Non Permanent Form Solutions in the Hamiltonian Formulation of Surface Water Waves

Using the KAM method, we exhibit some solutions of a finite dimensional approximation of the Zakharov Hamiltonian formulation of gravity water waves (Zakharov, 1968), which are spatially periodic, quasi-periodic in time, and not permanent form travelling waves. For this Hamiltonian, which is the total energy of the waves, the canonical variables are some complex quantities an and a ∗ n ( n ∈ Z)), which are linear combinations of the Fourier components of the free surface elevation and the velocity potential evaluated at the surface. We expose the method for the case of a system with a finite number of degrees of freedom, the Zufiria model (Zufiria, 1988), with only 3 modes interacting.