Numerical Integration of Schrödinger Maps via the Hasimoto Transform

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2024-01-31 DOI:10.1137/22m1531555

Valeria Banica, Georg Maierhofer, Katharina Schratz

{"title":"Numerical Integration of Schrödinger Maps via the Hasimoto Transform","authors":"Valeria Banica, Georg Maierhofer, Katharina Schratz","doi":"10.1137/22m1531555","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 322-352, February 2024. <br/> Abstract. We introduce a numerical approach to computing the Schrödinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schrödinger (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 realization 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.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"121 1","pages":""},"PeriodicalIF":2.8000,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Numerical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/22m1531555","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 322-352, February 2024.
Abstract. We introduce a numerical approach to computing the Schrödinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schrödinger (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 realization 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.

查看原文本刊更多论文

通过 Hasimoto 变换对薛定谔图进行数值积分

SIAM 数值分析期刊》第 62 卷第 1 期第 322-352 页，2024 年 2 月。摘要。我们介绍了一种基于 Hasimoto 变换的薛定谔图（SM）数值计算方法，该变换将薛定谔图流与立方非线性薛定谔（NLS）方程联系起来。利用这种非线性变换，我们能够为 SM 方程引入第一个完全明确的无条件稳定对称积分器。我们的方法由两部分组成：对 NLS 方程进行积分，然后对 Hasimoto 变换进行数值评估。出于研究 SM 方程粗糙解的愿望，我们还为 NLS 方程引入了一种新的对称低规则积分器。它与我们新颖的快速低规则性哈希莫托（FlowRH）变换相结合，基于对马格努斯展开中共振结构的定制分析和基于块-托普利兹分区的快速实现，产生了一种高效的 SM 方程低规则性积分器。与之前提出的方法相比，这一方案尤其能让我们在更一般的情况下（即在低规则性假设下）获得 SM 的近似值。我们的方法在理论收敛分析和数值实验中都表现出了良好的特性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.