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Peakons and their bifurcation in a generalized Camassa-Holm equation. (English) Zbl 1090.37554

Summary: R. Camassa and D. D. Holm [Phys. Rev. Lett. 71, 1661–1664 (1993; Zbl 0972.35521)] recently derived a new dispersive shallow water equation known as the Camassa-Holm equation. They showed that it also has solitary wave solutions which have a discontinuous first derivative at the wave peak and thus are called “peakons”.

In this paper, from the mathematical point of view, we study the peakons and their bifurcation of the following generalized Camassa-Holm equation

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

with $a>0$, $k\in ℝ$, $m\in ℕ$ and the integral constants taken as zero. Using the bifurcation method of the phase plane, we first give the phase portrait bifurcation, then give the integral expressions of peakons through the bifurcation curves and the phase portraits, and finally obtain the peakon bifurcation parameter value and the number of peakons. For $m=1,2,3$, we give the explicit expressions for the peakons. It seems that the bifurcation method of phase planes is good for the study of peakons in nonlinear integrable equations.

##### MSC:
 37K50 Bifurcation problems (infinite-dimensional systems) 35Q51 Soliton-like equations 35Q53 KdV-like (Korteweg-de Vries) equations