High-order long-time asymptotics for small solutions to the one-dimensional nonlinear Schr\"odinger equation

TL;DR

This paper studies the long-time behavior of small solutions to the 1D nonlinear Schrödinger equation, establishing global existence, modified scattering, and detailed asymptotic expansions using space-time resonance analysis.

Contribution

It provides the first rigorous derivation of high-order asymptotics for small solutions, accounting for long-range effects in a low-regularity setting.

Findings

Global well-posedness for small initial data

Modified scattering with asymptotic expansion at arbitrary order

Analysis based on space-time resonance method

Abstract

We investigate the global well-posedness and modified scattering for the one-dimensional Schr\"odinger equation with gauge-invariant polynomial nonlinearity. For small localized initial data of finite energy in a low-regularity class, we establish global existence of solution together with persistence of the localization of the associated profile. We further provide a rigorous derivation of the asymptotic expansion at arbitrary order of such solutions, taking into account long-range effects induced by the cubic component of the nonlinearity. Our analysis relies on the space-time resonance method.