Some mathematical problems in the Ginzburg-Landau theory of superconductivity

In agreement with the Landau theory of phase transitions, a superconductor is described macroscopically by the Ginzburg-Landau equations (1950). These are nonlinear partial differential equations for a complex-valued function, $y$ (the order parameter), and a vector-field A (the vector potential). The equations contain a parameter, λ, which determines whether they describe a superconductor of the first kind (λ < 1), or of the second kind (λ > 1). It was observed by Abrikosov (1957) that a highly-symmetric family of solutions known as n-vortices plays a central role in the theory. These solutions are classified by their integer topological degree, $n\in Z$.; The principal goal of this thesis is to establish the stability properties of the n-vortex. The stability question is studied for three types of evolution equations: a gradient flow, a nonlinear wave equation, and a nonlinear Schrödinger equation. Our main result determines the dependence of the stability of n-vortices on the topological degree, n, and on the parameter, λ. Specifically, we prove that for λ < 1, all vortices are stable, while for λ > 1, n-vortices are stable if n = ±1 and unstable if |n| ≥ 2. Previous work on vortex stability (Taubes (1980), Stuart (1994)) has focused on the special case λ = 1, in which the Ginzburg-Landau equations reduce to the first-order Bogomolnyi equations. In particular, our result resolves a long-standing conjecture, first rigorously formulated by Jaffe and Taubes (1980).