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.

翻译：我们提出了一种基于Hasimoto变换的Schrödinger映射（SM）数值计算方法，该变换将SM流与立方非线性Schrödinger（NLS）方程联系起来。通过利用这一非线性变换，我们首次引入了SM方程的全显式无条件稳定对称积分器。该方法包含两个部分：NLS方程的积分以及Hasimoto变换的数值评估。受研究SM方程粗解需求的驱动，我们还为NLS方程引入了一种新型对称低正则性积分器。结合基于Magnus展开中共振结构针对性分析及块Toeplitz分区快速实现的快速低正则性Hasimoto（FLowRH）变换，我们为SM方程构建了高效的低正则性积分器。该方案特别允许我们在比现有方法更一般的正则性假设下获得SM的近似解。理论收敛性分析与数值实验均展示了所提方法的优越性能。