Large deviations principle for the cubic NLS equation with slowly decaying data

TL;DR

This paper establishes a sharp large deviations principle for the cubic nonlinear Schrödinger equation with Gaussian initial data, improving decay conditions and advancing understanding of probabilistic behavior in Fourier Lebesgue spaces.

Contribution

It proves a precise large deviations principle for the NLS with Gaussian data, refining decay conditions from exponential to  decay in Fourier Lebesgue spaces.

Findings

Established a sharp large deviations principle for the cubic NLS.

Improved decay condition from exponential to  decay.

Enhanced understanding of probabilistic properties of NLS solutions.

Abstract

In this note, we prove a sharp large derivation principle (LDP) for the cubic nonlinear Schr\"odinger equation with Gaussian random initial data in Fourier Lebesgue spaces. As a consequence, we improve the exponential decay condition in [M.A. Garrido, R. Grande, K.M. Kurianski, G. Staffilani. Commun. Pure Appl. Math. 76 (2023), 4087--4136] to $\ell^1$ decay.