Numerical integration of Schr\"odinger maps via the Hasimoto transform

TL;DR

This paper presents a novel numerical method for Schr"odinger maps using the Hasimoto transform, enabling explicit, stable, and low-regularity integrators that improve efficiency and applicability to rough solutions.

Contribution

It introduces the first fully explicit unconditionally stable symmetric integrators for Schr"odinger maps based on the Hasimoto transform, including a low-regularity scheme for rough solutions.

Findings

The new integrator is unconditionally stable and explicit.

It effectively handles low-regularity solutions.

Numerical experiments confirm theoretical convergence.

Abstract

We introduce a numerical approach to computing the Schr\"odinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schr\"odinger (NLS) equation. In exploiting this nonlinear transform we are able to introduce the first fully explicit unconditionally stable symmetric integrators for the SM equation. Our approach consists of two parts: an integration of the NLS equation followed by the numerical evaluation of the Hasimoto transform. Motivated by the desire to study rough solutions to the SM equation, we also introduce a new symmetric low-regularity integrator for the NLS equation. This is combined with our novel fast low-regularity Hasimoto (FLowRH) transform, based on a tailored analysis of the resonance structures in the Magnus expansion and a fast realisation based on block-Toeplitz partitions, to yield an efficient low-regularity integrator for the SM equation. This scheme in particular allows us to obtain approximations to the SM in a more general regime (i.e. under lower regularity assumptions) than previously proposed methods. The favorable properties of our methods are exhibited both in theoretical convergence analysis and in numerical experiments.