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

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

2008-05-16

arxiv.org/abs/0805.2558v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

33 pages, 3 figures

Scientific paper

10.1016/j.physd.2008.08.016

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.

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