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arXiv · 2609.24447

Higher-order estimates of highest waves of the Whitham equation

Robin Østern Lien
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Abstract

The Whitham equation has given its name to a wider family of nonlocal, nonlinear equations with very weak dispersion, which all feature highest waves. Recent work by Ehrnström, Maehlen and Varholm establishes leading-order asymptotics at the crest of solutions to such Whitham-type equations, and conjectures similar expansions for all derivatives of the solution. By extending their techniques we confirm the conjecture for a range of equations with highest waves of Hölder regularity $C^{s}$ for $s\in[0.35,1)$. We do this by strong induction, redistributing difference operators to deal with singularities in higher-order derivatives arising from integral convolution kernels. The restriction $s\geq0.35$ arises solely from the separate argument establishing the zeroth-order asymptotics, which requires a uniform sign estimate that we verify using rigorous interval arithmetic. The higher-order induction itself applies for every $s\in(0,1)$, so any extension of the zeroth-order result immediately yields the corresponding higher-order expansions.

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Robin Østern Lien. 2026-09-21. Higher-order estimates of highest waves of the Whitham equation. https://arxiv.org/abs/2609.24447

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