An integrable normal form for water waves in infinite depth

We consider the Birkhoff normal form for the water wave problem posed in a fluid of infinite depth, with the starting point of our analysis a version of the Hamiltonian given by Zakharov. We verify that in the fourth order normal form, the coefficients vanish for all non-generic resonant terms, and we show that the resulting truncated system is completely integrable. In contrast we show that there are resonant fifth order terms with nonvanishing coefficients, answering to the negative the conjecture of Dyachenko & Zakharov on the integrability of free surface hydrodynamics. We analyse all solutions of the fourth order integrable system. In particular, the bifurcation structures of standing wave and travelling wave solutions are remarkably explicit due to the normal form. © 1995 Elsevier Science B.V. All rights reserved.