New Types of Nonlinear Waves and Bifurcation Phenomena in Schamel-Korteweg-de Vries Equation

Abstract

We study the nonlinear waves described by Schamel-Korteweg-de Vries equation $u_t+\left({au}^{1/2}+bu\right)u_x+{\delta u}_{xxx}=0$. Two new types of nonlinear waves called compacton-like waves and kink-like waves are displayed. Furthermore, two kinds of new bifurcation phenomena are revealed. The first phenomenon is that the kink waves can be bifurcated from five types of nonlinear waves which are the bell-shape solitary waves, the blow-up waves, the valley-shape solitary waves, the kink-like waves, and the compacton-like waves. The second phenomenon is that the periodic-blow-up wave can be bifurcated from the smooth periodic wave.