The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem

2020/11/26 by Anne Boutet de Monvel, Iryna Karpenko, de Monvel, Anne Boutet +3 · 2 citations
Mathematics · Physics and Astronomy · #35B40 #35Q15 #35Q51 #37K40 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Primary: 35Q53 #Secondary: 37K15

paper · pdf · doi:10.48550/arxiv.2011.13235

openalex publication_date 2020/11/26 · openalex created_date 2020/12/07 · openalex updated_date 2026/07/28

Abstract

This paper deals with the Cauchy problem for the modified Camassa-Holm (mCH) equation \beginalignat*4 &m_t+((u²-u_x²)m)_x=0,& &m:= u-u_xx,& &t>0,& &-∞0), where the leading asymptotic term of the deviation of the solution from the background is nontrivial: this term is given by modulated (with parameters depending on (x)/(t)), decaying (as t^-1/2) trigonometric oscillations.

Citations

• The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

Cited by

• The modified Camassa-Holm equation on the half line: a Riemann--Hilbert approach
• On asymptotic approximation of the modified Camassa-Holm equation in different space-time solitonic regions

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• Asymptotics of step-like solutions for the Camassa-Holm equation
• On the long-time asymptotic of the modified Camassa-Holm equation with nonzero boundary conditions in space-time solitonic regions