On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations

2010/09/13 by Ying Fu, Yue Liu, Fu, Ying +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1009.2466

openalex publication_date 2010/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Considered herein are the generalized Camassa-Holm and Degasperis-Procesi equations in the spatially periodic setting. The precise blow-up scenarios of strong solutions are derived for both of equations. Several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established. The exact blow-up rates are also determined. Finally, geometric descriptions of these two integrable equations from non-stretching invariant curve flows in centro-equiaffine geometries, pseudo-spherical surfaces and affine surfaces are given.

Citations

• Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation
• Wave breaking for nonlinear nonlocal shallow water equations
• The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

Cited by

• Global weak solutions to a weakly dissipative μHS equation

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