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

2018/06/17 by Wanfeng Yan, Yan, W. P.
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1806.06391

openalex publication_date 2018/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Global Conservative Solutions of the Camassa–Holm Equation
• Wave breaking for nonlinear nonlocal shallow water equations
• The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

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