The Cauchy problem and continuation of periodic solutions for a generalized Camassa-Holm equation

TL;DR

This paper investigates a three-parameter family of nonlinear equations, including the Camassa-Holm equation, establishing global existence, preventing wave breaking, and exploring unique continuation properties for specific parameter choices.

Contribution

It introduces a generalized family of equations extending the Camassa-Holm model and provides new results on global solutions and wave breaking prevention.

Findings

Global existence of solutions under certain parameters

Wave breaking scenarios are prevented for specific cases

Unique continuation properties are established for some parameter values

Abstract

We consider a three-parameter family of non-linear equations with $(p+1)-$order non-linearities. Such family includes as a particular member the well-known $b-$equation, which encloses the famous Camassa-Holm equation. For certain choices of the parameters we establish a global existence result and show a scenario that prevent the wave breaking of solutions. Also, we explore unique continuation properties for some values of the parameters.