Analysis of the formation of an inner solution near the crest of steep surface waves of infinite depth

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Surface periodic waves of infinite depth are investigated. The boundary value problem is formulated in the parametric plane with respect to the Zhukovsky function. By making use of the discrete Fourier transform, the problem is reduced to a finite system of nonlinear transcendental equations. It is shown that with an increase in the steepness of the waves, an inner solution is formed near the crest, and under the corresponding scaling of the sought function this solution is independent of the steepness. It is shown that the numerical reproduction of the inner solution is a key factor for accurate calculations of the almost-highest gravity waves.