The bifurcations of solitary and kink waves described by the Gardner equation

Yiren  Chen; Zhengrong Liu

doi:10.3934/dcdss.2016067

Article Contents

2016, Volume 9, Issue 6: 1629-1645. Doi: 10.3934/dcdss.2016067

This issue Previous Article Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity Next Article Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion

The bifurcations of solitary and kink waves described by the Gardner equation

Yiren Chen^1, and
Zhengrong Liu^2,

1.
School of Mathematics, South China University of Technology, Guangzhou 510640
2.
Department of Mathematics, South China University of Technology, Guangzhou 510640

Received: June 30, 2015

Revised: August 31, 2016

Published: November 2016

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Abstract

In this paper, we investigate the bifurcations of nonlinear waves described by the Gardner equation $u_{t}+a u u_{x}+b u^{2} u_{x}+\gamma u_{xxx}=0$. We obtain some new results as follows: For arbitrary given parameters $b$ and $\gamma$, we choose the parameter $a$ as bifurcation parameter. Through the phase analysis and explicit expressions of some nonlinear waves, we reveal two kinds of important bifurcation phenomena. The first phenomenon is that the solitary waves with fractional expressions can be bifurcated from three types of nonlinear waves which are solitary waves with hyperbolic expression and two types of periodic waves with elliptic expression and trigonometric expression respectively. The second phenomenon is that the kink waves can be bifurcated from the solitary waves and the singular waves.

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Mathematics Subject Classification: Primary: 34A20, 34C35, 35B65; Secondary: 58F05, 76B25.

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