Small Dispersion Limit of the Camassa-Holm Equation

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• Christian Klein, Université de Bourgogne, Dijon
• Friday 27 November 2009, 14:00-15:00
• MR11, CMS.

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The small dispersion limit of solutions to the Camassa-Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the one-phase solution to the CH equation, where the branch points of the corresponding elliptic curve depend on the physical coordinates via the Whitham equations. We present a conjecture for the phase of the asymptotic solution. A numerical study of this limit for smooth hump-like initial data provides strong evidence for the validity of this conjecture. We present a quantitative numerical comparison between the CH and the asymptotic solution. We illustrate differences to the well known small dispersion limit of the Korteweg-de Vries equation.

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