Initial Value Problem of the Whitham Equations for the Camassa-Holm Equation

Abstract

We study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham equation matches the Burgers solution, which exists outside the cusp.

33 pages, 3 figures

References in corpus (2)

• An integrable shallow water equation with peaked solitons
• Self-Similar Solutions of the Non-Strictly Hyperbolic Whitham Equations for the KdV Hierarchy

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• Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions