A Constructive Existence Proof for the Extreme Stokes Wave

摘要: Stokes conjectured in 1880 that an extreme gravity wave on water (or ‘wave of greatest height’) exists, has sharp crests included angle 2π/3 and a boundary is convex between successive crests. These three conjectures have all been proved recently, but by diverse methods are not conspicuously direct. The present paper proceeds from first approximate solution the form integral equation due to Nekrasov, contraction mapping for related governs new dependent variable space L 2(0,π). This method provides: (a) constructive approach with predicted Stokes; (b) rather accurate second approximation. However, led (so far, at least) convexity.