Conformal variables in the numerical simulations of long-crested rogue waves

The recently suggested theoretical model for highly nonlinear potential long-crested water waves is discussed, where weak three-dimensional effects are included as small corrections to exact two-dimensional equations written in terms of the conformal variables [V.P. Ruban, Phys. Rev. E 71, 055303(R) (2005)]. Some numerical results based on this theory are presented, which describe spontaneous formation of rogue waves on the deep water for different initial conditions. In particular, the given numerical examples describe: (i) nonlinear stage of the modulational instability, (ii) breathing rogue wave in a random wave field, and (iii) freak wave in a weakly crossing sea state.