Dynamical behavior near self-similar blowup waves for the generalized b-equation

Abstract

In this paper, we consider the explicit wave-breaking mechanism and its dynamical behavior near this singularity for the generalized b-equation. This generalized b-equation arises from the shallow water theory, which includes the Camassa-Holm equation, the Degasperis-Procesi equation, the Fornberg-Whitham equation, the Korteweg-de Vires equation and the classical b-equation. More precisely, we find that there exists an explicit self-similar blowup solution for the generalized b-equation. Meanwhile, this self-similar blowup solution is asymptotic stability in a parameters domain, but instability in other parameters domain.