On decaying properties of nonlinear Schr\"odinger equations

TL;DR

This paper establishes that solutions to the 3D cubic defocusing nonlinear Schrödinger equation decay at the same rate as linear solutions, even with lower initial data regularity, and explores the impact of randomization on decay estimates.

Contribution

It provides quantitative decay estimates for nonlinear Schrödinger solutions with minimal initial data regularity and introduces randomization to relax data assumptions.

Findings

Nonlinear solutions decay at the same rate as linear solutions.

Lower regularity initial data still yields decay estimates.

Randomization replaces the need for $L^1$-data assumption.

Abstract

In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing Nonlinear Schr\"odinger equation with various initial data, deterministic and random. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see \cite{fan2021decay} and the references therein) and moreover we quantify the decay, which is another novelty of this work. Furthermore, we show that the (physical) randomization of the initial data can be used to replace the $L^1$-data assumption (see \cite{fan2022note} for the necessity of the $L^1$-data assumption). At last, we note that this method can be also applied to derive decay estimates for other nonlinear dispersive equations.