The Existence and Bifurcation of Peakon and Anti-Peakon to the n-Degree b-Equation

The dynamical model of a nonlinear wave is governed by a partial differential equation which is a special case of the [Formula: see text]-family equation. Its traveling system is a singular system with a singular straight line. On this line, there exist two degenerate nodes of the associated regular system. By using the method of dynamical systems and the theory of singular traveling wave systems, in this paper we show that, corresponding to global level curves, this wave equation has global periodic wave solutions and anti-solitary wave solutions. We obtain their exact representations. Specially, we discover some new phenomena. (i) Infinitely many periodic orbits of the traveling wave system pass through the singular straight line. (ii) Inside some homoclinic orbits of the traveling wave system there is not any singular point. (iii) There exist periodic wave bifurcation and double anti-solitary waves bifurcation.

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Bifurcations of Solitary Waves of a Simple Equation

Yang

Liu

Chen

2020

Int. J. Bifurcation Chaos

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In this paper, we consider a simple equation which involves a parameter [Formula: see text], and its traveling wave system has a singular line. Firstly, using the qualitative theory of differential equations and the bifurcation method for dynamical systems, we show the existence and bifurcations of peak-solitary waves and valley-solitary waves. Specially, we discover the following novel properties: (i) In the traveling wave system, there exist infinitely many periodic orbits intersecting at a point, or two points and passing through the singular line, and there is no singular point inside a homoclinic orbit. (ii) When [Formula: see text], in the equation there exist three types of bifurcations of valley-solitary waves including periodic wave, blow-up wave and double solitary wave. (iii) When [Formula: see text], in the equation there exist two types of bifurcations of valley-solitary wave including periodic wave and blow-up wave, but there is no double solitary wave bifurcation. Secondly, we perform numerical simulations to visualize the above properties. Finally, when [Formula: see text] and the constant wave speed equals [Formula: see text], we give exact expressions to the above phenomena.

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The Existence and Bifurcation of Peakon and Anti-Peakon to the n-Degree b-Equation

Cited by 4 publications

References 40 publications

Blow-up solutions for a case of b-family equations

Blow-up solutions for a case of b-family equations

Bifurcations and Exact Solutions in a Nonlinear Wave Equation

Bifurcations of Solitary Waves of a Simple Equation

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