Dual free boundaries for Stokes waves

Classically, gravity waves on the surface of a two-dimensional layer of per fect fluid are described in potential theory by a free-boundary problem for the (harmonic) stream function psi on an unknown domain Omega subset of R- 2. Many authors independently have transformed this problem by a conformal mapping into a quasi-linear equation (W) for a function w of a single varia ble. Equation (W) is non-local, as it involves the Hilbert transform, and i t has variational structure, being an Euler-Lagrange equation. Here we deal with periodic waves in which case w is periodic. Our aim is to introduce a new non-local variational equation (V) to be satisfied by another periodic function v. The connection between (V) and (W) is one of duality, in three related senses. First, there is a system of partial differential equations with variational structure for a vector-function ((v) over tilde, (w) over tilde) on the unit disc: Solving this system partially yields either w(t) : = (w) over tilde (exp(it)), a solution of (W), or v(t) := (v) over tilde (e xp(it)), a solution of (V). Second, the duality can be defined in terms of a Riemann-Hilbert problem on the unit circle (RH). Third, there is a direct relation between the variational principles for (V) and (W). Most signific antly, there is a potential-theoretic problem corresponding to v. It is als o a free-boundary problem for a (different) harmonic function <(<psi>)over cap>, and the free streamline does not coincide with the free streamline of psi. The problems for psi and psi correspond only via duality. However v i s a function of w, and vice versa. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.