Highest Cusped Waves for the Burgers-Hilbert equation

Dahné, Joel; Gómez-Serrano, Javier

doi:10.48550/arxiv.2205.00802

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2023

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“…A very large number of terms is required, as the map involved is just barely a contraction. A recent preprint in the same direction for the Burgers-Hilbert equation is [10].…”

Section: Introductionmentioning

confidence: 99%

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

Ehrnström¹,

Mæhlen²,

Varholm³

2023

Preprint

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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°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 [14,15,34]. 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.

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“…A very large number of terms is required, as the map involved is just barely a contraction. A recent preprint in the same direction for the Burgers-Hilbert equation is [10].…”

Section: Introductionmentioning

confidence: 99%