On the wave-breaking phenomena and global existence for the generalized periodic Camassa-Holm equation

TL;DR

This paper investigates wave-breaking and global solutions for a generalized periodic Camassa-Holm equation, establishing conditions for finite-time wave-breaking, existence of permanent waves, and global weak solutions.

Contribution

It provides new sufficient conditions for wave-breaking, constructs permanent wave solutions, and proves the existence of global weak solutions for the generalized equation.

Findings

Finite-time wave-breaking under certain conditions

Existence of strong permanent waves

Global weak solutions in the energy space

Abstract

Considered herein is the initial-value problem for the generalized periodic Camassa-Holm equation which is related to the Camassa-Holm equation and the Hunter-Saxton equation. Sufficient conditions guaranteeing the development of breaking waves in finite time are demonstrated. On the other hand, the existence of strong permanent waves is established with certain initial profiles depending on the linear dispersive parameter in a range of the Sobolev spaces. Moreover, the admissible global weak solution in the energy space is obtained.