# Algebro-Geometric Solutions for a Discrete Integrable Equation.

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 Backlund 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 [26-29]. 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

[mathematical expression not reproducible], (1)

which is the simple subalgebra of the Lie algebras [A.sub.1], and corresponding loop algebras can be expressed as

[mathematical expression not reproducible]. (2)

According to the loop algebras, we introduce the following discrete spectral problems

[mathematical expression not reproducible], (3)

where

[mathematical expression not reproducible]. (4)

Thus

[mathematical expression not reproducible], (5)

where

[mathematical expression not reproducible]. (6)

According to the following stationary discrete zero curvature equation for [V.sub.n],

(E[V.sub.n]) [U.sub.n] = [U.sub.n][V.sub.n], (7)

[mathematical expression not reproducible]. (8)

Substituting (6) into (8) yields

[mathematical expression not reproducible], (9)

where [DELTA] = E - 1.

We choose the initial values [a.sub.0] = -1/2, [b.sub.0] = 0 and need to select zero constants for the inverse operation of the difference operator [DELTA] in computing [a.sub.n], n [greater than or equal to] 1. On this condition, recursion relations (9) uniquely determine [a.sub.n], [b.sub.n], [c.sub.n], n [greater than or equal to] 1. Then, we obtain the first few quantities

[mathematical expression not reproducible] (10)

From

[mathematical expression not reproducible], (11)

we have the discrete zero curvature equation

[mathematical expression not reproducible], (12)

where

[mathematical expression not reproducible]. (13)

Thus, we obtain the following integrable discrete hierarchy

[mathematical expression not reproducible]. (14)

And

[mathematical expression not reproducible], (15)

with m = 1. Equation (15) can be read as

[mathematical expression not reproducible]. (16)

It is easy to find that the Lax pair of (15) is given by

[mathematical expression not reproducible], (17)

where

[mathematical expression not reproducible]. (18)

In the following, we express Lenard's gradient sequences [S.sub.j] (0 [less than or equal to] j [member of] Z), by the recursion equation

[mathematical expression not reproducible], (19)

with two operators

[mathematical expression not reproducible], (20)

where [S.sub.j](n) = [([s.sup.(1).sub.j], [s.sup.(2).sub.j], [s.sup.(3).sub.j]).sup.T].

From the equation [J.sub.n][S.sub.-1](n) = 0 and (19), respectively,

[mathematical expression not reproducible]. (21)

Equation (19) implies that

[mathematical expression not reproducible]. (22)

The discrete integrable hierarchical (14) could be rewritten as generation of the following, so spectrum problem is

[mathematical expression not reproducible], (23)

where

[mathematical expression not reproducible]. (24)

From the compatibility conditions of the discrete Lax pair (23), we can read that the hierarchical equation is

[mathematical expression not reproducible]. (25)

Thus, we also have

[mathematical expression not reproducible]. (26)

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 [psi](n) = [([[psi].sup.(1)](n), [[psi].sup.(1)](n)).sup.T], [phi](n) = [([[phi].sup.(1)](n), [[phi].sup.(1)](n)).sup.T], and we define a Lax matrix [W.sub.n] as follows:

[mathematical expression not reproducible]. (27)

and [W.sub.n] should meet the following equations:

[mathematical expression not reproducible]. (28)

It is easy to see that (28) can be written as

[mathematical expression not reproducible], (29)

where

[mathematical expression not reproducible]. (30)

Substituting (30) into (29) yields

[mathematical expression not reproducible], (31)

where [G.sub.j](n) = [([h.sub.j](n), [g.sub.j](n), [f.sub.j](n)).sup.T].

It is evident that

[G.sub.-1] = [[alpha].sub.0][S.sub.-1] (n), (32)

where [[alpha].sub.0] is a constant.

Acting with [J.sup.-1.sub.n] [K.sub.n] and [K.sup.-1.sub.n][J.sub.n], respectively, on (32) yields

[G.sub.0](n) = [[alpha].sub.0][S.sub.0](n) + [[alpha].sub.1][S.sub.-1] (n). (33)

Thus

[mathematical expression not reproducible], (34)

where [[alpha].sub.0], [[alpha].sub.1], ..., [[alpha].sub.k+1] are constants.

Substituting (34) into the [K.sub.n][G.sub.N-1] (n) = 0 gives the following discrete stationary equation:

[[alpha].sub.0][X.sub.N](n) + [[alpha].sub.1][X.sub.N-1] (n) + ... + [[alpha].sub.N][X.sub.0] (n) = 0. (35)

According to (31), we have ([[alpha].sub.0] = 1)

[mathematical expression not reproducible]. (36)

Define the elliptic coordinates [[mu].sub.j] and [v.sub.j] by expressing g(n) and h(n):

[mathematical expression not reproducible], (37)

where we denote [mathematical expression not reproducible], respectively.

By comparing coefficients of the same power for [lambda], we get

[mathematical expression not reproducible]. (38)

Equation (38) can be rewritten as

[mathematical expression not reproducible], (39)

by making use of (37).

Thus, (16) can be written as

[mathematical expression not reproducible]. (40)

Consider the function det [W.sub.n] which is a (4N + 2)th-order polynomial in [lambda]:

[mathematical expression not reproducible]. (41)

Substituting (30) into (41) and comparing coefficients of the same powers of [lambda] read

[mathematical expression not reproducible] (42)

and deduce that

[mathematical expression not reproducible], (43)

by taking m = 1.

Hence, it follows that

[mathematical expression not reproducible]. (44)

Again from (37) and (44), we have

[mathematical expression not reproducible]. (45)

Similarly, when m = 2

[mathematical expression not reproducible]. (46)

Thus

[mathematical expression not reproducible]. (47)

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 r of the hyperelliptic curve with genus N:

[mathematical expression not reproducible]. (48)

which has two infinite points [[infinity].sub.1] and [[infinity].sub.2], not branch point of [GAMMA]. We fix a set of regular cycle paths: [a.sub.1], ..., [a.sub.N]; [b.sub.1], ..., [b.sub.N], which are independent and have the intersection numbers:

[mathematical expression not reproducible]. (49)

We choose the holomorphic differentials, on [GAMMA]

[mathematical expression not reproducible], (50)

and define

[mathematical expression not reproducible], (51)

where A = [([A.sub.ij]).sub.NxN], B = [([B.sub.ij]).sub.NxN].

Thus, we denote the matrices C and [tau] by

[mathematical expression not reproducible] (52)

and verify that [tau] is symmetric and has positive defined imaginary part.

By normalizing [[??].sub.j] into the new basis [[omega].sub.j],

[mathematical expression not reproducible], (53)

which meets

[mathematical expression not reproducible]. (54)

The Abel map A(p) is introduced as

[mathematical expression not reproducible], (55)

and the Able-Jacobi coordinates are defined as

[mathematical expression not reproducible], (56)

where

[mathematical expression not reproducible], (57)

and [p.sub.0] is a chosen base point on [GAMMA].

The components of the Abel-Jacobi coordinates in (56) are

[mathematical expression not reproducible], (58)

where [??]([p.sub.0]) is the local coordinate of [p.sub.0].

We infer that

[mathematical expression not reproducible]. (59)

Similarly, we have

[mathematical expression not reproducible]. (60)

Let the fundamental solution matrix of (3) be of the form

[mathematical expression not reproducible]. (61)

It is easy to obtain that

[Q.sub.n+1] = [U.sub.n][U.sub.n-1] ... [U.sub.0], (62)

from which we have

[mathematical expression not reproducible], (63)

Suppose that [delta] is eigenvalue of the Lax matrix [W.sub.n] in the solution space of equation [psi](n + 1) = [U.sub.n][psi](n), which is invariant under the action of [W.sub.n] due to (E[W.sub.n])[U.sub.n] = [U.sub.n][W.sub.n]. The corresponding eigenfunction is [psi](n) that can be called the Baker function which satisfies that

[mathematical expression not reproducible]. (64)

It is easy to check that

[mathematical expression not reproducible], (65)

which has two eigenvalues [[delta].sub.[+ or -]] = [+ or -][delta], where

[mathematical expression not reproducible]. (66)

The corresponding Baker function can be taken as

[mathematical expression not reproducible], (67)

where

[mathematical expression not reproducible]. (68)

Let [p.sup.[+ or -]] (n, [lambda]), [q.sup.[+ or -]](n, [lambda]) be the components of the Baker functions [[phi].sup.[+ or -]](n) and [[??].sup.[+ or -]](n), respectively. Actually, starting from

[mathematical expression not reproducible], (69)

we can infer that

[mathematical expression not reproducible]. (70)

Similarly, we have

[mathematical expression not reproducible], (71)

where [mathematical expression not reproducible].

5. Algebro-Geometric Solutions

The well-known Riemann theta function of [GAMMA] is defined by

[mathematical expression not reproducible]. (72)

where [xi] = [([[xi].sub.1], ..., [[xi].sub.N]).sup.T], <[xi], z> = [[summation].sup.N.sub.j=1] [[xi].sub.j][z.sub.j].

According to the Riemann theorem, there exists a constant [M.sup.(i)] [member of] [C.sup.N] so that

(i) [F.sub.1] = [theta](A(p) - [[rho].sup.(1)](n) - [M.sup.(1)]) has exactly N zeros at [mathematical expression not reproducible];

(ii) [F.sub.2] = [theta](A(p) - [[rho].sup.(2)](n) - [M.sup.(2)]) has exactly N zeros at [mathematical expression not reproducible].

We have the inversion formula

[mathematical expression not reproducible], (73)

with the constant [mathematical expression not reproducible]. Through a standard treatment, we arrive at

[mathematical expression not reproducible], (74)

where [mathematical expression not reproducible].

Substituting (74) into (40) yields

[mathematical expression not reproducible]. (75)

Thus

[mathematical expression not reproducible], (76)

where

[mathematical expression not reproducible]. (77)

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.

https://doi.org/10.1155/2017/5258375

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).

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Mengshuang Tao and Huanhe Dong

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Huanhe Dong; mathsdong@126.com

Received 25 April 2017; Revised 4 August 2017; Accepted 24 October 2017; Published 14 November 2017

Academic Editor: Chris Goodrich
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