Global bifurcation for the whitham equation

Abstract

We prove the existence of a global bifurcation branch of 2π-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 Hölder class Cα, α < 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. © EDP Sciences, 2013.