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Based on a general Riemann theta function and Hirota’s bilinear forms, we devise a straightforward way to explicitly construct double periodic wave solution of

It is always important to investigate the exact solutions for nonlinear evolution equations, which play an important role in the study of nonlinear models of natural and social phenomena. Nonlinear wave phenomena appears in various scientific and engineering areas, such as fluid mechanics, theory of solitons, hydrodynamics, and theory of turbulence, optical fibers, chaos theory, biology, and chemical physics. In the last three decades, various powerful methods have been presented, such as extended tanh method [

The Hirota’s bilinear method provides a powerful way to derive soliton solutions to nonlinear integrable equations and its basis is the Hirota bilinear formulation. Once the corresponding bilinear forms are obtained, multisoliton solutions and rational solutions to nonlinear differential equations can be computed in quite a systematic way. In 1980s, based on Hirota bilinear forms, Nakamura proposed a comprehensive method to construct a kind of multiperiodic solutions of nonlinear equations in his papers [

In recent years, Hon et al. have extended this method to investigate the discrete Toda lattice [

Our aim in the present work is to improve the main steps of the existing methods of Fan and Chow in [

The organization of this paper is as follows. In Section

In this section we briefly present the notation that will be used in this paper. Here the bilinear operators

The Hirota bilinear operators

Then, one would like to consider a general Riemann theta function and discuss its periodicity; the Riemann theta function reads

In the definition of the theta function (

A function

In particular,

The theta function

The meromorphic functions

By using (

Differentiating (

Assuming that

In general, for a polynomial operator

Making use of the formula (

Formula (

In this section, we will focus on the following

We consider a variable transformation

The constant

When

By introducing the notations as

This system admits an explicit solution

The double periodic wave solution (

It has a single phase variable

It has two fundamental periods 1 and

The speed parameter of

It has only one wave pattern for all time and it can be viewed as a parallel superposition of overlapping one-soliton waves, placed one period apart.

Now, we further consider the asymptotic properties of the double periodic wave solution. The relation between the periodic wave solution (

If the vector

It implies that the double periodic solution (

We explicitly expand the coefficients of system (

In this paper, based on the Hirota’s bilinear method, combining the theory of a general Riemann theta function, we have derived a method of constructing double periodic wave solutions for

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Fundamental Research Funds for the Central Universities (2013XK03), the National Natural Science Foundation of China (Grant no. 11371361), and the National Natural Science Foundation of China (Grant no. 11271008).