Abstract
Stokes conjectured in 1880 that an extreme gravity wave on water (or ‘wave of greatest height’) exists, has sharp crests of included angle 2π/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,π). 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.
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Communicated by C. A. Stuart
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Fraenkel, L.E. A Constructive Existence Proof for the Extreme Stokes Wave. Arch Rational Mech Anal 183, 187–214 (2007). https://doi.org/10.1007/s00205-006-0003-y
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DOI: https://doi.org/10.1007/s00205-006-0003-y