Bifurcation of peakons and periodic cusp waves for the generalization of the Camassa–Holm equation

In this paper, we investigate the generalization of the Camassa–Holm equation ut+K(um)x−(un)xxt=[((un)x)22+un(un)xx]x, where KK is a positive constant and m,n∈Nm,n∈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=m+n1+ncn−m+1n 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≥2,m=n+1n≥2,m=n+1, when KK 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. To the best of our knowledge, up until now, this phenomenon has not appeared in any other literature. By further studying the cause of this phenomenon, we show that this planar system has some different characters from the previous Camassa–Holm systems. What is more, we obtain some periodic cusp wave solutions in the form of polynomial functions, which are different from those in the form of exponential functions. Some previous results are extended.