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A differential-difference Davey-Stewartson system with self-consistent sources is constructed using the source generation procedure. We observe how the resulting coupled discrete system reduces to the identities for determinant by presenting the Gram-type determinant solution and Casorati-type determinant solution.

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 [

The Davey-Stewartson system is an integrable

Since the pioneering works of Mel’nikov [

In [

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 [

The differential-difference Davey-Stewartson system reads [

where

If we apply the dependent variables transformations

Equations (1)-(3) can be transformed into the following bilinear Equations [

where, as usual, the bilinear operators

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

where

with

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

with

Using Equations (10)-(11), we can calculate the

where

Other functions appearing in Equations (5)-(7) such as

Substituting Equations (15), (17) and

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

where

If we introduce two new fields

then we have shown that

In the same way, substituting (15) (17) and

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

If we introduce another two new fields

then we have shown that

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

and

for

and

Similarly, bilinear equations

and

for

and

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

So bilinear Equations (7), (21), (25) and (28)-(31) for

the bilinear Equations (7), (21), (25) and (28)-(31) for

It is shown in [

Let us introduce the following double-Casorati determinant:

where for

in which

with

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

where the pfaffian elements are defined by

in which

Proof: The double Casorati determinants in (11)-(13) can be expressed by pfaffians [

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

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 [

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).