Higher order asymptotics for the nonlinear Schr\"odinger equation

TL;DR

This paper derives detailed long-term behavior of solutions to the defocusing nonlinear Schrödinger equation using advanced mathematical techniques, assuming initial data with specific regularity and decay properties.

Contribution

It introduces higher order asymptotic analysis for the nonlinear Schrödinger equation via the $ar{ ext{d}}$-nonlinear steepest descent method, extending previous results.

Findings

Explicit higher order asymptotics derived

Applicable to initial data in weighted Sobolev spaces

Enhanced understanding of long-time solution behavior

Abstract

In this paper we compute the higher order long time asymptotics of the defocussing nonlinear Schr\"odinger equation using the $\overline{\partial}$-nonlinear steepest descent method. We assume initial condition in weighted Sobolev space with finite order of regularity and decay.