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This article concerns the water wave problem in a three-dimensional domain ofinfinite depth and examines the modulational regime for weakly nonlinearwavetrains. We use the method of normal form transformations near theequilibrium state to provide a new derivation of the Hamiltonian Dystheequation describing the slow evolution of the wave envelope. A precisecalculation of the third-order normal form allows for a refined reconstructionof the free surface. We test our approximation against direct numericalsimulations of the three-dimensional Euler system and against predictions fromthe classical Dysthe equation, and find very good agreement.

Hamiltonian Dysthe equation for 3D deep-water gravity waves