2013/06/08 by Andrew N. W. Hone, Andrew Hone, Stéphane Lafortune +1 · 15 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Applied mathematics #Bifurcation #Camassa–Holm equation #Computer science #Derivative (finance) #Dispersion (optics) #Integrable system #Materials science #Mathematical analysis #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Peakon #Physics #Quantum mechanics #Range (aeronautics) #Soliton #Stability (learning theory) #math.AP #nlin.PS
paper · pdf · doi:10.1016/j.physd.2013.11.006
published in Physica D Nonlinear Phenomena 269, 28-36 (Elsevier BV)
arxiv created 2013/06/08 · openalex publication_date 2013/11/21 · arxiv updated 2015/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The Camassa-Holm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory. Among its many interesting mathematical properties, which include complete integrability, perhaps the most striking is the fact that in the case where linear dispersion is absent it admits weak multi-soliton solutions - "peakons" - with a peaked shape corresponding to a discontinuous first derivative. There is a one-parameter family of generalized Camassa-Holm equations, most of which are not integrable, but which all admit peakon solutions. Numerical studies reported by Holm and Staley indicate changes in the stability of these and other solutions as the parameter varies through the family. In this article, we describe analytical results on one of these bifurcation phenomena, showing that in a suitable parameter range there are stationary solutions - "leftons" - which are orbitally stable.