Birkhoff Normal form for Gravity Water Waves

We consider the gravity water waves system with a one-dimensional periodic interface in infinite depth, and present the proof of the rigorous reduction of these equations to their cubic Birkhoff normal form (Berti et al. in Birkhoff normal form and long-time existence for periodic gravity Water Waves. arXiv:1810.11549, 2018). This confirms a conjecture of Zakharov–Dyachenko (Phys Lett A 190:144–148, 1994) based on the formal Birkhoff integrability of the water waves Hamiltonian truncated at degree four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size ε in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order ε−3.