Stephen James Gustafson
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Mathematical analysis of fundamental properties (e.g. stability and dynamical behaviour) of topological solitons in physical systems such as ferromagnets.
Graduate Student Supervision
Doctoral Student Supervision (Jan 2008 - May 2019)
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.
This thesis concerns the stationary solutions and their stability for some evolution equations from physics. For these equations, the basic questionsregarding the solutions concern existence, uniqueness, stability and singularity formation. In this thesis, we consider two different classes of equations: the Landau-Lifshitz equations, and nonlinear Dirac equations. There are two different definitions of stationary solutions. For the Landau-Lifshitz equation, the stationary solution is time-independent, while for the Dirac equation, the stationary solution, also called solitary wave solution or ground state solution, is a solution which propagates without changing its shape.The class of Landau-Lifshitz equations (including harmonic map heat flow and Schrödinger map equations) arises in the study of ferromagnets (andanti-ferromagnets), liquid crystals, and is also very natural from a geometric standpoint. Harmonic maps are the stationary solutions to these equations. My thesis concerns the problems of singularity formation vs. global regularity and long time asymptotics when the target space is a 2-sphere. We consider maps with some symmetry. I show that for m-equivariant maps with energy close to the harmonic map energy, the solutions to Landau-Lifshitz equations are global in time and converge to a specific family of harmonic maps for big m, while for m =1, a finite time blow up solution is constructed for harmonic map heat flow. A model equation for Schrödinger map equations is also studied in my thesis. Global existence and scattering for small solutions and local well-posedness for solutions with finite energy are proved. The existence of standing wave solutions for the nonlinear Dirac equationis studied in my thesis. I construct a branch of solutions which is a continuous curve by a perturbation method. It refines the existing results that infinitely many stationary solutions exist, but with uniqueness and continuity unknown. The ground state solutions of nonlinear Schrodinger equations yield solutions to nonlinear Dirac equations. We also show that this branchof solutions is unstable. This leads to a rigorous proof of the instability of the ground states, confirming non-rigorous results in the physical literature.