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A note on solitary waves for modified forms of Camassa-Holm and Degasperis-Procesi equations. (English) Zbl 1203.35234

Summary: Consider the mCH and mDP equations given by \(u_t - u_{xxt}+3u^{2}u_x=2u_xu_{xx}+uu_{xxx}\), and \(u_t - u_{xxt}+4u^{2}u_x=3u_xu_{xx}+uu_{xxx}\), respectively. A new characteristic of solitary wave solutions, bell-shaped solitary wave and peakon coexist for the same wave speed, is found. Previous some results are extended. A conjecture is clarified.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C08 Soliton solutions
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