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

在一个连续体（微磁）描述中，经典铁磁体的平衡构型和动力学行为是由Landau-Lifshitz方程控制的。这个非线性偏微分方程系统显示了类似Schrödinger（色散波）和类似热量（扩散）的行为，并具有显著的几何结构：它自然地将线性热量和Schrödinger方程推广到在2球中取值的映射。******本建议的目的是获得关于解的行为的解析（和数值）信息。在应用方向上，目标是研究物理相关设置，如二维薄膜，包括Dzyaloshinskii-Moriya相互作用（手性铁磁体），寻求(a)在物理文献中预测和实验观察到的“拓扑孤子”构型（如skyrmions， skyrmions晶格和漩涡）的存在和性质的结果；(b)这些构型在能量和动力意义上的稳定性；(c)更一般的时间相关解的定性性质，如坍缩。在理论方面，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）的存在性和性质，变分学的经典工具，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。对称约简、谱理论和微扰理论可以用来评估平衡的稳定性。时间相关解的研究需要几何变换、（哈密顿）动力系统理论中的工具，以及最近为各种非线性色散方程的稳定性、渐近行为和奇点形成问题开发的许多分析工具。******拓扑磁孤子（例如，手性孤子）在物理文献中引起了强烈的关注，已经在实验中观察到，并且可能具有重要的技术应用（例如，磁数据存储）。该提案旨在补充这些不同的物理/数值和实验观测与严格的（和数值）数学结果对这些对象的关键属性。虽然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Schrödinger和热流，但这里提出的更物理的模型的数学理论和结果仍然非常缺乏。在探索这些令人兴奋的最新发展的所有含义方面，严谨的分析是一个重要的机会，可以发挥关键作用。它不应该被错过。

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.