The exact solutions and approximate analytic solutions of the (2+1)-dimensional KP equation based on symmetry method.

Litao Gai,Yingmo Jie,Sudao Bilige

doi:10.1186/s40064-016-2908-8

Abstract

In this paper, we successfully obtained the exact solutions and the approximate analytic solutions of the (2 + 1)-dimensional KP equation based on the Lie symmetry, the extended tanh method and the homotopy perturbation method. In first part, we obtained the symmetries of the (2 + 1)-dimensional KP equation based on the Wu-differential characteristic set algorithm and reduced it. In the second part, we constructed the abundant exact travelling wave solutions by using the extended tanh method. These solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions respectively. It should be noted that when the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions. Finally, we apply the homotopy perturbation method to obtain the approximate analytic solutions based on four kinds of initial conditions.

Highlights

The nonlinear phenomenon has been extensively appeared in the fields of mathematical physics and engineering technology
A multitude of research focuses have been changed from linear problems to nonlinear ones. These problems can be ascribed to the research of nonlinear partial differential equations (NLPDE), as the complexity of equation, it becomes hard to get the exact solutions
In this paper, we studied that construct the exact solutions and the approximate analytic solutions of NLPDE by using the Lie symmetry, the extended tanh method and the homotopy perturbation method

Summary

Introduction

The nonlinear phenomenon has been extensively appeared in the fields of mathematical physics and engineering technology. We investigate the applications of the symmetry method in the boundary value problem of the nonlinear PDEs based on Wu-differential characteristic set algorithm and use the symmetry method and the homotopy analytic method to solve the boundary value problem (Sudao et al 2014; Sudao 2011).

Results

Conclusion