Stokes wave is the fully nonlinear gravity wave propagating with the constant velocity [1,2].
We consider Stokes wave in the conformal variables which maps the
domain occupied by fluid into the lower complex half-plane. Then
Stokes wave can be described through the position and the type of
complex singularities in the upper complex half-plane. Similar idea
was exploited for other hydrodynamic systems in different
approximations, see e.g. [3-6]. We studied fully nonlinear problem
and 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 located in
that second (nonphysical) sheet of the Riemann surface in the lower
half-plane. As the nonlinearity increases, both singularities
approach the real line forming the classical Stokes solution
(limiting Stokes wave) with the branch point of power 2/3. We
reformulated Stokes wave equation through the integral over jump at
the branch cut which provides the efficient way for finding of the
explicit form of Stokes wave [7].

References

[1] G.~G. Stokes, Transactions of the Cambridge
Philosophical Society {\bf 8}, 441
(1847).

[2] G.~G. Stokes, Mathematical and Physical Papers {\bf
1}, 197 (1880).

[3] E. A. Kuznetsov, M. D. Spector, and V. E. Zakharov,
Phys. Rev. E 49, 1283 (1994).

[4] E. Kuznetsov, M. Spector, and V. Zakharov, Physics
Letters A 182, 387 (1993).

[5] M. Mineev-Weinstein, P. B. Wiegmann, and A. Zabrodin,
Phys. Rev. Lett. 84, 5106 (2000).

[6] P. M. Lushnikov, Physics Letters A 329, 49 (2004).

[7] S.A. Dyachenko, P.M. Lushnikov, and A.O. Korotkevich. JETP Letters {\bf 98}, 675-679 (2014).