The Stokes and Krasovskii Conjectures for the Wave of Greatest Height

The integral equation:
φ_μ(s) = (1/3 π)∫^π ₀((sin φ_μ(t))/(μ ⁻¹+ ∫^t ₀sin φ_μ(u) d u )) (log((sin½( s + t ))/ (sin½( s − t )))d t
was derived by Nekrasov to describe waves of permanent form on the surface of a nonviscous, irrotational, infinitely deep flow, the function φ_μ giving the angle that the wave surface makes with the horizontal. The wave of greatest height is the singular case μ=∞, and it is shown that there exists a solution φ_∞ to the equation in this case and that it can be obtained as the limit (in a specified sense) as μ→∞ of solutions for finite μ. Stokes conjectured that φ_∞( s )→⅙π as s ↓0, so that the wave is sharply crested in the limit case; and Krasovskii conjectured that sup _{s ∈[0,π]}φ_μ( s )≤⅙π for all finite μ. Stokes' conjecture was finally proved by Amick, Fraenkel, and Toland, and the present article shows Krasovskii's conjecture to be false for sufficiently large μ, the angle exceeding ⅙π in what is a boundary layer.