Travelling Wave Solutions of Wu–Zhang System via Dynamic Analysis

Hang Zheng,Guo Lei,Yonghui Xia,Yuzhen Bai

doi:10.1155/2020/2845841

Abstract

In this paper, based on the dynamical system method, we obtain the exact parametric expressions of the travelling wave solutions of the Wu–Zhang system. Our approach is much different from the existing literature studies on the Wu–Zhang system. Moreover, we also study the fractional derivative of the Wu–Zhang system. Finally, by comparison between the integer-order Wu–Zhang system and the fractional-order Wu–Zhang system, we see that the phase portrait, nonzero equilibrium points, and the corresponding exact travelling wave solutions all depend on the derivative order α. Phase portraits and simulations are given to show the validity of the obtained solutions.

Highlights

Many authors made some efforts on nonlinear partial differential equations (NPDEs)
Where ε and u denote the surface velocity of water along the x-direction and the y-direction, respectively, and v means the elevation of water. rough some transformations, equation (1) reduces into the (1 + 1)-dimensional dispersive long wave equation (Wu–Zhang system) as follows: ut − uux − vx, (2)
We summarize as follows: the first integral method [16], extended tanh-function method [17], characteristic function method [18], modified Conte’s invariant Painleveexpansion method and truncation of the WTC’s approach [19,20,21], elliptic function rational expansion method [22], generalized extended tanh-function method [23], generalized extended rational expansion method [24], and so on

Summary

Introduction

Many authors made some efforts on nonlinear partial differential equations (NPDEs) (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14]). The exact periodic wave solutions of equation (2) (see Figure 2(b)) can be written as

Results

Conclusion