Wave-breaking analysis and weak multi-peakon solutions for a generalized cubic–quintic Camassa–Holm type equation
We consider the Cauchy problem and multi-peakon solutions of a generalized cubic–quintic Camassa–Holm (gcqCH) equation, which is actually an extension of the cubic CH equation [alias the Fokas–Olver–Rosenau–Qiao equation in the literature] and the quintic CH equation, and possesses the Hamiltonian structure and conversation law. We first present the blow-up criteria and the precise blow-up quantity in terms of the Moser-type estimate in Sobolev spaces. Then, by using the blow-up quantity and the characteristics associated with the gcqCH equation, we obtain two kinds of sufficient conditions on the initial data to guarantee the occurrence of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon as well as global N-peakon solutions of the gcqCH equation are also investigated. Particularly, we study the two-peakon dynamical system with the time evolution of their elastic collisions.
Weng, W., Qiao, Z. & Yan, Z. Wave-breaking analysis and weak multi-peakon solutions for a generalized cubic–quintic Camassa–Holm type equation. Monatsh Math (2022). https://doi.org/10.1007/s00605-022-01699-w