Scattering and uniform in time error estimates for splitting method in NLS

TL;DR

This paper establishes uniform in time error estimates for a splitting method applied to the defocusing nonlinear Schrödinger equation, leveraging scattering theory techniques to improve long-term accuracy analysis.

Contribution

It introduces a novel approach using a classical scattering theory vectorfield to achieve uniform in time error bounds for splitting methods in NLS.

Findings

Uniform error estimates hold over long time intervals.

The method applies to mass-(super)critical and energy-subcritical NLS.

Technical modifications extend previous error analysis techniques.

Abstract

We consider the nonlinear Schr{\"o}dinger equation with a defocusing nonlinearity which is mass-(super)critical and energy-subcritical. We prove uniform in time error estimates for the Lie-Trotter time splitting discretization. This uniformity in time is obtained thanks to a vectorfield which provides time decay estimates for the exact and numerical solutions. This vectorfield is classical in scattering theory, and requires several technical modifications compared to previous error estimates for splitting methods.