Almost sure scattering for the one dimensional nonlinear Schr\"odinger equation

TL;DR

This paper establishes almost sure scattering and global well-posedness for the one-dimensional nonlinear Schrödinger equation with superlinear nonlinearity, using measure-theoretic techniques to describe the evolution of initial data.

Contribution

It introduces a novel measure-based approach to analyze the nonlinear Schrödinger equation, enabling quantitative asymptotics and scattering results.

Findings

Global well-posedness for p>1

Almost sure scattering for p>3

Description of measures with linear and nonlinear evolution

Abstract

We consider the one-dimensional nonlinear Schr\"odinger equation with a nonlinearity of degree $p>1$. We exhibit measures on the space of initial data for which we describe the non trivial evolution by the linear Schr\"odinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. We deduce from this precise description the global well-posedness of the equation for $p>1$ and scattering for $p>3$. To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties) for the nonlinear evolution.