Abstract: Under reasonable conditions, the study of water waves can be reduced to the study of an elliptic system with a highly nonlinear free boundary condition. In the irrotational setting ('the water does not curl'), uniqueness is an immediate consequence of the maximum principle for harmonic functions. Using the theory for eigenfunctions of the Laplacian we show uniqueness for a class of rotational water waves: the surface profile and the vorticity function determines the fluid motion.

Due to the difficulties inherent in the exact equation linearization is widely used. If the solution is also approximated to the first order, a classic result states that the particle paths of individual fluid particles are closed orbits. We show that the exact solution of the linearized problem does not exhibit this property. By the study of dynamical systems,  it can be seen that every particle travels along an *almost* closed elliptic curve, moving slightly forward with the current.

This is joint work with A. Constantin and G. Villari.