Non-uniform continuity of periodic Holm-Staley b-family of equations

2010/10/19 by Ognyan Christov, Christov, Ognyan, Sevdzhan Hakkaev +3
Mathematics · Physics and Astronomy · #35Q35 #Advanced Mathematical Physics Problems #Differential Equations and Numerical Methods #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1010.3807

openalex publication_date 2010/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a family of non-evolutionary partial differential equations known as Holm - Staley b - family which includes the integrable Camassa-Holm and Degasperis-Procesi equations. We show that the solution map is not uniformly continuous. The proof relies on a construction of smooth periodic travelling waves with small amplitude.

Citations

• The Cauchy problem for quasi-linear symmetric hyperbolic systems
• Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation
• Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation
• Wave breaking for nonlinear nonlocal shallow water equations
• On the Blow-Up of Solutions of a Periodic Shallow Water Equation

Related

• On the well-posedness of the Holm-Staley b-family of equations
• Non-uniform dependence on initial data for the Camassa--Holm equation in Besov spaces: Revisited
• Non-uniform continuity on initial data for the two-component b-family system in Besov space
• Non-uniform continuity of the generalized Camassa-Holm equation in Besov spaces
• Non-uniform dependence for the two-component Camassa-Holm shallow water system