A Hilbert-transform formulation of the Stokes wave problem and its consequences

John Toland

University of Bath, Department of Mathematical Sciences Claverton Down, Bath BA2 7AY, U.K.

                      jft@maths.bath.ac.uk

It will be shown how a 2π-periodic solution u of the non-linear equation

Cu′ = λ{u + uCu′ + C(uu′)}

gives rise to a solution of the classical periodic water-wave problem on a flow of infinite depth. Here C: L_{2}(S^{1}) ⟶ L_{2}(S^{1}) denotes the Hilbert transform and the prime denotes differentiation. A brief survey of the properties of this equation will be given.