Resonant large deviations principle for the beating NLS equation

TL;DR

This paper establishes a large deviations principle for the beating NLS equation with random initial data, revealing how nonlinear resonance can increase the probability of extreme wave events.

Contribution

It introduces a large deviations framework for the beating NLS equation and demonstrates the impact of resonant energy exchange on extreme wave formation.

Findings

Resonant energy exchange increases likelihood of extreme waves.

Nonlinear focusing causes tail fattening in solution probability measures.

Large deviations principle applies to solutions with random initial Fourier modes.

Abstract

We prove a large deviations principle for the solution to the beating NLS equation on the torus with random initial data supported on two Fourier modes. When these modes have different initial variance, we prove that the resonant energy exchange between them increases the likelihood of extreme wave formation. Our results show that nonlinear focusing mechanisms can lead to tail fattening of the probability measure of the sup-norm of the solution to a nonlinear dispersive equation.