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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=652727
• 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)描述下，经典铁磁体的平衡构型和动力学行为受Landau-Lifshitz方程支配。这个非线性偏微分方程组表现出类薛定谔(色散波)和类热(扩散)的行为，并具有显著的几何结构：它自然地将线性热方程和薛定谔方程推广到在2-球面上取值的映射。*本建议的目的是获得关于解的行为的分析(和数值)信息。在应用方面，目标是研究与物理相关的环境，如2D薄膜，包括Dzyaloshinskii-Moriya相互作用(手性铁磁体)，寻求(A)在物理文献中预测并通过实验观察到的“拓扑孤子”组态的存在和性质的结果；(B)这些组态在能量和动力学意义上的稳定性；以及(C)更一般依赖于时间的解的定性性质，例如崩塌。在理论上，我们的目标是解释一般目标流形的性质，如曲率，对动力学的定性性质的影响。为了证明静态构型(能量临界点)的存在和性质，经典的变分工具，如集中紧性，是有用的。另一种方法是基于各向同性情况的微扰理论，由于尺度不变性，这是一个微妙的、非标准的挑战。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。依赖于时间的解的研究需要几何变换、(哈密顿)动力系统理论的工具，以及最近发展起来的许多分析工具，用于各种非线性色散方程的稳定性、渐近性和奇性形成问题。*拓扑磁孤子(例如手性Skyrmions)在物理文献中引起了极大的关注，已经在实验上被观测到，并可能有重要的技术应用(例如磁数据存储)。该提议旨在用关于这些天体关键性质的严格(和数值)数学结果来补充这些不同的物理/数值和实验观测结果。尽管最近在某些特殊情况的数学分析--特别是各向同性薛定谔和热流--方面取得了惊人的进展，但这里提出的更多物理模型的数学理论和结果仍然非常缺乏。在探索这些令人振奋的最新事态发展的所有影响方面，严格分析发挥关键作用是一个重大机会。这是不应该错过的。