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

TL;DR

This paper investigates the initial value problem for the Whitham equations associated with the Camassa-Holm equation, revealing self-similar solutions for step initial data and cusp-shaped solutions for smooth initial data, with boundary matching Burgers solutions.

Contribution

It provides a detailed analysis of the initial value problem for the Whitham equations in the context of the Camassa-Holm equation, including solution behaviors for different initial conditions.

Findings

Self-similar solutions for step initial data

Existence of cusp-shaped solutions for smooth initial data

Boundary solutions match Burgers solutions outside the cusp

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.