The regularity and local bifurcation of steady periodic water waves

Steady periodic water waves on infinite depth, satisfying exactly the kinem atic and dynamic boundary conditions on the free surface, with or without s urface tension, are given by solutions of a rather tidy nonlinear pseudo-di fferential operator equation for a 2 pi-periodic function of a real variabl e. Being an Euler-Lagrange equation, this formulation has the advantage of gradient structure, but is complicated by the fact that it involves a non-l ocal 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 theor y for gradient operators is used to prove the existence of (non-zero) small amplitude waves near every eigenvalue (irrespective of multiplicity) of th e linearised problem. Finally it is shown that when surface tension is abse nt there are no sub-harmonic bifurcations or turning points at the outset o f the branches of Stokes waves which bifurcate from the trivial solution.