On the precise cusped behaviour of extreme solutions to Whitham-type equations

We prove exact leading-order asymptotic behaviour at the origin for nontrivial solutions of two families of nonlocal equations. The equations investigated include those satisfied by the cusped highest steady waves for both the uni- and bidirectional Whitham equations. The problem is therefore analogous to that of capturing the $120\unicode{xb0}$ interior angle at the crests of classical Stokes' waves of greatest height. In particular, our results partially settle conjectures for such extreme waves posed in a series of recent papers. Our methods may be generalised to solutions of other nonlocal equations, and can moreover be used to determine asymptotic behaviour of their derivatives to any order.