Further Results about Traveling Wave Exact Solutions of the (2+1)-Dimensional Modified KdV Equation

Yang Yang,Xue-Hua Tang,Jian-Ming Qi,Yong-Yi Gu

doi:10.1155/2019/3053275

Abstract

We used the complex method and the exp(-ϕ(z))-expansion method to find exact solutions of the (2+1)-dimensional mKdV equation. Through the maple software, we acquire some exact solutions. We have faith in that this method exhibited in this paper can be used to solve some nonlinear evolution equations in mathematical physics. Finally, we show some simulated pictures plotted by the maple software to illustrate our results.

Highlights

Nonlinear science is basic science to study the generality of nonlinear phenomena
It can be seen from the above analysis that the complex method and exp(−φ(z))-expansion method are powerful tools for solving the exact solutions of nonlinear evolution equations
The general meromorphic solutions of (2+1)dimensional mKdV equation are obtained by the complex method, and we found eight solutions of (2+1)-dimensional mKdV equation

Summary

Introduction

Nonlinear science is basic science to study the generality of nonlinear phenomena. It is a comprehensive discipline which has been gradually developed by various branch disciplines characterized by nonlinearity since the 1960s [1]. By the above lemma and results, we introduce complex method to find exact solutions of some PDEs. The detailed five steps are as follows: (1) Put the transform T : u(x, t) 󳨀→ w(z), (x, t) 󳨀→ z into a given PDE to produce a nonlinear ODE. (3) By determinant relation equation (33)–(35) we will, respectively, find the elliptic, rational and periodic solutions u(z) of (24) or (32) with pole at z = 0. Consider that a nonlinear partial differential equation (PDE) in the following form:. In (39), P is a polynomial with an unknown function μ(x, y, t) and its derivatives in which nonlinear terms and highest order derivatives are involved By solving the algebraic equations, we get the values of Cn ≠ 0, δ, and μ, and we put these into (34) along with(43)-(49) to get the determination of the solutions of (39)

Proof of Theorems 1 and 2

Computer Simulations

Conclusions