Elsevier

Physics Letters A

Volume 347, Issues 4–6, 5 December 2005, Pages 215-227
Physics Letters A

Multiple Riccati equations rational expansion method and complexiton solutions of the Whitham–Broer–Kaup equation

https://doi.org/10.1016/j.physleta.2005.08.015Get rights and content

Abstract

A series of complexiton solutions of the Whitham–Broer–Kaup equation are found through a multiple Riccati equations rational expansion method presented in this Letter. Many new types of complexiton solutions such as various combination of trigonometric periodic and hyperbolic function solutions, various combination of trigonometric periodic and rational function solutions, various combination of hyperbolic and rational function solutions, etc., are obtained.

Introduction

There have been a great amount of activity aiming to find as many and general as possible exact solutions of nonlinear partial differential equations (PDEs). The tanh function method is considered to be one of the most straightforward and effective algorithm to obtain solitary wave solutions for lots of nonlinear PDEs. 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 [1], [2], [3], [4], [5]. However, to our knowledge, the complexiton solutions have not been found by existing various improved and extended tanh function methods, in which hyperbolic (solitary) function and triangular periodic functions can appear in a solution at the same time. In [6], with the aid of symbolic computation, the Riccati equation rational expansion (RERE) method was devised for constructing rational formal solitary wave solutions and triangular periodic wave solutions. The present work is motivated by the desire to extend the RERE method and present a multiple Riccati equations rational expansion (MRERE) method to find complexiton solutions. The key step in the MRERE method is that solutions of two Riccati equations with different parameters are used as two variable in the components of finite rational expansion. The MRERE method is a straightforward and pure algebraic algorithm implemented in a computer algebraic system. It is available for integrable systems and nonintegrable systems. For illustration, we apply the MRERE method to the Whitham–Broer–Kaup equation and find many new types of complexiton solutions such as various combination of trigonometric periodic and hyperbolic function solutions, various combination of trigonometric periodic and rational function solutions, various combination of hyperbolic and rational function solutions, etc., are obtained.

Section snippets

Summary of the multiple Riccati equations rational expansion method

In the following we would like to outline the main steps of our method.

Step 1

Given a system of polynomial PDE with constant coefficients, with some physical fields ui(x,y,t) in three variables x, y, t,Δ(ui,uit,uix,uiy,uitt,uixt,uiyt,uixx,uiyy,uixy,)=0, use the wave transformationui(x,y,t)=Ui(ξ),ξ=k(x+ly+λt), where k, l and λ are constants to be determined later. Then the nonlinear partial differential system (2.1) is reduced to a nonlinear ordinary differential system:Θ(Ui,Ui,Ui,)=0.

Step 2

We introduce

Complexiton solution of the Whitham–Broer–Kaup equation

Let us consider the Whitham–Broer–Kaup (WBK) equation,ut+uux+vx+βuxx=0,vt+(uv)x+αuxxxβvxx=0, where α, β0 are all constants. Under Boussinesq approximation, Whitham [7], Broer [8] and Kaup [9] obtained nonlinear WBK equation. It is not difficult to see that when parameters α and β take different constants, system (3.1) includes many important mathematical and physical equations, such as when α=0, β0 system (3.1) becomes classical long wave equation that describe shallow water with dispersive

Conclusion

There are few works in soliton theory to construct the complexiton solutions. Recently, Ma [16] found the complexiton solutions to the KdV equation thought its bilinear form. Lou et al. [17] used the mapping relation to solve the (n+1)-dimensional sine-Gordon field equation and many new types of complexiton solutions are found. Although our method cannot recover all of complexiton solutions obtained by Ma's method and Lou's method, other new types of complexiton solutions obtained in this

Acknowledgements

The author would like to thank the referees very much for their careful reading of the manuscript and many valuable suggestions. The work was supported by China Postdoctoral Science Foundation, Nature Since Foundation of Zhejiang Province of China (Y604056) and Ningbo Doctoral Foundation of China (2005610030).

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