Branch cuts of Stokes wave

Complex analytical structure of Stokes wave for two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth is studied both analytically and numerically. Stokes wave is the fully nonlinear periodic gravity wave propagating with the constant veloci1ty. Simulations with the quadruple (32 digits) and variable precisions (more than 200 digits) are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with 120 degrees angle on the crest. A conformal map is used which maps a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half-plane. The Stokes wave is fully characterized by the complex singularities in the upper complex half-plane. These singularities are addressed by rational (Padé) interpolation of Stokes wave in the complex plane. Convergence of Padé approximation to the density of complex poles with the increase of the numerical precision and subsequent increase of the number of approximating poles reveals that the only singularities of Stokes wave are branch cuts. We identified that this singularity is the square-root branch point. That branch cut defines the second sheet of the Riemann surface if we cross the branch cut. Second singularity is also the square-root and located in that second (nonphysical) sheet of the Riemann surface in the lower half-plane. Crossing corresponding branch cut in second sheet one arrives to the third sheet of Riemann surface with another singularity etc forming infinite number of sheets. As the nonlinearity increases, all singularities approach the real line forming the classical Stokes solution (limiting Stokes wave) with the branch point of power 2/3.