Universal nature of the nonlinear stage of modulational instability

TL;DR

This paper analyzes the long-time behavior of modulational instability in the focusing nonlinear Schrödinger equation, revealing a universal asymptotic structure characterized by constant regions and an oscillatory central zone.

Contribution

It provides a rigorous characterization of the universal nonlinear stage of modulational instability for a broad class of initial conditions in the focusing NLS equation.

Findings

The solution splits into constant and oscillatory regions over time.

The oscillatory region is described by slow modulations of periodic solutions.

The long-time behavior is universal across different initial perturbations.

Abstract

We characterize the nonlinear stage of modulational instability (MI) by studying the long-time asymptotics of focusing nonlinear Schrodinger (NLS) equation on the infinite line with initial conditions that tend to constant values at infinity. Asymptotically in time, the spatial domain divides into three regions: a far left field and a far right field, in which the solution is approximately equal to its initial value, and a central region in which the solution has oscillatory behavior and is described by slow modulations of the periodic traveling wave solutions of the focusing NLS equation. These results demonstrate that the asymptotic stage of MI is universal, since the long-time behavior of a large class of perturbations is described by the same asymptotic state.