Asymptotic stability of small ground states for NLS under random perturbations

TL;DR

This paper investigates the asymptotic stability of small ground states in the cubic Schrödinger equation with a potential, demonstrating persistence under rough perturbations and constructing a large measure set of solutions with stability behavior.

Contribution

It introduces a novel probabilistic approach combining critical-weighted strategies and local energy decay estimates to analyze stability under rough perturbations.

Findings

Existence of a large measure set of asymptotically stable solutions

Extension of probabilistic global well-posedness to perturbed settings

Development of a distorted Fourier transform for stability analysis

Abstract

We consider the cubic Schr\"odinger equation on the euclidean space perturbed by a short-range potential $V$. The presence of a negative simple eigenvalue for $-\Delta+V$ gives rise to a curve of small and localized nonlinear ground states that yield some time-periodic solutions known to be asymptotically stable in the energy space. We study the persistence of these coherent states under rough perturbations. We shall construct a large measure set of small scaling-supercritical solutions below the energy space that display some asymptotic stability behavior. The main difficulty is the need to handle the interactions of localized and dispersive terms in the modulation equations. To do so, we use a critical-weighted strategy to combine probabilistic nonlinear estimates in critical spaces based on $U^p, V^q$ (controlling higher order terms) with some local energy decay estimates (controlling lower order terms). We also revisit in the perturbed setting the analysis of B\'enyi, Oh and Pocovnicu on the probabilistic global well-posedness and scattering for small supercritical initial data. We use a distorted Fourier transform and semiclassical functional calculus to generalize probabilistic and bilinear Strichartz estimates.