Asymptotic stability and instability of explicit self-similar waves for a class of nonlinear shallow water equations

Abstract

This paper considers the dynamical behavior of solutions near explicit self-similar singularity for a class of nonlinear shallow water models including the Dullin–Gottwald–Holm equation, the Korteweg–de Vires equation, the Camassa–Holm equation, the dispersive rod equation and the Benjamin–Bona–Mahony equation. We first show there are explicit self-similar solutions for those five nonlinear shallow water equations, then we find that those explicit self-similar solutions for the Dullin–Gottwald–Holm equation, the Camassa–Holm equation and the dispersive rod equation are asymptotic stable, but for the Korteweg–de Vries equation and the Benjamin–Bona–Mahony equation being unstable.