1 Introduction

It is an important task in soliton theory to find integrable lattice system such as those connecting with well-known physical meaning equations. In past decades, lots of nonlinear integrable lattice soliton systems have been obtained and discussed successfully, e.g. the Ablowitz–Ladik lattice,^[1] the Toda lattice,^[2] the differential-difference KdV equation,^[3] the Blaszakl–Marciniak lattice,^[4–5] and so on.^[6–10] It is well known, there are many systematic approaches to obtain explicit solutions of lattice soliton systems, such as the inverse scattering transformation,^[11] the Hirota technique,^[12] the algebra-geometric method,^[13] the Darboux transformation,^[14–15] etc. Among them, it has been proved that Darboux transformation is one of the most fruitful algorithmic procedures to get explicit solutions of nonlinear partial differential equations from a trivial seed.^[16–22]

In this paper, we consider a discrete spectral problem

where , are potentials, and λ is the spectral parameter and . For a lattice function , the shift operator E, the inverse of E and difference operator D are defined as follows

If let , , then the spctral problem (1) can be changed into the spectral problem

which was considered by Ding et al., where only a positive discrete hierarchy and 1-fold Darboux transformation were obtained.^[23] So in this paper, we would like to further consider a negative discrete hierarchy and its properties associated with spectral problem (1).

The organization of this paper is as follows. In Sec. 2, we first establish a negative discrete hierarchy related to the spectral problem (1). In Sec. 3, it is shown that the hierarchy is integrable in Liouville sense and possesses bi-Hamiltonian structure. In Sec. 4, an N-fold Darboux transformation for negative discrete hierarchy is established with the help of gauge transformations of Lax pair. As an application, some exact solutions for a discrete equation in the negative hierarchy are given in Sec. 5.

2 A New Negative Discrete Hierarchy

In order to get the discrete integrable system, we first proceed to solve the stationary discrete zero curvature equation

where

From Eq. (5), it is easy to obtain the following recursion relations

Substituting

into Eq. (6) and comparing each power of λ yields

If we choose the initial values as

then , , () are determined from the recursion relations (7). In particular, the first set is

For any integer , we choose

Direct calculation shows that

Consider auxiliary spectral problem

then compatibility condition between Eqs. (1) and (12) gives zero curvature equation

which is equivalent to integrable discrete hierarchy

We can give the first two discrete integrable systems in the hierarchy:

When m = 0, the hierarchy (13) gives

When m = 1, the hierarchy (13) gives

Eq. (14) is a new negative discrete system, whose time part of the Lax pairs is

3 Bi-Hamiltonian Structures of the Hierarchy

Define

where

By implying the discrete trace identity

we have

with ε to be determined later. Substituting expressions

into Eq. (20) and comparing the coefficient of , we obtain

Taking m = 1 in Eq. (21), a direct calculation shows that ε = 0, so we get

The hierarchy (13) can be re-written in the form

where

where

It is easy to verify that J and K are all skew-symmetric operators, i.e. , . Then substituting Eq. (22) into Eq. (23) gives the bi-Hamitonian structure for the hierarchy (13)

We can further prove the following theorem:

4 -Fold Darboux Transformation

As an illustrative example, we construct the N-fold Darboux transformation for the system (14) based on its Lax pair

In fact, the Darboux transformation of the system (14) is a special gauge transformation of Lax pair (26) and (27). Consider the following gauge transformation

where T_n is a Darboux matrix. The eigenfunction should satisfy the following new Lax pair (26) and (27) with U_n and replaced by and respectively

We can construct the Darboux matrix T_n as follows

where , , , are given by a linear algebraic system

where are two basic solutions of Lax pair (26) and (27), and are parameters suitably chosen so that the determinant of the coefficients of the system (31) is nonzero. According to Eqs. (30)–(32), we can show that is a (2N)-th order polynomial of λ

By using above fact, we can prove the following proposition.

By using above facts, we can obtain the following theorem:

5 The Exact Solutions

In this section, we will give some exact solutions of system (14) via Darboux transformation (39). Taking the trivial solution , , we obtain two kinds of exact solutions of the Lax pair (26) and (27) with as follow

According to Eqs. (32) and (36), we have

By use of Cramer’s rule, the linear algebraic system (41) leads to

with

Next, we give some exact solutions of the system (14) via Darboux transformation (39) for two cases when N = 1 and N = 2 respectively.