New Exact Analytical Solutions for the General KdV Equation with Variable Coefficients

Bao-Jian Hong,Dian-Chen Lu

doi:10.3390/mca19030194

Abstract

In this paper, a general algebraic method based on the generalized Jacobi elliptic functions expansion method, the improved general mapping deformation method and the extended auxiliary function method with computerized symbolic computation is proposed to construct more new exact solutions of a generalized KdV equation with variable coefficients. As a result, eight families of new generalized Jacobi elliptic function wave solutions and Weierstrass elliptic function solutions of the equation are obtained by using this method, some of these solutions are degenerated to soliton-like solutions, trigonometric function solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the general method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

Highlights

Nonlinear partial differential equations (NLPDEs) are widely used to describe complex physical phenomena arising in the world around us and various fields of science
The investigation of exact solutions of NLPDEs plays an important role in the study of these phenomena such as the nonlinear dynamics and the mechanism behind the phenomena
The solution procedure of this method, by the help of Matlab or Mathematica, is of the utmost simplicity, and this method can be extended to all kinds of NLPDEs

Summary

INTRODUCTION

Nonlinear partial differential equations (NLPDEs) are widely used to describe complex physical phenomena arising in the world around us and various fields of science. In [14][15], Hong proposed a generalized Jacobi elliptic functions expansion method to obtain generalized exact solutions of NLPDEs. In [16], Hong et al proposed an. Lu improved general mapping deformation method to obtain generalized exact solutions of the general KdV equation with variable coefficients (GVKDV). The solution procedure of this method, by the help of Matlab or Mathematica, is of the utmost simplicity, and this method can be extended to all kinds of NLPDEs. In this work, we will proposed the general algebraic method to obtain several new families of exact solutions for the GVKDV equations.

SUMMARY OF THE GENERAL ALGEBRAIC METHOD

EXACT SOLUTIONS TO THE GVKDV EQUATION

4.CONCLUSION