#### Abstract

Integrable coupling system of a lattice soliton equation hierarchy is deduced. The Hamiltonian structure of the integrable coupling is constructed by using the discrete quadratic-form identity. The Liouville integrability of the integrable coupling is demonstrated. Finally, the discrete integrable coupling system with self-consistent sources is deduced.

#### 1. Introduction

Many physical problems may be modeled by soliton equation. The Hamiltonian structures of many systems have been obtained by the famous trace identity [1–6]. The study of integrable couplings of integrable systems has become the focus of common concern in recent years. It originates from the investigations on the symmetry problems and associated centerless Virasoro algebras [7]. Many integrable coupling systems have been constructed by using the methods of a direct method [8], perturbations [9], enlarging spectral problems [10, 11], creating new loop algebras [12, 13], and semidirect sums of Lie algebras [14, 15]. The Hamiltonian structures of the integrable couplings of lattice equations can be constructed by means of the discrete quadratic-form identity [16, 17].

Since Mel’Nikov proposed a new kind of integrable model which was called soliton equations with self-consistent sources [18] in 1983, many soliton equations with self-consistent sources [19–23] have been presented in recent years. For applications, these kinds of systems are usually used to describe interactions between different solitary waves. In this paper, we deduce a hierarchy of discrete integrable coupling system with self-consistent sources which are few compared with the continuous ones.

The paper will be organized as follows. We first get a hierarchy of integrable lattice soliton equation with self-consistent sources in Section 2. In Section 3, a hierarchy of discrete integrable coupling system is derived by making use of the discrete zero curvature representation. By means of the discrete quadratic-form identity we establish the Hamiltonian structures of the hierarchy. Further, the resulting Hamiltonian equations are all proved to be integrable in Liouville sense. Finally, we give the integrable coupling systems with self-consistent sources.

#### 2. A Hierarchy of Integrable Lattice Soliton Equations with Self-Consistent Sources

We first briefly describe our notations. Assume is a lattice function; the shift operator and the inverse of are defined by

A system of discrete equations is said to have a discrete Lax pair if it is equivalent to the compatibility condition In [16], a Lie algebra is presented as where Set and ; it is easy to see that , , and construct three Lie algebra, and So is an Abelian ideal of the Lie algebra . The corresponding loop algebra is defined by

In [15], a new discrete matrix spectral problem has been proposed: by solving the stationary discrete zero curvature equation where and introducing the auxiliary spectral problems associated with the spectral problem (9) a hierarchy of integrable lattice soliton equations with a potential has been presented: where Equation (13) possesses the following Hamiltonian forms [15]: where

Next, we will construct a hierarchy of integrable lattice soliton equations (13) with self-consistent sources. For distinct real , consider the auxiliary linear problem Based on the results in [24], we show the following equation: where According to the approach proposed in [24–26], through a direct computation, we obtain the discrete integrable hierarchy with self-consistent sources as follows:

Taking in the above system, under , we can obtain the following equation with self-consistent sources:

#### 3. A Hierarchy of Discrete Integrable Coupling System with Self-Consistent Sources

First, we will give out the integrable couplings of the hierarchy (13). Consider the discrete isospectral problem in which is the potential, and are real functions defined over , is a spectral parameter, , and is the eigenfunction vector.

We solve the stationary discrete zero curvature equation where Equation (23) gives Substituting the expansions into (25), we can get the recursion relation The initial values are taken as Note that the definition of the inverse operator of does not yield any arbitrary constant in computing and , . Thus, the recursion relation (27) uniquely determines and the first few quantities are given by Set so Take , , and let We introduce the auxiliary spectral problems associated with the spectral problem (22): The compatibility conditions of (22) and (34) are which give rise to the following hierarchy of integrable lattice equations:

So (35) is the discrete zero curvature representation of (36); the discrete spectral problems (22) and (34) constitute the Lax pairs of (36), and (36) are a hierarchy of Lax integrable nonlinear lattice equations. It is easy to verify that the first nonlinear lattice equation in (36), when , under , is

In (36) the first lattice equations constitute a hierarchy of integrable lattice soliton equations with a potential ; in the view of integrable coupling theory [7, 13, 17], (36) are integrable coupling systems of (13) or (15).

In what follows, we would like to establish the Hamiltonian structures for the integrable coupling systems (36).

Set , , and . We define a map Following [16], we introduce the matrix

It is easy to verify that meets . Under the definition of the quadratic-form function we have and . Set ; through a direct calculation, we get By the discrete quadratic-form identity [16] with being a constant to be determined, we have By the substitution of into (44) and comparing the coefficients of in (44), we get When in (46), a direct calculation shows that . So we have Set

Now we can rewrite those lattice equations in (36) as where is a local difference operator defined by where Obviously, the operator is a skew-symmetric operator; that is, . Moreover, we can prove that the operator satisfies the Jacobi identity

So we have the following facts.

Proposition 1. *is a discrete Hamiltonian operator.*

Set From the recursion relation (27) we can get the recursion operator in (53).

Therefore, we have So (49) are a family of Hamiltonian systems. The hierarchy of lattice equations (36) possesses Hamiltonian structures (54). Furthermore, a direct calculation shows that It is easy to verify that the operator is a skew-symmetric operator; that is, . So we have the following.

Proposition 2. * defined by (48) forms an infinite set of conserved functionals of the hierarchy (36), and , , are involution in pairs with respect to the Poisson bracket.*

*Proof. *We can find that . Namely, , and then . Hence
Similarly, we get
This implies that
Thus

In summary, we obtain the following theorem.

Theorem 3. *The lattice equations in (36) or the discrete Hamiltonian equations in (49) are all discrete Liouville integrable Hamiltonian systems.*

Now we search for the integrable coupling systems with self-consistent sources. For distinct real , consider the auxiliary linear problem Based on the results in [24], we show the following equation: where According to the approach proposed in [24–26], through a direct computation, we get the discrete integrable hierarchy with self-consistent sources as follows: When in the above system, under , we can obtain the following coupling equations with self-consistent sources:

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the Global Change and Air-Sea Interaction (no. GASI-03-01-01-02), National Natural Science Foundation of China (no. 61304074), the Nature Science Foundation of Shandong Province of China (no. ZR2013AQ017), the Science and Technology Plan Project of Qingdao (no. 14-2-4-77-JCH), the Open Fund of the Key Laboratory of Ocean Circulation and Waves, the Chinese Academy of Science (no. KLOCAW1401), the Open Fund of the Key Laboratory of Data Analysis and Application, and the State Oceanic Administration (no. LDAA-2013-04).