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

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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.