Convexity of Whitham's highest cusped wave

TL;DR

This paper proves the existence of a convex, highest cusped periodic traveling wave in the Whitham equation, confirming a long-standing conjecture about its shape and regularity at stagnation points.

Contribution

It establishes the convexity of Whitham's highest cusped wave, a key geometric property previously conjectured but not proven.

Findings

Existence of a convex highest cusped wave in the Whitham equation

Confirmation of the wave's cusp regularity at $C^{1/2}$

Resolution of the convexity conjecture by Ehrnström and Wahlén

Abstract

We prove the existence of a periodic traveling wave of extreme form of the Whitham equation that has a convex profile between consecutive stagnation points, at which it is known to feature a cusp of exactly $C^{1/2}$ regularity. The convexity of Whitham's highest cusped wave had been conjectured by Ehrnstr\"om and Wahl\'en.