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 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.

Exact explicit solitary wave and periodic wave solutions and their dynamical behaviors for the Schamel–Korteweg–de Vries equation*

“…The Schamel-Korteweg-de Vries (Schamel-KdV) equation [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] can be written as…”

“…Using a extended mapping method and the availability of symbolic computation, several classes of exact solutions are expressed by various JEFs and hyperbolic functions. [13] Wu and Liu [14] found two types of nonlinear wave solutions called compacton-like wave and kink-like wave solutions. The hyperbolic function solutions and the trigonometric function solutions with free parameters have been obtained in Ref.…”

“…The present results more completely answer the above problems and enrich the results of Refs. [11][12][13][14][15][16][17][18].…”

Exact Solutions for the Wick-Type Stochastic Schamel-Korteweg-de Vries Equation

“…In [1], Lee and Sakthivel used the exp-function method to obtain some exact travelling waves solutions for the following Schamel-Korteweg-de Vries equation [2][3][4].…”

New Exact Explicit Nonlinear Wave Solutions for the Broer-Kaup Equation

“…In this paper, we employ the bifurcation method and qualitative theory of dynamical systems [11][12][13][14][15][16][17][18][19][20][21] to investigate the nonlinear wave solutions for (1), and we obtain many exact explicit expressions of nonlinear wave solutions for (1). These nonlinear wave solutions contain solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions, most of which, to our knowledge, are newly obtained.…”