New Traveling Wave Solutions and Interesting Bifurcation Phenomena of Generalized KdV-mKdV-Like Equation

Yiren Chen,Shaoyong Li,Mohammad Mirzazadeh

doi:10.1155/2021/4213939

Yiren Chen, Shaoyong Li + Show 1 more

Open Access

https://doi.org/10.1155/2021/4213939

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Abstract

Using the bifurcation method of dynamical systems, we investigate the nonlinear waves and their limit properties for the generalized KdV-mKdV-like equation. We obtain the following results: (i) three types of new explicit expressions of nonlinear waves are obtained. (ii) Under different parameter conditions, we point out these expressions represent different waves, such as the solitary waves, the 1-blow-up waves, and the 2-blow-up waves. (iii) We revealed a kind of new interesting bifurcation phenomenon. The phenomenon is that the 1-blow-up waves can be bifurcated from 2-blow-up waves. Also, we gain other interesting bifurcation phenomena. We also show that our expressions include existing results.

Highlights

Most relationships in nature and human society are intrinsically nonlinear rather than linear in nature, so many phenomena in nature and human society can be described by nonlinear equations, such as automatic control, meteorology, engineering calculation, engineering budget, economy, and finance [1, 2]
Many authors have been interested in the study of the many forms of KdV-like equations [22,23,24,25], and there are several explicit solutions results of the generalized KdV-mKdV-like equation based on the significant physical background
The second phenomenon is that the trivial waves can be bifurcated from the solitary waves

Summary

Introduction

Most relationships in nature and human society are intrinsically nonlinear rather than linear in nature, so many phenomena in nature and human society can be described by nonlinear equations, such as automatic control, meteorology, engineering calculation, engineering budget, economy, and finance [1, 2]. Many scientists are very interested in nonlinear equations and their solutions and have done a lot of related work [3,4,5]. Many authors have been interested in the study of the many forms of KdV-like equations [22,23,24,25], and there are several explicit solutions results of the generalized KdV-mKdV-like equation based on the significant physical background. We study the nonlinear wave solutions and the bifurcation phenomena for Eq (1). The second phenomenon is that the trivial waves can be bifurcated from the solitary waves.

Our Main Results

The Derivation of Main Results

Conclusion