In this talk I consider the gravity-capillary water waves system for a bi-dimensional fluid with a periodic one-dimensional free surface. First, I present you how the equations are derived, their Hamiltonian structure and the main result of the quadratic lifespan for small solutions of the Cauchy problem. Then, I would like to show one step of the formal Birkhoff normal form and analyze the 3-waves resonance. Due to their presence for general values of the parameters, namely gravity, surface tension and depth, such normal form may be non trivial and exhibit a chaotic dynamics, which is usually referred in fluid dynamics as Wilton ripples. Nevertheless, by exploiting that these 3-waves resonances are finitely many for all the values of the parameters and by the Hamiltonian nature of the Birkhoff normal form, it is possible to perform the energy estimates for proving the main result.

This is a joint work with Massimiliano Berti and Roberto Feola.