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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

${u}_{t}+2k{u}_{x}-{u}_{xxt}+a{u}^{m}{u}_{x}=2{u}_{x}{u}_{xx}+u{u}_{xxx}$

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 of PDE in closed form 35Q51 Soliton-like equations