Global bifurcation for the Whitham equation

TL;DR

This paper establishes the existence of a global bifurcation branch of periodic traveling-wave solutions for the Whitham equation, characterizes their properties, and provides numerical methods and results including the highest cusped wave.

Contribution

It proves the existence of a global bifurcation branch for the Whitham equation and introduces a spectral scheme for computing these solutions.

Findings

Existence of a global bifurcation branch of solutions.

Presence of a highest, cusped wave in the branch.

Numerical confirmation of bifurcation properties and wave shapes.

Abstract

We prove the existence of a global bifurcation branch of $2\pi$-periodic, smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset of solutions in the global branch contains a sequence which converges uniformly to some solution of H\"older class $C^{\alpha}$, $\alpha < \frac{1}{2}$. Bifurcation formulas are given, as well as some properties along the global bifurcation branch. In addition, a spectral scheme for computing approximations to those waves is put forward, and several numerical results along the global bifurcation branch are presented, including the presence of a turning point and a `highest', cusped wave. Both analytic and numerical results are compared to traveling-wave solutions of the KdV equation.