# American Institute of Mathematical Sciences

December  2016, 9(6): 1629-1645. doi: 10.3934/dcdss.2016067

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

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

Received  July 2015 Revised  September 2016 Published  November 2016

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.
Citation: Yiren Chen, Zhengrong Liu. The bifurcations of solitary and kink waves described by the Gardner equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1629-1645. doi: 10.3934/dcdss.2016067
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