Non-resonant interacting water waves in 2+1 dimensions

We investigate the interaction among small amplitude water waves, when the fluid motion is in a basin of arbitrary, uniform depth. Waves are supposed to be non-resonant, i.e., with different group velocities that are not close to each other. Starting from the isotropic pseudo-differential Milewski–Keller equation and using an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling, we show that the amplitude slow modulation of Fourier modes can be described by a model system of non-linear evolution equations. We demonstrate that the system is C-integrable, i.e., can be linearized through an appropriate transformation of the dependent and independent variables. A subclass of solutions gives rise to non-localized line-solitons and localized solitons (dromions). Each soliton propagates with the group velocity and during a collision maintains its shape, the only change being a phase shift.