Algebro-Geometric Solutions for a Discrete Integrable Equation

With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.


Introduction
As we all know, the generation of integrable system, determination of exact solution, and the properties of the conservation laws are becoming more and more rich [1][2][3][4][5]; in particular, the discrete integrable systems have many applications in statistical physics, quantum physics, and mathematical physics [6][7][8][9][10][11].It is worth discussing the properties of discrete integrable systems, such as Darboux transformations [12,13], Hamiltonian structures [14][15][16], exact solutions [17], and the transformed rational function method [18].In the past decades, some methods have been proposed to gain explicit solutions of the continuous soliton equations, for instance, the algebro-geometric method [19,20], the inverse scattering transformation [21], the B ä cklund transformation [22], and the sine-cosine method [23].However, it is very hard to obtain algebro-geometric solutions for discrete soliton equations due to the treatment of discrete variables.In 1975, Its and Matveev first presented the algebro-geometric approaches [24], which permitted us to seek out a class of exact solutions to the soliton equations.The elliptic functions and multisoliton solutions may be acquired by these degenerated solutions [25].Recently, Qiao et al. further improved the algebrogeometric methods by making use of the nonlinearization theory [26][27][28][29].Trigonal curves are also systematically used to construct algebro-geometric solutions [30,31].But we note that there is few research to focus on the algebro-geometric solutions of discrete soliton equations.
In this paper, we will generate the algebro-geometric solutions of the discrete integrable system by taking advantage of the Riemann-Jacobi inversion theorem and Abel coordinates.In Sections 2 and 3, we will construct a new discrete integrable system by using Lie algebra and spectral problem.By introducing Abel-Jacobi coordinates, straightening out of the continuous and discrete flows will be given and placed in Section 4. Section 5 will be devoted to derive the algebro-geometric solutions of the abovementioned discrete integrable equation by utilizing the Riemann theta function.

The Discrete Integrable Hierarchy
We consider the algebra which is the simple subalgebra of the Lie algebras  1 , and corresponding loop algebras can be expressed as According to the loop algebras, we introduce the following discrete spectral problems where Thus where According to the following stationary discrete zero curvature equation for   , we get Substituting ( 6) into (8) yields where Δ =  − 1.
We choose the initial values  0 = −1/2,  0 = 0 and need to select zero constants for the inverse operation of the difference operator Δ in computing   ,  ≥ 1.On this condition, recursion relations (9) uniquely determine   ,   ,   ,  ≥ 1.Then, we obtain the first few quantities From we have the discrete zero curvature equation where Thus, we obtain the following integrable discrete hierarchy And with  = 1.Equation ( 15) can be read as It is easy to find that the Lax pair of ( 15) is given by where ) .
Equation ( 19) implies that The discrete integrable hierarchical ( 14) could be rewritten as generation of the following, so spectrum problem is where From the compatibility conditions of the discrete Lax pair (23), we can read that the hierarchical equation is Thus, we also have ) . (26)
Hence, it follows that Again from (37) and (44), we have Similarly, when  = 2 Thus

Straightening out of the Continuous and Discrete Flows
In order to acquire the algebro-geometric solutions of systems ( 16), we first introduce the Riemann surface Γ of the hyperelliptic curve with genus : which has two infinite points ∞ 1 and ∞ 2 , not branch point of Γ.We fix a set of regular cycle paths:  1 , . . .,   ;  1 , . . .,   , which are independent and have the intersection numbers: We choose the holomorphic differentials, on and define where  = (  ) × ,  = (  ) × .Thus, we denote the matrices  and  by and verify that  is symmetric and has positive defined imaginary part.By normalizing ω into the new basis   , which meets The Abel map A() is introduced as and the Able-Jacobi coordinates are defined as where and  0 is a chosen base point on Γ.
Remark 1.We have concluded the algebro-geometric solutions of the discrete system (16).It is significance of a major work for investigating numerical solutions of the discrete integrable system ( 16) like the way presented in [32].
Comparing the numerical solutions and algebro-geometric solutions about the discrete integrable system, we can get lots of useful properties.These problems will be studied in the future.