Complexity
Volume 2019 (2019), Article ID 5984356, 10 pages
https://doi.org/10.1155/2019/5984356

Research Article

Explicit Solutions and Conservation Laws for a New Integrable Lattice Hierarchy

Qianqian Yang, Qiulan Zhao, and Xinyue Li

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590 Shandong, China

Correspondence should be addressed to Qiulan Zhao; qlzhao@sdust.edu.cn

Received 15 February 2019; Accepted 28 May 2019; Published 17 June 2019

Academic Editor: Eulalia Martínez

Copyright © 2019 Qianqian Yang 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.

An integrable lattice hierarchy is derived on the basis of a new matrix spectral problem. Then, some properties of this hierarchy are shown, such as the Liouville integrability, the bi-Hamiltonian structure, and infinitely many conservation laws. After that, the Darboux transformation of the first integrable lattice equation in this hierarchy is constructed. Eventually, the explicitly exact solutions of the integrable lattice equation are investigated via graphs.

1. Introduction

Investigation of nonlinear integrable lattice equations [1–6] is an active topic in the field of nonlinear science because its solutions can explain some physical phenomena [7–10]. Researchers in soliton field have proposed many valuable nonlinear integrable lattice equations [11–15] until now, but there are still many integrable lattice equations that have not been discovered. Generally speaking, nonlinear integrable lattice equations are related to their matrix spectral problems, respectively. Therefore, finding suitable matrix spectral problems is of great significance for us to derive integrable lattice equations and investigate their explicitly exact solutions.

It is well known that it is not easy to get solutions for integrable lattice systems. Currently, there are several useful approaches which can help us obtain explicit solutions, for instance, the binary nonlinearization [16–18], Lie symmetry analysis [19, 20], and the Darboux transformation [21–29]. The Darboux matix method is one of important and effective techniques to obtain explicitly exact solutions of integrable lattice equations [30–34].

In this paper, we will focus on this spectral problem:

and its auxiliary problem,

where , and are potentials, and is a spectral parameter independent of t. According to the above spectral problem, an integrable lattice hierarchy in Liouville sense is presented firstly. Then, we investigate its bi-Hamiltonian structures using the discrete trace identity [35] proposed by G. Z. Tu. Its infinitely many conservation laws are also presented. Finally, the explicitly exact solutions of the integrable lattice equation derived from this matrix spectral problem are discussed by DT method.

The structure of the article is as follows. In Section 2, we will obtain a new integrable hierarchy in Liouville sense and present its bi-Hamiltonian structure in accordance with the matrix spectral problem. In Section 3, we will discuss conservation laws of the integrable hierarchy proposed in Section 2. In Section 4, its N-fold DT will be constructed and its exact solutions will be obtained. In Section 5, some conclusions are given.

2. Integrability and Bi-Hamiltonian Structures of a Lattice Hierarchy

Let us first introduce a shift operator , which obeys the following operations:

where is a lattice function.

In order to obtain a hierarchy of integrable lattice equations, we assume that

By solving the stationary discrete zero curvature equation

the following relationships will be derived:

Substituting

into (6) follows the recursion relations

By calculation, we find and is a arbitrary constant. For simplicity of calculation, we take . Then, the recursion relations Equation (8) could uniquely determine , , , and . The first few quantities are given by

and so on. Now, for any integer , we define

In order to derive the associated integrable lattice hierarchy, we give a modification ,

Letting

we can obtain that

Then, the discrete zero-curvature equation

is able to bring about the following integrable lattice hierarchy:

and compose the Lax representation of the integrable lattice hierarchy Equation (15). When m=1, (15) is reduced to the following:

in which is

Next, we try to construct the bi-Hamiltonian structures of (15). In order to achieve this goal, we denote and stands for the trace of the product of two square matrices of the same orders M and N. Then,

The discrete trace identity [35] plays an important role in constructing Hamiltonian structures of integrable lattice equations and studying their integrability. Here, we try to construct Hamiltonian structure of (15) by using

Substituting the expansions and into (19), we have

To get the value of the constant , we simply set in (20) and then find . Therefore we have that

So we obtain the desired Hamiltonian form of (15),

where

In (23), the operator is a Hamiltonian operator for the hierarchy Equation (15), and the operator is a recursion operator for the hierarchy Equation (15). These two operators and satisfy the following relationships:

where stands for the formal conjugation of and

Remark. The hierarchy of lattice equations (15) is an integrable Hamiltonian system in Liouville sense.

The bi-Hamiltonian structure Equation (23) of the integrable lattice equation exists.

3. Conservation Laws of (16)

Based on Lax representation of the hierarchy of integrable lattice equations (15), we can obtain its conversation laws. Substituting into the spectral problem (1), we have

which demonstrates that

where we have .

Then, if we suppose that , the following two recursion relationships will be found. One is

and the other is

Based on (2) and (17), a direct calculation gives

Then, we discuss conversation laws of the hierarchy of integrable lattice equations (15) according to the above two different recursion relationships, respectively. Substituting into (34), equating the coefficients of on both sides of the equation, we can achieve an infinite number of conversation laws of (16) based on the (32). The first two of them are

Similarly, we also have another infinite conservation laws of (16) based on (33). The first two of them are

4. The N-Fold DT and Exact Solutions of Eq. (16)

If a integrable lattice equation with Lax pair and exists a gauge transformation

where is a reversible matrix and

The new potentials and possess the same form with the old potentials and , respectively. The gauge transformation in (37) composes Darboux transformation of an integrable lattice equation with the transformation between old potential functions , and new potential functions   .

Based on (37) and (38), we are also able to obtain the relations, i.e.,

4.1. The N-Fold DT of Eq. (16)

The selection of is very important in constructing the Darboux transformation. The solution of the soliton equation can be obtained much more easily by an appropriate Darboux matrix. For this integrable lattice equation (16), we take

with and being all functions with respect to and and being a positive integer number. It is easy to find that is a (2N)-th order polynomial with respect to . If we assume that the 2N parameters () are the roots of the , can be expressed as

Setting and being the column vectors of , we find that the column vectors of are linear dependent because of . Then for , according to (37), we have

in which

and the parameters and are so chosen that the denominators in (43) are nonzero.

Theorem 1. The matrix defined by (38) has the same form as , i.e.,

where the transformation from the old potentials and into the new potentials and is given by

Proof. Let and

It can be see that and are (2N-1)-th order polynomials with respect to , and and are (2N)-th order polynomials with respect to . From (42), we have

Moreover, (46) can be described as

together with

where are roots of , is a 2-order square matrix, and are the functions with respect to n and t. Thus, (48) can be written as

Equating the coefficients of and on both sides of (50), we find

From (51), we can see that . The proof is completed.

Theorem 2. Under the transformation (37) and (45), the matrix defined by (38) has the same form as , i.e.,

Proof. Let and

in which all are the functions of n and t. Through a series of calculations, we can figure out that , , and are (2N+1)-th order polynomials with respect to , and is (2N)-th order polynomial with respect to . We can verify from (47) that are the roots of . So, we know

together with

where all the are the functions with respect to and . Thus, (54) can be rewritten as

Equating the coefficients of and on both sides of (56), we find that

From (38) and (57), we are able to see that . The proof is completed.

4.2. Exact Solutions of (16)

In this section, the explicit solutions of the integrable lattice equation (16) will be discussed under its Darboux transformation. We can find is a trivial solution of (16) easily. Starting from this trivial solution, the solutions of the Lax Pair Equations (1) and (17) can be obtained,

in which

According to (43), we have

By using of the Cramer rules, solving (42) follows

together with

and is obtained from by replacing its (2N)-th column with , i.e.,

By using (41), (45), and (62), the solution of (16) can be derived, and it is expressed as

In order to investigate the structure of the solutions (65) fully, we plot their structure figures as shown in Figures 1–4 when and .

(a)

(b)

(c) density of

(d) t=-5

(e) t=0

(f) t=5

Figure 1: A structure of soliton solution for in (66) with . Figures (a) and (b) are three-dimensional graphs of . Figure (c) is the density plot of and Figures (d), (e), and (f) are structures of the solution at different time.

(a)

(b)

(c) density of

(d) t=-5

(e) t=0

(f) t=5

Figure 2: A structure of soliton solution for in (66) with . Figures (a) and (b) are three-dimensional graphs of . Figure (c) is the density plot of and Figures (d), (e), and (f) are structures of the solution at different time.

(a)

(b)

(c) density of

(d) t=-5

(e) t=0

(f) t=5

Figure 3: A structure of soliton solution for in (68) with . Figures (a) and (b) are three-dimensional graphs of , Figure (c) is the density plot of , and Figures (d), (e), and (f) are structures of the solution at different time.

(a) 1

(b) 2

(c) density of

(d) t=-5

(e) t=0

(f) t=5

Figure 4: A structure of soliton solution for in (68) with . Figures (a) and (b) are three-dimensional graphs of , Figure (c) is the density plot of , and Figures (d), (e), and (f) are structures of the solution at different time.

(I) In particular, when , solving (42) leads to

where

Figures 1 and 2 show the structures of the soliton solutions and in (65).

(II) When N=2, solving (42) leads to

where

Figures 3 and 4 show the structures of the soliton solutions and in (68).

5. Conclusions

In this article, we have derived an integrable lattice hierachy on the basis of a new matrix spectral problem. Then, some properties of this hierarchy have been shown, such as the Liouville integrability, the bi-Hamiltonian structures, and infinitely many conservation laws. Eventually, the Darboux transformation of the first integrable equation in this hierarchy has been constructed and the exact solutions of the integrable equation have been investigated via graphs. The binary nonlinearization [36–38] is also a useful method to investigate explicit solutions. We will investigate the binary nonlinearization with regard to (16) associated with spectral problems (1) and (2) in subsequent papers.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors greatly appreciates Professor X. Y. Wen for his patient guidance and discussion. The work was supported by the Nature Science Foundation of China (No. 11701334) and the Science and Technology Plan Project of the Educational Department of Shandong Province of China (No. J16LI12).

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