Application of Zakharov equation in three dimensions to deep water gravity waves

A numerical method, based on the Zakharov equation in three dimensions, is developed to study the nonlinear dynamics of deep water gravity waves. We focus primarily on systems describing the evolution of hydrodynamic surface waves, considering a thick incompressible fluid spanning across all space available and subject to gravity. The reference height to the resting state is z = 0. In the case of hydrostatic equilibrium, the bottom of the water is located at a distance H from the water surface. This depth is considered constant. The height of the surface wave is denoted  x,y,t) and the volume occupied by the fluid is restricted by –H < z < (x,y,t). This study shows that rogue waves are generated by primary waves whose directions of propagation are nearly parallel (i.e. 0 < /6). This ensures their unexpected character (expressed by the quasi-spontaneous passage from a calm situation to a sea greatly agitated).