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: $ 1.31万
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=676836
• 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.

经典铁磁体的平衡配置和动力学行为，在连续（微磁）的描述，由朗道-Lifshitz方程。这个非线性偏微分方程组表现出薛定谔（色散波）和热（扩散）的行为，并拥有显着的几何结构：它自然地将线性热和薛定谔方程推广到在2-球中取值的映射。本提案的目的是获得有关解的行为的分析（和数值）信息。在应用方向，目标是研究物理相关的设置，如二维薄膜，包括Dzyaloshinskiii-Moriya相互作用（手征铁磁体），寻求（a）关于"拓扑孤立子”结构（如skyrmion、skyrmion晶格和涡旋）的存在和性质的结果，这些结构已在物理学文献中预测并在实验中观察到;（B）这些组态在能量和动力学意义上的稳定性;（c）更一般的含时解的定性性质，如坍缩。从理论上讲，目标是解释一般目标流形的性质（如曲率）对动力学定性性质的影响。** 为了证明静态组态的存在性和性质（能量临界点），经典的变分法工具，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下，一个微妙的，非标准的挑战，由于标度不变性。对称性约化、谱理论和微扰理论可以用来评估平衡点的稳定性。时间相关的解决方案的研究需要几何变换，工具（哈密顿）动力系统理论，以及许多分析工具最近开发的问题的稳定性，渐近行为，奇异性形成在各种非线性色散方程。****** 拓扑磁孤子（例如，手性Skyrmions）在物理学文献中引起了强烈的关注，已经在实验上观察到，并且可能具有重要的技术应用（例如，磁数据存储）。该提案的目的是补充这些各种物理/数值和实验观测与严格的（和数值）数学结果的关键属性，这些对象。虽然最近在某些特殊情况的数学分析上取得了惊人的进展，特别是各向同性薛定谔和热流的数学理论和结果，这里提出的更多的物理模型仍然非常缺乏。有一个重大的机会，严格的分析发挥关键作用，探索所有的影响，这些令人兴奋的最新发展。它不应该被错过。