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The sub-harmonic bifurcation of Stokes waves. (English) Zbl 0962.76012

The behaviour of steady periodic water waves on water of infinite depth, that satisfy exactly the kinematic and dynamic boundary conditions on the free surface of water, with or without surface tension, are given by solutions of a nonlinear pseudo-differential operator equation for a \(2\pi\)-periodic function of a real variable. The study is complicated by the fact that the equation is quasilinear, and it involves a non-local operator in the form of a Hilbert transform. Although this equation is exact, it is quadratic with no higher order terms, and the global structure of its solution set can be studied using elements of the theory of real analytic varieties, and using the variational technique. The purpose of this paper is to show that uniquely defined arc-wise connected set of solutions with prescribed minimal period bifurcates from the first eigenvalue of the linearized problem. Although the set is not necessarily maximal as a connected set of solutions, and may possibly self-intersect, it has a local real analytic parametrization that contains a wave of greatest height in its closure. The authors also examine the dependence of the solution on the Froude number in relation to Stokes waves.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76E99 Hydrodynamic stability
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