Abstract
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.
1. 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 [1–5]; in particular, the discrete integrable systems have many applications in statistical physics, quantum physics, and mathematical physics [6–11]. It is worth discussing the properties of discrete integrable systems, such as Darboux transformations [12, 13], Hamiltonian structures [14–16], 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 Bcklund 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 algebro-geometric methods by making use of the nonlinearization theory [26–29]. 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.
2. The Discrete Integrable Hierarchy
We consider the algebra which is the simple subalgebra of the Lie algebras , and corresponding loop algebras can be expressed as
According to the loop algebras, we introduce the following discrete spectral problemswhere Thuswhere
According to the following stationary discrete zero curvature equation for ,we getSubstituting (6) into (8) yieldswhere .
We choose the initial values , and need to select zero constants for the inverse operation of the difference operator in computing . On this condition, recursion relations (9) uniquely determine . Then, we obtain the first few quantities
From we have the discrete zero curvature equation where Thus, we obtain the following integrable discrete hierarchyAndwith . Equation (15) can be read asIt is easy to find that the Lax pair of (15) is given bywhere
In the following, we express Lenard's gradient sequences , by the recursion equationwith two operators where .
From the equation and (19), respectively, Equation (19) implies that
The discrete integrable hierarchical (14) could be rewritten as generation of the following, so spectrum problem iswhere
From the compatibility conditions of the discrete Lax pair (23), we can read that the hierarchical equation isThus, we also have
3. Decomposition of the Differential-Difference Equations
In this section, we shall resolve the discrete systems (16) into solvable ordinary differential equations. We assume that (23) has two basic solutions , , and we define a Lax matrix as follows: and should meet the following equations:
It is easy to see that (28) can be written aswhereSubstituting (30) into (29) yieldswhere .
It is evident thatwhere is a constant.
Acting with and , respectively, on (32) yields Thuswhere are constants.
Substituting (34) into the gives the following discrete stationary equation: According to (31), we have
Define the elliptic coordinates and by expressing and :where we denote , , and with , , and , respectively.
By comparing coefficients of the same power for , we get
Equation (38) can be rewritten asby making use of (37).
Thus, (16) can be written as
Consider the function det which is a th-order polynomial in :Substituting (30) into (41) and comparing coefficients of the same powers of read and deduce that by taking .
Hence, it follows thatAgain from (37) and (44), we have
Similarly, when Thus
4. 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 of the hyperelliptic curve with genus : which has two infinite points and , not branch point of . We fix a set of regular cycle paths: ; , 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 into the new basis , which meets The Abel map is introduced as and the Able-Jacobi coordinates are defined aswhere and is a chosen base point on .
The components of the Abel-Jacobi coordinates in (56) arewhere is the local coordinate of .
We infer that Similarly, we have
Let the fundamental solution matrix of (3) be of the form It is easy to obtain that from which we have
Suppose that is eigenvalue of the Lax matrix in the solution space of equation , which is invariant under the action of due to . The corresponding eigenfunction is that can be called the Baker function which satisfies that
It is easy to check that which has two eigenvalues , whereThe corresponding Baker function can be taken aswhere
Let , be the components of the Baker functions and , respectively. Actually, starting from we can infer that
Similarly, we have where .
5. Algebro-Geometric Solutions
The well-known Riemann theta function of is defined by where , .
According to the Riemann theorem, there exists a constant so that(i) has exactly zeros at ;(ii) has exactly zeros at
We have the inversion formula with the constant . Through a standard treatment, we arrive atwhere .
Substituting (74) into (40) yields Thuswhere which is the algebro-geometric solution to (16).
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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by National Natural Science Foundation of China (no. 61402265 and no. 11701334), Open Fund of the Key Laboratory of Ocean Circulation and Waves, Chinese Academy of Sciences (no. KLOCAW1401), and the SDUST Research Fund (2014TDJH102).