Large deviations principle for the cubic NLS equation

TL;DR

This paper establishes a Large Deviations Principle for the cubic nonlinear Schrödinger equation on the torus, analyzing rare events like rogue wave formation and identifying criticality thresholds.

Contribution

It introduces a notion of criticality and proves an LDP for subcritical and critical cases, linking probabilistic rare events to rogue wave formation.

Findings

Proved LDP for subcritical and critical regimes

Identified most likely initial conditions for rogue waves

Provided numerical and theoretical insights into rogue wave formation

Abstract

In this paper, we present a probabilistic study of rare phenomena of the cubic nonlinear Schr\"odinger equation on the torus in a weakly nonlinear setting. This equation has been used as a model to numerically study the formation of rogue waves in deep sea. Our results are twofold: first, we introduce a notion of criticality and prove a Large Deviations Principle (LDP) for the subcritical and critical cases. Second, we study the most likely initial conditions that lead to the formation of a rogue wave, from a theoretical and numerical point of view. Finally, we propose several open questions for future research.