Initial value problem of the Whitham equations for the Camassa–Holm equation

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–tx–t plane. On the boundary of the cusp, the Whitham solution matches the Burgers solution, which exists outside the cusp.