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A constructive existence proof for the extreme stokes wave


Reference:

Fraenkel, L. E., 2007. A constructive existence proof for the extreme stokes wave. Archive for Rational Mechanics and Analysis, 183 (2), pp. 187-214.

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Official URL:

http://dx.doi.org/10.1007/s00205-006-0003-y

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.

Details

Item Type Articles
CreatorsFraenkel, L. E.
DOI10.1007/s00205-006-0003-y
DepartmentsFaculty of Science > Mathematical Sciences
RefereedYes
StatusPublished
ID Code6991
Additional InformationID number: ISI:000244023300001

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