Traveling Wave Solution of the Olver–Rosenau Equation Solved by Dynamics System

Mei Xiong,Na Yang,Longwei Chen

doi:10.1155/2020/9071587

Abstract

Olver–Rosenau equations presented by Olver and Rosenau can be rewritten to the dynamic system by the wave transformation. The system is a Hamiltonian system with the first integral, and its phase-space and equilibrium point analysis are given in different parameter spaces in detail. On this basis, we can derive various solutions of the original equation relating these orbits in different phase-space planes, and the theoretical basis of the numerical solution is provided for engineering application and production practice.

Highlights

In recent years, nonlinear partial differential equations (NLPDEs) are more and more extensively used to engineering application and production practice
A lot of solving methods for the nonlinear partial differential equation are discussed to be applied in engineering and practice areas, such as the sine-Gordon expansion method [6] and the travelling wave method and its conservation laws [7], and so many examples are in this regard
The equation was derived in 2013 [10]. e resulting Hamiltonian equations are considered by the dynamical system theory and a phasespace analysis of their singular points. ose results of the study proved that the equations can support double compacton solutions. ey found that the new Olver–Rosenau compactons are different from the well-known Rosenau–Hyman compacton and Cooper–Shepard–Sodano compacton

Summary

Introduction

Nonlinear partial differential equations (NLPDEs) are more and more extensively used to engineering application and production practice. A lot of solving methods for the nonlinear partial differential equation are discussed to be applied in engineering and practice areas, such as the sine-Gordon expansion method [6] and the travelling wave method and its conservation laws [7], and so many examples are in this regard. E resulting Hamiltonian equations are considered by the dynamical system theory and a phasespace analysis of their singular points. Ey found that the new Olver–Rosenau compactons are different from the well-known Rosenau–Hyman compacton and Cooper–Shepard–Sodano compacton It was recently introduced by Li in [11], but Li did not give the solution because of the complexity of the integral. We discuss bifurcations and phase portraits of the system in all parameters.

Phase Portraits of the System

Conclusion