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

TL;DR

This paper analyzes the dynamical behavior near self-similar blowup solutions for the generalized b-equation, revealing explicit solutions and their stability properties relevant to shallow water wave models.

Contribution

It identifies explicit self-similar blowup solutions for the generalized b-equation and characterizes their stability depending on parameters.

Findings

Existence of explicit self-similar blowup solutions.

Asymptotic stability in certain parameter domains.

Instability in other parameter domains.

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.