Partial mass dynamics of the defocussing nonlinear Schr\"odinger equation

TL;DR

This paper investigates the long-term behavior of solutions to the defocusing nonlinear Schrödinger equation, utilizing refined scattering techniques to describe the evolution of solution zeros under less restrictive conditions.

Contribution

It introduces a novel approach to analyze the long-time dynamics of defocusing NLS by revisiting the scattering map with weaker decay and regularity assumptions.

Findings

Asymptotic description of solution zeros over time

Reduced regularity and decay requirements for analysis

Application of uniform resolvent bounds and Riemann-Lebesgue lemma

Abstract

We study the long time dynamics of the defocussing NLS equation. Compared with previous literature, we revisit the direct and inverse scattering map to obtain asymptotics in some weighted energy space that requires less restrictive decay and regularity assumptions. The main result is derived from an application of uniform resolvent bound and an approximation argument in the spirit of Riemann-Lebesgue lemma. As a consequence, our result depicts the long time dynamics of the zeros of the solution to the defocussing NLS equation.