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The regularity and local bifurcation of steady periodic water waves. (English) Zbl 0959.76010
The behaviour of steady periodic water waves on water of infinite depth, that satisfy exactly kinematic and dynamic boundary conditions on the free surface of water, with or without surface tension, is given by solutions of a nonlinear pseudo-differential operator equation for $$2\pi$$-periodic functions of 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. Bifurcation theory is used to prove the existence of small amplitude waves near every eigenvalue of the linearized problem. It is also shown that in the absence of surface tension there are no sub-harmonic bifurcations or turning points at the outset of branches of Stokes waves which bifurcate from the trivial solution.

##### MSC:
 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76B45 Capillarity (surface tension) for incompressible inviscid fluids 76E17 Interfacial stability and instability in hydrodynamic stability
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