A constructive existence proof for the extreme stokes wave

L. E. Fraenkel

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

Stokes conjectured in 1880 that an extreme gravity wave on water (or 'wave of greatest height') exists, has sharp crests of included angle 2 pi/3 and has a boundary that is convex between successive crests. These three conjectures have all been proved recently, but by diverse methods that are not conspicuously direct. The present paper proceeds from a first approximate solution of the extreme form of the integral equation due to Nekrasov, to a contraction mapping for a related integral equation that governs a new dependent variable in the space L (2)(0,pi). This method provides: (a) a constructive approach to an extreme wave with the sharp crests predicted by Stokes; and (b) a rather accurate second approximation. However, the method has not led (so far, at least) to the convexity.
Original languageEnglish
Pages (from-to)187-214
Number of pages28
JournalArchive for Rational Mechanics and Analysis
Volume183
Issue number2
DOIs
Publication statusPublished - 2007

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