Analysis and applications of geometric Schrodinger equations: topological solitons and dynamics in ferromagnets

几何薛定谔方程的分析和应用：拓扑孤子和铁磁体动力学

• 批准号: RGPIN-2018-03847
• 负责人: Gustafson, Stephen
• 金额: $ 2.62万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758518
• 关键词: Analysis; applications; geometric; Schrodinger; equations

Equilibrium configurations and dynamical behaviour in classical ferromagnets, within a continuum (micromagnetic) description, are governed by the Landau-Lifshitz equations. This system of nonlinear partial differential equations exhibits both Schrödinger (dispersive wave)-like and heat (diffusion)-like behaviour, and boasts remarkable geometric structure: it naturally generalizes the linear heat and Schrödinger equations to maps taking values in in the 2-sphere.The objective of this proposal is to obtain analytical (and numerical) information about behaviour of solutions. In the applied direction, the goal is to study physically relevant settings such as 2D thin-films, including Dzyaloshinskii-Moriya interactions (chiral ferromagnets), seeking (a) results on existence and properties of ``topological soliton” configurations such as skyrmions, skyrmion lattices, and vortices, which have been predicted in the physics literature and experimentally observed; (b) the stability of these configurations in the energetic and dynamical senses; and (c) qualitative properties of more general time-dependent solutions, such as collapse. In theoretical terms, the goal is to explain the effects of properties of a general target manifold, such as curvature, on the qualitative properties of the dynamics. To prove existence and properties of static configurations (energy critical points), classical tools of the calculus of variations, such as concentration-compactness, are useful. Another approach is perturbation theory, based on the isotropic case, a delicate, non-standard challenge due to the scaling invariance. Symmetry reduction, spectral theory, and perturbation theory can be used to assess the stability of equilibria. The study of time-dependent solutions requires geometric transformations, tools from (Hamiltonian) dynamical systems theory, as well as many analytical tools developed recently for problems of stability, asymptotic behaviour, and singularity formation in various nonlinear dispersive equations. Topological magnetic solitons (e.g., chiral skyrmions) have attracted intense attention in the physics literature, have been observed experimentally, and may have significant technological applications (e.g., magnetic data storage). The proposal aims to complement these various physical/numerical and experimental observations with rigorous (and numerical) mathematical results on the key properties of these objects. Though there has been spectacular recent progress on the mathematical analysis of certain special cases particularly the isotropic Schrödinger and heat-flows mathematical theory and results for the more physical models proposed here are still sorely lacking. There is a major opportunity for rigorous analysis to play a crucial role in exploring all the implications of these exciting recent developments. It should not be missed.

在连续介质（微磁）描述中，经典铁磁体的平衡配置和动态行为由朗道-利夫什茨方程控制。该非线性偏微分方程组表现出类薛定谔（色散波）和类热（扩散）行为。 ，并拥有非凡的几何结构：它自然地将线性热方程和薛定谔方程推广到取值的映射2-领域。该提案的目的是获得有关解决方案行为的分析（和数值）信息。在应用方向，目标是研究物理相关设置，例如 2D 薄膜，包括 Dzyaloshinskii-Moriya 相互作用（手性）。铁磁体），寻求（a）关于“拓扑孤子”构型（例如斯格明子、斯格明子晶格和涡旋）的存在和性质的结果，这些在物理学中已被预测文献和实验观察到的；（b）这些构型在能量和动力学意义上的稳定性；以及（c）更一般的与时间相关的解决方案的定性特性，例如塌陷。为了证明静态配置（能量临界点）的存在和属性，变分计算的经典工具（例如浓度紧性）是有用的。另一种方法是微扰理论，基于各向同性的情况，由于尺度不变性而产生的微妙的非标准挑战，谱理论和微扰理论可用于评估平衡解的稳定性。时间相关解的研究需要几何。变换、（哈密尔顿）动力系统理论的工具，以及最近为各种非线性色散方程中的稳定性、渐近行为和奇点形成问题开发的许多分析工具。 （例如，手性斯格明子）在物理文献中引起了强烈关注，已经通过实验观察到，并且可能具有重要的技术应用（例如，磁数据存储）。该提案旨在以严格的（尽管最近在某些特殊情况的数学分析方面取得了惊人的进展，特别是各向同性薛定谔和热流数学理论，但这里提出的更多物理模型的结果仍然存在。严格的分析在探索这些令人兴奋的近期发展的所有影响方面发挥着至关重要的作用，这是一个不容错过的机会。