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The explicit nonlinear wave solutions and their bifurcations of the generalized Camassa-Holm equation. (English) Zbl 1258.35050

Summary: We study explicit nonlinear wave solutions and their bifurcations of the generalized Camassa-Holm equation

u t +2ku x -u xxt +3u 2 u x =2u x u 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.

MSC:
35C08Soliton solutions of PDE
35C07Traveling wave solutions of PDE
35K55Nonlinear parabolic equations
34A05Methods of solution of ODE
34B15Nonlinear boundary value problems for ODE
34C37Homoclinic and heteroclinic solutions of ODE
34C23Bifurcation (ODE)