Instability of peaked travelling wave solutions of the b-family of Camassa-Holm equations

Abstract

The aim of this work is to study instability of peaked travelling waves of the b-family of the generalized Camassa-Holm equations. This family includes the integrable Camassa-Holm and Degasperis-Procesi equations for b = 2 and b = 3 respectively. We first review previous results on the existence and stability of peaked travelling waves on the infinite line and in the periodic domain. Next, we prove instability of the peaked solitary waves under suitable assumptions. The instability is obtained by the methods of characteristics and comparison theory for differential equations. We give some precise results on instability of the peaked periodic waves in the Camassa-Holm equation. Finally, we review open problems in the stability theory of peaked periodic waves in the Degasperis-Procesi equation.