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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 782436, 7 pages
http://dx.doi.org/10.1155/2013/782436
Research Article

The Hamiltonian Structure and Algebrogeometric Solution of a 1 + 1-Dimensional Coupled Equations

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 20 May 2013; Accepted 24 August 2013

Academic Editor: Jinde Cao

Copyright © 2013 Hongfei Pan and Tiecheng Xia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A 1 + 1-dimensional coupled soliton equations are decomposed into two systems of ordinary differential equations. The Abel-Jacobi coordinates are introduced to straighten the flows, from which the algebrogeometric solutions of the coupled 1 + 1-dimensional equations are obtained in terms of the Riemann theta functions.

1. Introduction

The study of explicit solutions for soliton equations is very important in modern mathematics, physics, and other sciences. There are several systematic approaches to obtain explicit solutions of the soliton equations, such as the inverse scattering transformation, the algebrogeometric method, and the Hirota bilinear method (see, e.g., [14] and references therein). Algebrogeometric (or quasiperiodic) solutions are very important explicit solutions for soliton equations, recently, based on the nonlinearization technique of Lax pairs and direct method proposed by Cao [5]. This new scheme is further shown to be a very powerful tool, through which algebrogeometric solutions of -dimensional and -dimensional continuous and discrete soliton equations can be obtained; see Cao et al. [6, 7], Geng et al. [8, 9], and Dai and Fan [10].

In this paper, we will construct the Hamiltonian structure and search for the algebrogeometric solution of the following coupled -dimensional soliton equations:

Our purpose is to construct the Hamiltonian structure and give the algebrogeometric solutions of the coupled -dimensional soliton equations based on its obtained Lax pairs. The paper is organized as follows. In Section 2, we use Lenard operator pairs to derive another form of the coupled -dimensional soliton equations. In Section 3, based on the trace identity [11, 12], we construct the Hamiltonian structure of the coupled -dimensional soliton equations. In Section 4, based on the Lax pairs of the coupled -dimensional soliton equations, variable separation technique is used to translate the solution of the coupled -dimensional soliton equations to solve ordinary differential equations. In Section 5, a hyperelliptic Riemann surface of genus and Abel-Jacobi coordinates are defined to straighten the associated flows. Jacobi's inverse problem is discussed, from which the algebrogeometric solutions of the coupled -dimensional soliton equations are constructed in terms of the Riemann theta functions.

2. The Hierarchy and Lax Pairs of the Coupled 1 + 1-Dimensional Soliton Equations

In this section, we introduce the Lenard gradient sequence to derive the hierarchy and its stationary hierarchy associated with (1) by the recursion relation: where and operators () A direct calculation gives from the recursion relation (2) thatConsider the spectral problem and the auxiliary problem where

Then the compatibility condition of (5) and (6) is , which is equivalent to the hierarchy of nonlinear evolution equations

In brief,

The first two nontrivial equations are

The second system is our coupled -dimensional soliton equations (1).

Let and be two basic solutions of the spectral equations (5) and (6). We define a matrix by in which , and are three functions. It is easy to calculate by (5) and (6) that which implies that . Equation (13) can be written as

We suppose that the functions , and are finite-order polynomials in :

Substituting (16) into (14) yields

It is easy to see that (17) implies and the equation has the general solution where is constant of integration. Therefore, if we take (19) as a starting point, then can be determined recursively by relation (17). In fact, noticing and acting with the operator upon (19), we obtain from (2) and (17) that where are integral constants. Substituting (20) into (17) yields a certain stationary evolution equation: where

This means that expression (16) is existent.

3. Hamiltonian Structure

Let where

It is easy to calculate

According to the trace identity [11, 12], we have

Comparing the coefficients of , we obtain we set and then get and where .

Thus the soliton equation (9) has a Hamiltonian structure: where

In speciality, the Hamiltonian structure of (1) is ():

4. Ordinary Differential Equations

In this section, (1) will be decomposed into two systems of solvable ordinary differential equations. Without loss of generality, let . From (2), (17), and (20), we have

By using (16), we can write and as the following finite products:

Equation (33) implies by comparing the coefficients of that Thus from (32) and (34), we obtain

Let us consider the function which is a th-order polynomial in with constant coefficients of the -flow and -flow: Substituting (16) into (36) and comparing the coefficients of yield which together with (32) gives

From (36) we see that

Again by using (14) and (33), we obtain which together with (39) gives

In a way similar to the above expression, by using (6) (), (15), and (39), we arrive at the evolution of and along the -flow:

Therefore, if the distinct parameters are given and let and be distinct solutions of ordinary differential equations (41), (42), and (43), then determined by (35) is a solution of the coupled -dimensional equations (1).

5. Algebrogeometric Solutions

In this section, we will give the algebrogeometric solutions of the coupled -dimensional equation (1). To this end, we first introduce the Riemann surface of the hyperelliptic curve with genus on . On there are two infinite points and , which are not branch points of . We equip with a canonical basis of cycles: which are independent and have intersection numbers as follows:

We will choose the following set as our basis: which are linearly independent of each other on , and let

It is possible to show that the matrices and are invertible matrices [13, 14]. Now we define the matrices and by . Then the matrix can be shown to be symmetric and it has a positive-definite imaginary part (Im ). If we normalize into the new basis then we have

Now we introduce the Abel-Jacobi coordinates as follows: where , and is the local coordinate of . From (42) and (50), we get which implies

With the help of the following equality in a similar way, we obtain from (50), (51), (41), (42), and (43) that

On the basis of these results, we obtain the following: where are constants and

Now we introduce the Abel map : and Abel-Jacobi coordinates: According to the Riemann theorem [13, 14], there exists a Riemann constant vector such that the function has exactly zeros at for or for . To make the function single valued, the surface is cut along all to form a simple connected region, whose boundary is denoted by . By [13, 14], the integrals are constants independent of and with

By the residue theorem, we have

Here we need only to compute the residues in (63). In a way similar to calculations in [10], we arrive at where and and are constants. Thus from (63) and (64), we arrive at

Substituting (66) into (35), we get an algebrogeometric solution for the coupled -dimensional soliton equations (1): where is arbitrary complex functions about variable .

Acknowledgments

This work is in part supported by the Natural Science Foundation of China (Grant nos. 11271008, 61072147, and 11071159) and the Shanghai University Leading Academic Discipline Project (A.13-0101-12-004).

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