A new coupled sub-equations expansion method and novel complexiton solutions of (2 + 1)-dimensional Burgers equation

Abstract

In this paper, based on symbolic computation and the idea of rational expansion method, a new coupled sub-equations method is devised to uniformly construct a series of exact complexiton solutions for nonlinear evolution equations. Compared with most existing tanh function methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions which include many new types of complexiton solutions. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equation.

Introduction

Since Gaedner, Greene, Kruskal and Miural solved the Korteweg–de Vries equation by means of the inverse scattering transformation approach in 1967 [1], the modern theory of soliton has been widely applied in physics and deeply studied in mathematics. There have been a great amount of activities aiming to find methods for exact solution of nonlinear evolution equations, such as Bäcklund transformation [2], Darboux transformation [3], Hirota bilinear method [4], Lie group method [5], variable separation approach [6], Painlevé analysis method [7], various tanh function methods [8], [9], [10], [11], [12] and so on. Among them, the tanh function method is considered to be one of the most straightforward and effective algebraic algorithm to obtain solitary wave solutions for lots of nonlinear evolution equations. In line with the development of computerized symbolic computation, much work has been concentrated on the various extensions and applications of the tanh function method [8], [9], [10], [11], [12]. In Ref. [12], Wang et al. firstly presented the rational expansion method which further exceeded the applicability of the tanh function methods and is more powerful than exiting straightforward constructive algorithms. The key idea of rational expansion method is that the solution sought is expressible as a finite series of rational form of terms whose derivatives can be expressed as polynomial of themselves.

Recently, Ma [13] found a novel class of explicit exact solutions to the Korteweg–de Vries equation through its bilinear form and defined the solutions as complexiton solutions. In Ref. [14], Lou et al. presented and answered the problem “Are there any exact explicit multiple periodic wave solutions and periodic–solitary wave solutions for the nSG equation” with help of the mapping relations among the sine-Gordon field equation and the cubic nonlinear Klein–Gordon fields. Such solutions possess singularities of combinations of trigonometric function waves and exponential function waves which have different travelling speeds of new type. Above mentioned solutions of nonlinear evolution equations have a common character: combination of trigonometric function waves and exponential function waves. For unification and conciseness, so we call the solutions obtained by Ma and the solutions obtained by Lou as complexiton solutions.

On the lines of the rational expansion thought, the present work is motivated by the desire to present a new coupled sub-equations to construct more types and general formal solutions which contain not only the results obtained by using the method [8], [9], [10], [11], [12] but also other types of solutions including many new types of complexiton solutions. For illustration, we apply the new method to solve (2 + 1)-dimensional Burgers equation and successfully construct new and more general complexiton solutions.

The rest of this paper is organized as follows. In Section 2, we illustrate the new coupled sub-equation method. In Section 3, we apply the generalized method to (2 + 1)-dimensional Burgers equation and obtain some new complexiton solutions for this model. A short summary and discussion are given in final.