We study the interaction of gravity waves on the surface of an infinitely
deep ideal fluid. Starting from Zakharov's variational formulation for water
waves we derive an expansion of the Hamiltonian to an arbitrary order, in a
manner that avoids a laborious series reversion associated with expressing the
velocity potential in terms of its value at the free surface. The expansion
kernels are shown to satisfy a recursion relation enabling us to draw some
conclusions about higher-order wave-wave interaction amplitudes, without
referring to the explicit forms of the individual lower-order kernels. In
particular, we show that unidirectional waves propagating in a two-dimensional
flow do not interact nonlinearly provided they fulfill the energy-momentum
conservation law. Switching from the physical variables to the so-called normal
variables we explain the vanishing of the amplitudes of fourth- and certain
fifth-order non-generic resonant interactions reported earlier and outline a
procedure for finding the one-dimensional wave vector configurations for which
the higher order interaction amplitudes become zero on the resonant
hypersurfaces.Comment: 13 page