When , then ; hence, the wave functions of Lax operator satisfyWe take the solution of system (61) as follows: where denotes the -exponential function satisfying with an equivalent form Noting , where are defined by (62), we get from (51) and (52)