The Fourier Coefficients of Stokes’ Waves

It is common to formulate the Stokes wave problem as Nekrasov’s nonlinear integral equation to be satisfied by a periodic function θ which gives the angle between the tangent to the wave and the horizontal. The function θ is odd for symmetric waves. In that case, numerical calculations using spectral methods reveal the coefficients in the sine series of θ to form a sequence of positive terms that converges monotonically to zero. In this paper, we prove that the Fourier sine coefficients of θ form a log-convex sequence that converges monotonically to zero. In harmonic analysis there are many very beautiful theorems about the behavior of functions whose Fourier sine series form a convex monotone sequence tending to zero.