Superregular solitonic solutions: A novel scenario for the nonlinear stage of modulation instability

We describe a general N-solitonic solution of the focusing nonlinear Schrödinger equation in the presence of a condensate by using the dressing method. We give the explicit form of one- and two-solitonic solutions and study them in detail as well as solitonic atoms and degenerate solutions. We distinguish a special class of solutions that we call regular solitonic solutions. Regular solitonic solutions do not disturb phases of the condensate at infinity by coordinate. All of them can be treated as localized perturbations of the condensate. We find a broad class of superregular solitonic solutions which are small perturbations at a certain moment of time. Superregular solitonic solutions are generated by pairs of poles located on opposite sides of the cut. They describe the nonlinear stage of the modulation instability of the condensate and play an important role in the theory of freak waves.