A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

Zhang, Ning; Xu, Xi-Xiang

doi:https://doi.org/10.1155/2021/9912387

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2021 | Article ID 9912387 | https://doi.org/10.1155/2021/9912387

A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

Ning Zhang¹and Xi-Xiang Xu²

Academic Editor: Rigoberto Medina

Received01 Apr 2021

Accepted14 Aug 2021

Published30 Aug 2021

Abstract

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

1. Introduction

Over the past three decades, the integrable nonlinear differential-difference systems (INDDEs) have received considerable attention. Many INDDEs have been proposed and studied [1–15]. Finding new INDDEs is still an important and difficult work. The discrete zero curvature representation is one of the most significant tools to generate the INDDEs. Furthermore, the Hamiltonian structure of the INDDEs can be established by discrete trace identity or discrete variational identity [4, 5]. For a family of INDDEsone of the interesting problems in the theory of lattice soliton and integrable systems is to look for a Hamiltonian operator and a sequence-conserved functional so that equation (1) may be represented as the following Hamiltonian form:where the Hamiltonian functional is . The variational derivative is . If we can discover infinitely many involutive conserved functionals for a family of discrete Hamiltonian system, the Liouville integrability of the the discrete Hamiltonian system (2) is proved [9–16].

For a lattice function , the shift operator and the inverse of are defined by

In this paper, we introduce the following spectral problem:where is the eigenfunction vector, is the spectral parameter and , is the potential vector, and depend on integer and real . Starting from spectral problem (4), a novel family of INDDEs are deduced. We are going to prove that the obtained family has a triple Hamiltonian structure (tri-Hamiltonian structure). Furthermore, its Louville integrability is presented. It should be pointed out that, in the theory of lattice soliton and integrable systems, the integrable families, which possess concise tri-Hamiltonian structure, are very rare [12–18]. Also, the symmetrical algebraic structure of equations is an important research direction for INDDEs [19–24]. Furthermore, the research of nonisospectral INDDEs has been widespread concern, the algebraic structure of isospectral and nonisospectral vector fields is established in [16–18], and the key of the theory is to derive the corresponding nonisospectral family of INDDEs.

This paper is organized as follows. In Section 2, starting from the matrix spectral problem (4), by means of the discrete zero curvature representation, we derive a family of INDDEs. In Section 3, we establish a triple Hamiltonian structure (tri-Hamiltonian structure) for the obtained integrable family through the discrete trace identity [4]. Infinitely many commuting symmetries and infinitely many commuting conserved functionals for the obtained family are given. The Louville integrability of the obtained family is demonstrated. In Section 4, a nonisospectral integrable family associated with the obtained family is deduced by solving an initial nonisospectral discrete zero curvature equation and the corresponding characteristic operator equation. Lie algebra of isospectral and nonisospectral vector fields is presented. Finally, in Section 5, there will be some conclusions and remarks.

2. The Family of Integrable Differential-Difference Equations

In this section, we shall derive a family of integrable differential-difference equations associated with eigenvalue problem (4). To this end, we first solve the following stationary discrete zero curvature equation:

Upon settingwe find that equation (5) becomes

Substituting expansionsinto (7) and comparing each power of in the equations of (7), we obtain the initial conditions:and the recursion relations:

Proposition 1. If the initial values are chosen bythen , which are solved by equation (10), are all local, and they are just rational functions in the two dependent variables and .

Proof. On the basis of second and third equations in equation (10), we see that and can be solved locally by , and . In order to obtain from the first equation in the equation (10), we need to use operator to solve the corresponding difference equation. In the following, we are going to show that may be deduced through an algebraic method rather than by solving the difference equations. From (5), we know thatThis tells us , where is an arbitrary function of time variable only [10, 14]. Furthermore, we select . Then, we obtain a recursion relation for :Therefore, can be determined locally by , and , and then are all local and they are just rational functions in the two dependent variables and .
The proof is completed.
In particular, we haveSetLet us introduce the following auxiliary spectral problems associated with the spectral problem (4):Then, the compatibility condition of (4) and (16)is equivalent to the discrete zero curvature equationswhich give rise to the family of integrable differential-difference equations:When , (19) becomes a trivial linear system:And, when , we obtain the first INDDE in family (19) as follows:

3. Tri-Hamiltonian Structure

Next, we shall establish a tri-Hamiltonian structure for integrable family (19). First, we introduce some concepts. The variational derivative, the Gateaux derivative, and the inner product are defined, respectively, bywhere are required to be rapidly vanished at infinity and denotes the standard inner product of and in the Euclidean space . The adjoint operator of is defined by . If an operator has the property , then is said to be skew-symmetric. If a skew-symmetric operator meets the Jacobi identity, i.e.,then operator is called a Hamiltonian operator. Based on a given Hamiltonian operator , we can define a Poisson bracket [4]:

Following [4], we set is defined as , where and are the some order square matrices. Hence,

Then, the discrete trace identity becomes

Substituting expansions into (28) and comparing the coefficients of , we arrive at

When in equation (29), through a direct calculation, we find that . Thus, equation (29) can be written aswhere

Moreover, we havewherewherewith

For three arbitrary constants , it is easy to verify that the operator is a skew-symmetric operator, i.e.,. Furthermore, by a straightforward and lengthy calculation, we can prove that the operator fulfills the Jacobian identity (22). So, we can get the following proposition.

Proposition 2. For all values of three arbitrary constants , is a Hamiltonian operator.
Furthermore, we can obtain that integrable family (19) possesses tri-Hamiltonian structureMoreover, from (7), we find the recursion relationwherewhere is a recursion operator. By means of the operator , we have

Now, we would like to prove the Liouville integrability of the discrete Hamiltonian systems (36). It is crucial to show the existence of infinite involutive conserved functionals.

Proposition 3. are conserved functionals of the whole family (19). And, they are in involution in pairs with respect to the Poisson bracket (24).

Proof. A direct calculation showsthat is,So,Repeating the above argumentation, we can get thatThen, combining the above two equations, we haveThe proof is finished.
Based on system (39) and Proposition 3, we can obtain the following theorem.

Theorem 1. Each INDDE in family (19) is Liouville-integrable discrete Hamiltonian system and has tri-Hamitonian structure (36).

4. Lie Algebraic Structure of Vector Fields

In Section 2, we have discussed the spectral problem (4) in the isospectral case, namely, . Next, we shall investigate the spectral problem (4) in the nonisospectral case. Now, let us suppose that in (4). First, we shall derive a nonisospectral integrable family using the method proposed in [23, 24]. The nonisospectral initial zero curvature equation is presented. In this equation, the corresponding solutions are easily solved as follows:

Now, we introduce the operator equation:

In the above equationhere, is the undetermined function. This (47) is called a characteristic operator equation [23, 24]. Through a straightforward computation, the following results are obtained:

Let us set

Then, we obtain that

Thus, we get a integrable family of nonisospectral discrete evolution equationsand based on (51), the integrable family has the following nonisospectral zero curvature representation

Following [23, 24], let be vector fields; be matrix operators; and be smooth functions, then the following Lie bracket can be defined bywhere

We start our calculation as follows:

By the above results, we can get

Moreover, it is easy to find that is an injective map. Using the above Lie algebra relations, we can arrive at the Lie algebra of vector fields for the isospectal and nonisospectal families (19) and (52) as follows:

From equations (58) and (59), we obtain

Hence, are all the master symmetries of . Furthermore, the vector field is the symmetries of . Here,constitute a symmetry algebra of Witt type.

5. Conclusions and Remarks

In this paper, we have deduced a family of integrable differential-difference equations through the discrete zero curvature equation. With the help of discrete zero curvature equations, we derive a family of INDDEs. A triple Hamiltonian structure (tri-Hamiltonian structure) for the obtained integrable family is established by the discrete trace identity. Louville integrability of the obtained family is proved. Then, a nonisospectral integrable family associated with the obtained family is derived by solving an initial nonisospectral discrete zero curvature equation and the corresponding characteristic operator equation. Ultimately, Lie algebra of isospectral and nonisospectral vector fields is obtained. In addition, there are other problems worth further investigation for the integrable family (9). For example, the inverse scattering transformation, Hamiltonian structures, binary nonlinearization of Lax pairs, and integrable coupling systems by semidirect sums of Lie algebra.

Data Availability

The data used to support the findings of this study are available from the corresponding author on reasonable request.

Conflicts of Interest

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

Acknowledgments

This work was supported by the Youth Science Foundation Project of China (Grant no. 11805114).

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Copyright © 2021 Ning Zhang and Xi-Xiang Xu. 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.

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