About shape of giant breather

The pulse of freak waves on the surface of deep water can be a breather-type solution of the Euler equation. The shape of surface is periodic function of time in a moving frame. Only in the limit of very small steepness its shape is described by the Nonlinear Shredinger Equation (NLSE). For moderately small steepness we derived more exact envelope nonlocal equation similar to well-known Dysthe equation (DE), which is more convenient for analytical and numerical study. We have found approximate solution of this equation by the use of the variational approach.