Erik Wahlén: On the highest wave for Whitham’s wave equation

Abstract:

In the 1960’s G. B. Whitham suggested a non-local version of the KdV equation as a model for water waves.
Unlike the KdV equation it is not integrable, but it has certain other advantages.
In particular, it has the same dispersion relation as the full water wave problem and it allows for
wave breaking. The equation has a family of periodic, travelling wave solutions for any given
wavelength. Whitham conjectured that this family contains a highest wave which has a cusp at the
crest. I will outline a proof of this conjecture using global bifurcation theory and precise information
about an integral operator which appears in the equation. The latter is related to Stieltjes functions
and completely monotone functions.