TY - JOUR

T1 - A constructive existence proof for the extreme stokes wave

AU - Fraenkel, L. E.

N1 - ID number: ISI:000244023300001

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

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

U2 - 10.1007/s00205-006-0003-y

DO - 10.1007/s00205-006-0003-y

M3 - Article

SN - 0003-9527

VL - 183

SP - 187

EP - 214

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 2

ER -