Stephen James Gustafson
Relevant Degree Programs
Mathematical analysis of fundamental properties (e.g. stability and dynamical behaviour) of topological solitons in physical systems such as ferromagnets.
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Graduate Student Supervision
Doctoral Student Supervision (Jan 2008 - April 2022)
This main focus of this thesis is on the existence and stability of topological solitons called chiral magenetic skyrmions, arising in planar ferromagnets including combinations of Zeeman, anisotropy, and chiral (Dzyoloshinskii-Moriya) interactions.The first part deals with the existence of skyrmions with co-rotational symmetry under small chiral interaction, which is a variational problem. By treating it as a perturbation problem, our results make precise the sense in which skyrmions are perturbations of bubbles (degree-one harmonic maps). The perturbation approach relies on delicate 2D resolvent expansion for a linearized operator with a zero-energy resonance. We show the uniqueness and (in a special case) the monotonicity of the skyrmion solution profile. By applying our precise bounds of the differences between skyrmions and harmonic maps, we obtain precise asymptotics of the skyrmion energy.In the second part, as an application of the first part, we study the spectrum of the second variation of the energy about the skyrmion, by perturbation theory in the presence of a threshold resonance. We first prove the linear (spectral) stability of chiral magnetic skyrmions against arbitrary perturbations; thus the skyrmion solution from the first part is a strict local minimizer (modulo translations) of the energy. We also prove the orbital stability of the skyrmions for the Landau-Lifshitz equation with dissipation, which is an alternate quantitative proof of the recent skyrmion stability result of Li-Melcher in Journal of Functional Analysis 2018 . These results are consistent with observations in the physics literature.
The present thesis is split in two parts. The first deals with the focusing Nonlinear Schrödinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish thedegenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations.The second considers the perturbed energy critical focusing NonlinearSchrödinger Equation in three dimensions. We construct solitary wave solutions for focusing subcritical perturbations as well as defocusing supercritical perturbations. The construction relies on the resolvent expansion, which is singular due to the presence of a resonance. Specializing to pure power focusing subcritical perturbations we demonstrate, via variational arguments, and for a certain range of powers, the existence of a ground state soliton, which is then shown to be the previously constructed solution. Finally, we present a dynamical theorem which characterizes the fate of radially-symmetric solutions whose initial data are below the action of the ground state. Such solutions will either scatter or blow-up in finite time depending on the sign of a certain function of their initial data.
The main focus of this thesis is on critical parabolic problems, in particular, theharmonic map heat from the plane to S2, and nonlinear focusing heat equationswith an algebraic nonlinearity. The focus of this work has been on long-time dynamics, stability and singularityformation, and the investigation of the role of special, soliton-like, solutions to theasymptotic behaviour of solutions. Harmonic Map Heat Flow: Flow: we consider m-corotational solutions to the harmonic map heat flow from R2 to S2. We first work in a class of maps with trivial topology and energy of the initial data below two times the energy of the stationary harmonic map solutions. We give a new proof of global existence and decay. The proof is based on the "concentration-compactness plus rigidity" approach of Kenig and Merle and relies on the dissipation of the energy and a profile decomposition. We also treat m-corotational maps (m greater than 3) with non-trivial topology and energy of the initial data less than three times the energyof the stationary harmonic map solutions. Through a new stability argument we rule out finite-time blow-up and show that the global solution asymptoticallyconverges to a harmonic map. Nonlinear Heat Equation: we also study solutions of the focusing energy-criticalnonlinear heat equation. We show that solutions emanating from initial data with energy and kinetic energy below those of the stationary solutionsare global and decay to zero. To prove that global solutions dissipate to zerowe rely on a refined small data theory, L2-dissipation and an approximation argument. We then follow the "concentration-compactness plus rigidity" roadmap of Kenig and Merle (and in particular the approach taken by Kenig and Koch for Navier-Stokes) to exclude finite-time blow-up.
The Schrödinger equation, an equation central to quantum mechanics, is a dispersive equation which means, very roughly speaking, that its solutions have a wave-like nature, and spread out over time. We will consider global behaviour of solutions of two nonlinear variations of the Schrödinger equation. In particular, we consider the nonlinear magnetic Schrödinger equation. [Formulas omitted] We show that under suitable assumptions on the electric and magnetic potentials, if the initial data is small enough in H¹, then the solution of the above equation decomposes uniquely into a standing wave part, which converges as t → ∞, and a dispersive part, which scatters. We also consider the Schrödinger map equation into the 2-sphere. We obtain a global well-posedness result for this equation with radially symmetric initial data without any size restriction on the initial data. Our technique involves translating the Schrödinger map equation into a cubic, non-local Schrödinger equation via the generalized Hasimoto transform. There, we also show global well-posedness for the non-local Schrödinger equation with radially-symmetric initial data in the critical space L²(ℝ²), using the framework of Kenig-Merle and Killip-Tao-Visan.
No abstract available.
- Co-rotational chiral magnetic skyrmions near harmonic maps (2021)
Journal of Functional Analysis, 280 (4), 108867