Lax integrability and the peakon problem for the modified Camassa-Holm equation

Abstract

Peakons are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the case of the modified Camassa-Holm equation studied in this paper is dictated by the distributional compatibility of its Lax pair and, as a result, it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the Burgers type. Subsequently, we give a complete construction of peakon solutions satisfying the modified Camassa-Holm equation in the sense of distributions; our approach is based on solving certain inverse boundary value problem the solution of which hinges on a combination of classical techniques of analysis involving Stieltjes' continued fractions and multi-point Padé approximations. We propose sufficient conditions needed to ensure the global existence of peakon solutions and analyze the large time asymptotic behaviour whose special features include a formation of pairs of peakons which share asymptotic speeds, as well as Toda-like sorting property.

42 pages

References in corpus (3)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• Degasperis-Procesi peakons and the discrete cubic string
• Complex oscillations in the delayed Fitzhugh-Nagumo equation

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• On asymptotic approximation of the modified Camassa-Holm equation in different space-time solitonic regions
• The Cauchy problem and wave-breaking phenomenon for a generalized sine-type FORQ/mCH equation
• A Riemann-Hilbert approach to the modified Camassa-Holm equation with step-like boundary conditions
• Orbital stability of two-component peakons
• The Calogero-Françoise integrable system: algebraic geometry, Higgs fields, and the inverse problem
• Non-uniform continuity of the Fokas-Olver-Rosenau-Qiao equation in Besov spaces