Abstract
We study the nonlinear waves described by Schamel-Korteweg-de Vries equationut+au1/2+buux+δuxxx=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.
Highlights
Introduction and PreliminaryConsider the following Schamel-Korteweg-de Vries (S-KdV) equation [1, 2]: ut + ux + δuxxx = 0, (1)where a, b, and δ are constants
We study the nonlinear waves described by Schamel-Korteweg-de Vries equation ut +ux + δuxxx = 0
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
Summary
The concept of compacton, soliton with compact support or strict localization of solitary waves, appeared in the work of Rosenau and Hyman [9] where a genuinely nonlinear dispersive equation K(n; n) is defined by ut + a(un)x + (un)xxx = 0. They found certain solitary wave solutions which vanish identically outside a finite core region. We study the nonlinear waves and their bifurcations in (1) by using the bifurcation method of dynamical systems [21–23].
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