Liu, Zhengrong; Liang, Yong The explicit nonlinear wave solutions and their bifurcations of the generalized Camassa-Holm equation. (English) Zbl 1258.35050 Int. J. Bifurcation Chaos Appl. Sci. Eng. 21, No. 11, 3119-3136 (2011). Summary: We study explicit nonlinear wave solutions and their bifurcations of the generalized Camassa-Holm equation \[ u_t+2ku_x-u_{xxt}+3u^2u_x=2u_xu_{xx}+uu_{xxx}. \] Not only are the precise expressions of the explicit nonlinear wave solutions obtained, but some interesting bifurcation phenomena are revealed. Firstly, it is verified that \(k = 3/8\) is a bifurcation parametric value for several types of explicit nonlinear wave solutions. When \(k < 3/8\), there are five types of explicit nonlinear wave solutions, which are (i) hyperbolic peakon wave solution, (ii) fractional peakon wave solution, (iii) fractional singular wave solution, (iv) hyperbolic singular wave solution, (v) hyperbolic smooth solitary wave solution. When \(k = 3/8\), there are two types of explicit nonlinear wave solutions, which are fractional peakon wave solution and fractional singular wave solution. When \(k > 3/8\), there is not any type of explicit nonlinear wave solutions. Secondly, it is shown that there are some bifurcation wave speed values such that the peakon wave and the anti-peakon wave appear alternately. Thirdly, it is displayed that there are other bifurcation wave speed values such that the hyperbolic peakon wave solution becomes the fractional peakon wave solution, and the hyperbolic singular wave solution becomes the fractional singular wave solution. Cited in 16 Documents MSC: 35C08 Soliton solutions 35C07 Traveling wave solutions 35K55 Nonlinear parabolic equations 34A05 Explicit solutions, first integrals of ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations Keywords:generalized Camassa-Holm equation; explicit nonlinear wave solution; bifurcation parametric value; bifurcation wave speed value PDFBibTeX XMLCite \textit{Z. Liu} and \textit{Y. Liang}, Int. J. Bifurcation Chaos Appl. Sci. 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