The regularity and local bifurcation of steady periodic water waves

Steady periodic water waves on infinite depth, satisfying exactly the kinematic and dynamic boundary conditions on the free surface, with or without surface tension, are given by solutions of a rather tidy nonlinear pseudo-differential operator equation for a 2 pi-periodic function of a real variable. Being an Euler-Lagrange equation, this formulation has the advantage of gradient structure, but is complicated by the fact that it involves a non-local operator, namely the Hilbert transform, and is quasi-linear. This paper is a mathematical study of the equation in question. First it is shown that its W-1,W-2 solutions are real analytic. Then bifurcation theory for gradient operators is used to prove the existence of (non-zero) small amplitude waves near every eigenvalue (irrespective of multiplicity) of the linearised problem. Finally it is shown that when surface tension is absent there are no sub-harmonic bifurcations or turning points at the outset of the branches of Stokes waves which bifurcate from the trivial solution.