---
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.'
address:
- 'Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway'
- 'Centre for Mathematical Sciences, Lund University, PO Box 118, 22100 Lund, Sweden\'
author:
- Mats Ehrnström
- Erik Wahlén
bibliography:
- 'highest.bib'
title: 'On Whitham’s conjecture of a highest cusped wave for a nonlocal dispersive equation'
---

Introduction
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In 1967, G.B. Whitham proposed in [@0163.21104] a nonlocal shallow water wave model for capturing the balance between 