# Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions - Mathematical Physics

Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions - Mathematical Physics - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: The small dispersion limit of solutions to the Camassa-Holm CH equation ischaracterized by the appearance of a zone of rapid modulated oscillations. Anasymptotic description of these oscillations is given, for short times, by theone-phase solution to the CH equation, where the branch points of thecorresponding elliptic curve depend on the physical coordinates via the Whithamequations. We present a conjecture for the phase of the asymptotic solution. Anumerical study of this limit for smooth hump-like initial data provides strongevidence for the validity of this conjecture. We present a quantitativenumerical comparison between the CH and the asymptotic solution. The dependenceon the small dispersion parameter $\epsilon$ is studied in the interior and atthe boundaries of the Whitham zone. In the interior of the zone, the differencebetween CH and asymptotic solution is of the order $\epsilon$, at the trailingedge of the order $\sqrt{\epsilon}$ and at the leading edge of the order$\epsilon^{1-3}$. For the latter we present a multiscale expansion whichdescribes the amplitude of the oscillations in terms of the Hastings-McLeodsolution of the Painlev\-e II equation. We show numerically that thismultiscale solution provides an enhanced asymptotic description near theleading edge.

Autor: S. Abenda, T. Grava, C. Klein

Fuente: https://arxiv.org/