I will discuss a joint work with M. Berti and R. Feola in which we give a rigorous proof of the conjecture of Zakharov and Dyachenko, also supported by works of Craig-Worfolk and Craig-Sulem, on the integrability up to order 4 of the gravity water waves Hamiltonian in 1 dimension.

A consequence of this results is a long-time existence theorem for

small spatially periodic solutions on time scales of $\mathcal{O}\left(\left(\text{size of data}\right)^{-3}\right)$.

If time permits, I will also discuss long-time existence results for periodic solutions of the full gravity-capillary problem in 3 dimensions.