On Fourier time-splitting methods for nonlinear Schrodinger equations in the semi-classical limit II. Analytic regularity

TL;DR

This paper analyzes Fourier time-splitting methods for nonlinear Schrödinger equations in the semi-classical limit, demonstrating preservation of WKB structure and error estimates independent of the Planck constant using analytic spaces.

Contribution

It establishes that both exact and numerical solutions maintain WKB structure over a uniform time interval and provides error bounds for quadratic observables independent of the Planck constant.

Findings

WKB structure is preserved over a time interval independent of Planck constant.

Quadratic observables can be accurately computed with a time step independent of Planck constant.

Analytic spaces help overcome regularity loss issues in the semi-classical limit.

Abstract

We consider the time discretization based on Lie-Trotter splitting, for the nonlinear Schrodinger equation, in the semi-classical limit, with initial data under the form of WKB states. We show that both the exact and the numerical solutions keep a WKB structure, on a time interval independent of the Planck constant. We prove error estimates, which show that the quadratic observables can be computed with a time step independent of the Planck constant. The functional framework is based on time-dependent analytic spaces, in order to overcome a previously encountered loss of regularity phenomenon.