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We consider the gravity water waves system with a periodic one-dimensionalinterface in infinite depth, and prove a rigorous reduction of these equationsto Birkhoff normal form up to degree four. This proves a conjecture ofZakharov-Dyachenko [55] suggested by the formal Birkhoff integrability of thewater waves Hamiltonian truncated at order four. As a consequence, we alsoobtain a long-time stability result: periodic perturbations of a flat interfacethat are of size $\varepsilon$ in a sufficiently smooth Sobolev space lead tosolutions that remain regular and small up to times of order$\varepsilon^{-3}$. This time scale is expected to be optimal. Maindifficulties in the proof are the presence of non-trivial resonant four-wavesinteractions, the so-called Benjamin-Feir resonances, the small divisorsarising from near-resonances and the quasilinear nature of the equations. Someof the main ingredients that we use are: (1) a reduction procedure to constantcoefficient operators up to smoothing remainders, that, together with theverification of key algebraic cancellations of the system, implies theintegrability of the equations at non-negative orders; (2) aPoincar\'e-Birkhoff normal form of the smoothing remainders that deals withnear-resonances; (3) an a priori algebraic identification argument of the abovePoincar\'e- Birkhoff normal form equations with the formal Hamiltoniancomputations of [55, 19, 27, 17], that allows us to handle the Benjamin-Feirresonances.

Birkhoff normal form and long time existence for periodic gravity water waves