Abstract
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.
The work was partially supported by the Russian Foundation for Basic Research (grant no. 01-01-00767).
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Dedicated to Professor O. A. Ladyzhenskaya
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Plotnikov, P.I., Toland, J.F. (2002). The Fourier Coefficients of Stokes’ Waves. In: Birman, M.S., Hildebrandt, S., Solonnikov, V.A., Uraltseva, N.N. (eds) Nonlinear Problems in Mathematical Physics and Related Topics I. International Mathematical Series, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0777-2_18
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DOI: https://doi.org/10.1007/978-1-4615-0777-2_18
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