A constructive existence proof for the extreme stokes wave

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.