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New ‘ansatz’ for obtaining wave solutions of the generalized Camassa-Holm equation. (English) Zbl 1069.35017

Summary: An alternate approach is proposed for obtaining periodic wave and peaked solitary wave solutions of the following nonlinear generalized Camassa-Holm equation \[ u_{t}+2ku_{x}-u_{xxt}+au^{m}u_{x} = 2u_{x}u_{xx}+uu_{xxx} \] For \(m\) = 1, 2, 3 we give the explicit expressions for the peakons. The ‘ansatz’, introduced in this paper, will also demonstrate the existence of a new class of discontinuous soliton solutions with infinite spikes.

MSC:

35C05 Solutions to PDEs in closed form
35Q51 Soliton equations
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