Paper detail

Global bifurcation for the Whitham equation

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 $C^α$, $α< \frac{1}{2}$. 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&#39;, cusped wave. Both analytic and numerical results are compared to traveling-wave solutions of the KdV equation.