Advances in Mathematical Physics
Volume 2016 (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

Hongxia Wu,1,2 Liangjuan Gao,¹ Jingxin Liu,¹ and Yunbo Zeng³

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 [1–17]. 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 [9–17]. 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 satisfy

Let be the formal inverse of such as . In general, the -deformed Leibnitz rule holds

where -number and -binomial are defined by

For a -PDO , we separate into the differential part and the integral part . The conjugate operation is given by

where .

The -exponential function is defined as

satisfying .

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 have

Differentiating both sides of (25) with respect to yields

From (23), (24), and (26), we have

This 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). Consider

Taking 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 have

From (15), a direct computation leads to

Noticing that

then , we obtain by the tedious computation

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

In 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 have

Noting , 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

where

then 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 to

From (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 consider

Noting (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 satisfy

We 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)

In addition, by Theorem 3, we obtain

where , and satisfy

Then (65a), (65b), (65c), (65d), and (65e) present one soliton solution of (12a), (12b), (12c), (12d), and (12e) with . In particular, when , where is an arbitrary constant, (65a), (65b), (65c), (65d), and (65e) can be reduced to one soliton solution to the first nontrivial equation of the -KdV hierarchy [7]. Certainly, we also use Theorem 4 and Proposition 5 to construct the multisoliton solution to (12a), (12b), (12c), (12d), and (12e). But owing to the complexity of the computation, we omit it here.

6. Summary

As -reduction of the extended -deformed KP hierarchy, -NKdVHSCS is explored in this paper. Two kinds of DBTs are constructed and the soliton solution to the first nontrivial equation of -KdVHSCS is also obtained. We find that one of the DBTs provides a nonauto Bäcklund transformation for the two -NKdVESCS with different degree, which enables us to obtain the new solution of -NKdVHSCS from the known solution of -NKdVH. Noting that we only investigate DBT and solution of -NKdVESCS, other integrable structures will be studied in our forthcoming paper such as infinite conservation law, tau function, and Hamiltonian structure.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant nos. 11201178 and 11171175), Fujian National Science Foundation (Grant no. 2012J01013), Fujian Higher College Special Project of Scientfic Research (Grant no. JK2012025), Fujian provincial visiting scholar program, and the Scientific Research Foundation of Jimei University, China.

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