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

TL;DR

This paper proves the existence of a highest cusped traveling wave solution for Whitham's nonlocal dispersive equation, confirming Whitham's conjecture and analyzing its properties and regularity.

Contribution

It establishes the existence of the highest wave as a limiting case of periodic solutions and analyzes its regularity and integral kernel properties.

Findings

The highest wave has optimal $C^{1/2}$-regularity.

The integral kernel is completely monotone.

An explicit representation formula for the kernel is provided.

Abstract

We consider the Whitham equation $u_t + 2u u_x+Lu_x = 0$, 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.