On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation

Abstract

We consider the Whitham equation $u_t+2u{u}_x+L{u}_x=0$, where L is the nonlocal Fourier multiplier operator given by the symbol $m(\xi )=\sqrt{\tanh \xi /\xi }$. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting case at the end of the main bifurcation curve of P-periodic solutions, and give several qualitative properties of it, including its optimal $C^{1/2}$-regularity. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol $m(\xi )$, and a following study of the highest wave. In particular, we show that the integral kernel corresponding to the symbol $m(\xi )$ is completely monotone, and provide an explicit representation formula for it. Our methods may be generalised.