Global Bifurcation for the Whitham Equation

¹ Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
² Department of Mathematics, University of Bergen Postbox 7800, 5020 Bergen, Norway

^⋆ Corresponding author. E-mail: henrik.kalisch@math.uib.no

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 C^α, 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.