On the nonlinear stage of Modulation Instability

TL;DR

This paper derives exact N-solitonic solutions to the focusing Nonlinear Schrödinger Equation to describe the nonlinear evolution of modulation instability in condensates, highlighting special classes of solutions that model localized perturbations.

Contribution

It introduces a general N-solitonic solution framework and identifies regular and superregular solutions that characterize the nonlinear stage of modulation instability.

Findings

Regular solitonic solutions do not disturb the condensate phases at infinity.

Superregular solutions are small perturbations representing the nonlinear stage.

Solutions describe localized perturbations evolving from modulation instability.

Abstract

We study the nonlinear stage of the modulation instability of a condensate in the framework of the focusing Nonlinear Schr\"{o}dinger Equation. We find a general N-solitonic solution of the focusing NLSE in the presence of a condensate by using the dressing method. We separate a special designated class of "regular solitonic solutions" that do not disturb phases of the condensate at infinity by coordinate. All regular solitonic solutions can be treated as localized perturbations of the condensate. We find an important class of "superregular solitonic solutions" which are small perturbations at certain a moment of time. They describe the nonlinear stage of the modulation instability of the condensate.