Global Bifurcation for the Whitham Equation

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 Holder class C-alpha, alpha < 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.