The complex singularity of a Stokes wave

S. A. Dyachenko^{+}, P. M. Lushnikov^{+*}, A. O. Korotkevich^{+*}

^{+}Department of Mathematics and Statistics, University of New Mexico, 87131-0001 NM, USA

^{*}Landau Institute for Theoretical Physics of the RAS, 119334 Moscow, Russia

**Abstract**
Two-dimensional potential flow of the ideal incompressible fluid with
free surface and infinite depth can be described by a conformal map of the
fluid domain into the complex lower half-plane.
Stokes wave is the fully nonlinear gravity wave propagating with the
constant velocity.
The increase of the scaled wave height H/λ from the linear limit
H/λ=0 to the critical value marks the transition from
the limit of almost linear wave to a strongly
nonlinear limiting Stokes wave. Here H is the wave height and λ is
the wavelength. We simulated fully nonlinear Euler equations, reformulated in
terms of conformal variables, to find Stokes
waves for different wave heights.
Analyzing spectra of these solutions we found in conformal variables, at each
Stokes wave height, the distance v_{c} from the lowest singularity in the upper
half-plane to the real line which corresponds to the fluid free surface.
We also identified that this singularity is the square-root branch point.
The limiting Stokes wave emerges as the singularity reaches the fluid surface.
From the analysis of data for v_{c}→ 0 we suggest a new power law scaling
as well as new estimate .