Abstract
We investigate a generalized Camassa-Holm equationC(3,2,2):ut+kux+γ1uxxt+γ2(u3)x+γ3ux(u2)xx+γ3u(u2)xxx=0. We show that theC(3,2,2)equation can be reduced to a planar polynomial differential system by transformation of variables. We treat the planar polynomial differential system by the dynamical systems theory and present a phase space analysis of their singular points. Two singular straight lines are found in the associated topological vector field. Moreover, the peakon, peakon-like, cuspon, smooth soliton solutions of the generalized Camassa-Holm equation under inhomogeneous boundary condition are obtained. The parametric conditions of existence of the single peak soliton solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for single peak soliton, kink wave, and kink compacton solutions of theC(3,2,2)equation.
Highlights
Mathematical modeling of dynamical systems processing in a great variety of natural phenomena usually leads to nonlinear partial differential equations (PDEs)
Two singular straight lines are found in the associated topological vector field
The parametric conditions of existence of the single peak soliton solutions are given by using the phase portrait analytical technique
Summary
Mathematical modeling of dynamical systems processing in a great variety of natural phenomena usually leads to nonlinear partial differential equations (PDEs). To understand the role of nonlinear dispersion in the formation of patters in liquid drop, Rosenau and Hyman [12] introduced and studied a family of fully nonlinear dispersion Korteweg-de Vries equations ut + (um)x + (un)xxx = 0 This equation, denoted by K(m, n), owns the property that, for certain m and n, its solitary wave solutions have compact support [12]. We can obtain bifurcation and smooth solutions of the nonlinear PDE (1) through studying the system (8), if the corresponding orbits are bounded and do not intersect with the vertical straight line φ = φs. It is worth of pointing out that traveling waves sometimes lose their smoothness during the propagation due to the existence of singular curves within the solution surfaces of the wave equation Most of these works are concentrated on the nonlinear wave equations with only a singular straight line [6,7,8,9].
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