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

### Mats Ehrnström

Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway### Erik Wahlén

Centre for Mathematical Sciences, Lund University, PO Box 118, 22100 Lund, Sweden

## Abstract

We consider the Whitham equation $u_{t}+2uu_{x}+Lu_{x}=0$, where *L* is the nonlocal Fourier multiplier operator given by the symbol $m(ξ)=tanhξ/ξ $. 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 $m(ξ)$, and a following study of the highest wave. In particular, we show that the integral kernel corresponding to the symbol $m(ξ)$ is completely monotone, and provide an explicit representation formula for it. Our methods may be generalised.

## Cite this article

Mats Ehrnström, Erik Wahlén, On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 6, pp. 1603–1637

DOI 10.1016/J.ANIHPC.2019.02.006