  This article concerns the water wave problem in a three-dimensional domain of
infinite depth and examines the modulational regime for weakly nonlinear
wavetrains. We use the method of normal form transformations near the
equilibrium state to provide a new derivation of the Hamiltonian Dysthe
equation describing the slow evolution of the wave envelope. A precise
calculation of the third-order normal form allows for a refined reconstruction
of the free surface. We test our approximation against direct numerical
simulations of the three-dimensional Euler system and against predictions from
the classical Dysthe equation, and find very good agreement.
