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Bifurcation of peakons and periodic cusp waves for the generalization of the Camassa-Holm equation. (English) Zbl 1218.35026
From the summary: We investigate the generalization of the Camassa-Holm equation $$u_t+K(u^m)_x-(u^n)_{xxt}= \bigg[\frac{\big((u^n)_x\big)^2}{2}+ u^n(u^n)_{xx} \bigg]_x,$$ where $K$ is a positive constant and $m,n\in\Bbb N$. The bifurcation and some explicit expressions of peakons and periodic cusp wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. Further, in the process of obtaining the bifurcation of phase portraits, we show that $$K=\frac{m+n}{1+n} c^{\frac{n-m+1}{n}}$$ is the peakon bifurcation parameter value for the equation. From the bifurcation theory, in general, the peakons can be obtained by taking the limit of the corresponding periodic cusp waves. However, we find that in the cases of $n\ge2$, $m=n+1$, when $K$ tends to the corresponding bifurcation parameter value, the periodic cusp waves will no longer converge to the peakons, instead, they will still be the periodic cusp waves.

35B32Bifurcation (PDE)
35B10Periodic solutions of PDE
Full Text: DOI
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