The study of discrete integrable system has become an active area of research for over thirty years. Various integrable discretization methods have been proposed to produce the discrete analogues of integrable systems. One powerful technique to find the integrable discretization is the Hirota’s bilinear method [1] -[6] . The traditional Hirota’s discretization of integrable equations relies on gauge invariance and soliton solutions, while the modi- fied Hirota’s approach [5] [6] emphasizes on discretizing integrable bilnear equations such that the resulting discrete bilinear equations have bilinear Bäcklund transformations.

The Davey-Stewartson system is an integrable -dimensional generalization of the nonlinear Schrödinger system. In [7] , the authors applied the modified Hirota’s approach to the Davey-Stewartson system to produce an integrable differential-difference Davey-Stewartson system which is characterized by determinant solutions, bilinear Bäcklund transformation and lax pair. This differential-difference Davey-Stewartson system also can be derived as a reduction of a -dimensional generalization of the Ablowitz-Ladik lattice [8] .

Since the pioneering works of Mel’nikov [9] , the soliton equations with self consistent sources have received considerable attention. Soliton equations with self consistent sources are integrable coupled generalization of the original soliton equations, and some of such type of equations have found important physical applications. A variety of methods have been proposed to deal with these soliton equations with sources, such as inverse scattering methods [9] -[13] , Darboux transformation methods [14] -[17] , Hirota’s bilinear method and Wronskian technique [18] -[28] etc. However, most results have been achieved in continuous case. Comparatively less work has been done in discrete case. In view of this unsatisfactory situation, it would be interesting to produce new discrete soliton equations with self consistent sources.

In [27] , a direct method, called the source generalization procedure, was proposed to construct and solve the soliton equations with self consistent sources. In this paper, we apply the source generalization procedure to construct and solve the differential-difference Davey-Stewartson system with self-consistent sources.

The outline of this paper is as follows. In Section 2, the differential-difference Davey-Stewartson system with self-consistent sources is produced and its Gram-type determinant solutions are presented. In Section 3, the Casorati-type determinant solutions to the differential-difference Davey-Stewartson system with self-consistent sources is derived. Finally, Section 4 is devoted to a conclusion.

In [7] , a differential-difference Davey-Stewartson system which is an integrable discretization of the DSI system is proposed, and the double-Casorati and Grammian determinants solutions to this discrete Davey-Stewartson system are derived. In this section, we first review the Grammian determinant solutions for the discrete Davey- Stewartson system and then apply the source generation procedure to this system to produce a differential-dif- ference Davey-Stewartson system with self-consistent sources.

The differential-difference Davey-Stewartson system reads [7]

where, , and are constants. In Equations (1)-(3) and in the following we always use a notational simplification for by writing explicitly a discrete variable only when it is shifted from its position. For example,

.

If we apply the dependent variables transformations

Equations (1)-(3) can be transformed into the following bilinear Equations [7] [8] :

where, as usual, the bilinear operators and [28] are defined as:

,

.

The Grammian determinant solutions for the differential-difference Davey-Stewartson system (5)-(7) is given by [7] :

where is a matrix, is a matrix of constant elements , is a matrix with block structure, and are column vectors

,

with, , , satisfying the following equations:

We are now in a position to construct the differential-difference Davey-Stewartson system with self-consistent sources by applying the source generation procedure. Firstly, we change Grammian determinant solutions (8)- (11) of Equations (5)-(7) to the following form:

where the matrix satisfies

with being an arbitrary function of, being a positive integer, and, are defined as before.

Using Equations (10)-(11), we can calculate the -derivatives of the in (12)-(13) in following way:

where denotes a matrix resulting from eliminating the th row and th column from the matrix, and, denote vectors resulting from eliminating the th element from, respectively.

Other functions appearing in Equations (5)-(7) such as, , , , , , , , , can also be expressed in terms of Grammian determinants which are the same as the results given in [7] .

Substituting Equations (15), (17) and, , , expressed by means of Gram- mian determinants given in [7] into the left side of Equation (6), and then applying the Jacobi identities for the determinants [28] , we finally obtain

Using the Jacobi identities for the determinants again, Equation (22) is equal to

where, denote matrices resulting from eliminating the rth row and jth column, respectively, from the matrix.

If we introduce two new fields for defined by

then we have shown that given in (12)-(13) and given in (20) satisfy the following bilinear equation:

In the same way, substituting (15) (17) and, , , expressed by means of Grammian determinants given in [7] into the left side of the Equation (6), and then applying the Jacobi identities for the determinants, we finally obtain

Using the Jacobi identities for the determinants again, Equation (22) is equal to

If we introduce another two new fields for defined by

then we have shown that given in (12)-(13) and given in (24) satisfy the following bilinear equation:

There are more quadratic relations between the fields introduced. For example, the determinant identities

and

for yield the bilinear equations

and

Similarly, bilinear equations

and

for can be derived from the determinant identities

and

The determinant identities (26)-(27) and (32)-(33) are special cases of the pfaffian identity [28] ,

So bilinear Equations (7), (21), (25) and (28)-(31) for construct the differential-difference Davey-Stewartson system with self-consistent sources, and functions F, G, H and for in Equations (3), (12), (20), (24) are the Gram-type determinant solutions of the differential-difference Davey- Stewartson system with self-consistent sources. Under the dependent variable transformations

,

the bilinear Equations (7), (21), (25) and (28)-(31) for are transformed into the following nonlinear equations:

It is shown in [7] that the differential-difference Davey-Stewartson system exhibits N-soliton solutions expressed by means of two types of determinants, double-Casorati and Grammian determinants. It is natural to consider if the differential-difference Davey-Stewartson system with self-consistent sources have two types of determinant solutions. In this section, we shall derive another class of determinant solutions, Casorati-type determinant solutions to the differential-difference Davey-Stewartson system with self-consistent sources (7), (21), (25) and (28)-(31) for.

Let us introduce the following double-Casorati determinant:

where for,

in which satisfies

with being an arbitrary function of, is an arbitrary constant and being a positive integer, and, , , satisfy the following equations:

From now on the determinant (42) will, for simplicity, be denoted as

Taking into account Equations (42)-(48), we can state the following Proposition:

Proposition 1 The solutions to Equations (7) (21) (25) and (28)-(31) for can be expressed as the following double-Casorati type determinants:

where the pfaffian elements are defined by

in which are integers, , , and in the pfaffians indicates deletion of the letter under it.

Proof: The double Casorati determinants in (11)-(13) can be expressed by pfaffians [28] in the following way:

where the pfaffian elements are given in (56)-(58).

We first show that functions (49)-(55) satisfy Equations (21) and (25). Using Equations (43)-(47), we can calculate the following differential and difference formula for:

Substitution of Equations (52)-(55) and (62)-(72) into Equations (21) and (25) yields the following determinant identities, respectively:

and

It is easy to show that (49)-(51) satisfy Equation (7). Now we prove that functions (49)-(55) satisfy Equations (28)-(31). From Equations (52)-(58), we can derive the difference formula for pfaffians, as follows:

Substituting Equations (59)-(60), (63)-(64), (71)-(72) and (75)-(80) into Equations (28)-(31), we obtain the following determinant identities, respectively:



In this paper, we apply the source generation procedure to the differential-difference Davey-Stewartson system (1)-(3) to generate a differential-difference Davey-Stewartson system with self-consistent sources (35)-(41), and clarify the algebraic structures of the resulting coupled discrete system by expressing the solutions in terms of two types of determinants, Casorati-type determinant and Gram-type determinant.

In [29] , a Davey-Stewartson equation with self-consistent sources is constructed. It would be of interest to find the proper reduction and certain continuous limits which give the Davey-Stewartson equation with self- consistent sources investigated in [29] from the differential-difference Davey-Stewartson system with self-con- sistent sources (35)-(41).

The author would like to express her sincere thanks to Prof. Xing-Biao Hu for his helpful discussions and encouragement. This work was supported by the program of higher-level talents of Inner Mongolia University (2011153) and the National Natural Science Foundation of China (Grant No. 11102212).