Normal Form Transformations and Dysthe’s Equation for the Nonlinear Modulation of Deep-Water Gravity Waves

A new Hamiltonian version of Dysthe’s equation is derived for two-dimensional weakly modulated gravity waves on deep water. A key ingredient in this derivation is a Birkhoff normal form transformation that eliminates all non-resonant cubic terms and allows for a refined reconstruction of the free surface. This modulational approximation is tested against numerical solutions of the classical Dysthe’s equation and against direct numerical simulations of Euler’s equations for nonlinear water waves. Very good agreement is found in the context of Benjamin–Feir instability of Stokes waves, for which an analysis is provided. An extension of our Hamiltonian model incorporating exact linear dispersion as well as an alternate spatial form are also proposed.