Quadratic Life Span of Periodic Gravity-capillary Water Waves

We consider the gravity-capillary water waves equations for a bi-dimensional fluid with a periodic one-dimensional free surface. We prove a rigorous reduction of this system to Birkhoff normal form up to cubic degree. Due to the possible presence of three-wave resonances for general values of gravity, surface tension, and depth, such normal form may be not trivial and exhibit a chaotic dynamics (Wilton ripples). Nevertheless, we prove that for all the values of gravity, surface tension, and depth, initial data that are of size ε in a sufficiently smooth Sobolev space leads to a solution that remains in an ε-ball of the same Sobolev space up times of order ε−2. We exploit that the three-wave resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form.