论文标题
Schrödinger图的数值整合通过Hasimoto变换
Numerical integration of Schrödinger maps via the Hasimoto transform
论文作者
论文摘要
我们引入了基于Hasimoto变换计算Schrödinger图（SM）的数值方法，该方法将SM流与立方非线性Schrödinger（NLS）方程相关联。在利用这种非线性变换时，我们能够为SM方程引入第一个完全明确的无条件稳定的对称积分器。我们的方法由两个部分组成：NLS方程的集成，然后对Hasimoto变换进行数值评估。由于希望研究SM方程的粗略解决方案的愿望，我们还为NLS方程式引入了一种新的对称低规范性积分器。这与我们新颖的快速低规范性Hasimoto（FlowRH）转换结合在一起，基于对Magnus膨胀中的共振结构的定制分析以及基于块状Toeplitz分区的快速实现，以产生有效的SMS方程的低规范性积分器。该方案特别使我们能够以比以前提出的方法更通用的制度（即在较低的规律性假设）中获得与SM的近似值。我们方法的有利特性在理论收敛分析和数值实验中都表现出来。
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 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.
