

Branch Cut Singularity of Stokes Wave
Date/Time: 12:20 07Aug2014
Abstract:
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 halfplane. Then Stokes wave can be described through the position and the type of complex singularities in the upper complex halfplane. Similar idea was exploited for other hydrodynamic systems in different approximations, see e.g. [36]. We studied fully nonlinear problem and identified that this singularity is the squareroot 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 halfplane. 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. MineevWeinstein, 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}, 675679 (2014).
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