Variational and other methods for nonlinear PDE

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This proposal aims to study structures known as topological defects, found in a range of physical phenomena on both very small scales (superconductors, superfluids, micromagnetic materials) and very large scales (models for hypothetical objects known as cosmic strings which, if they exist, may span galaxies). These topological defects share some properties with vortex filaments in ordinary fluids that one encounters in everyday existence, notably water and air. Examples of such vortex filaments include smoke rings, tornados, trailing vortices from airplane wings, and the vortices shed by a canoe paddle. The analogies between these classical vortex filaments and topological defects are strongest for the particular defects (known as quantized vortex filaments) found in quantum mechanical fluids.The proposal addresses three classes of problems:1. the derivation of effective laws that govern the behaviour of topological defects. Like a vortex filament in a fluid, a topological defect is nothing more than a particular type of localized disturbance that propagates within a dynamic medium. To capture its behaviour, one starts from equations that describe the evolution of the ambient medium, and the challenge is to give a rigorous mathematical description of the motion of this particular type of disturbance within the medium.For example, an equation called the binormal curvature flow (BCF) is widely believed to govern quantized vortex filaments in ideal superfluids. Providing a rigorous proof of this belief is a long-standing open problem that drives much of our research in this area. 2. the extension of ideas developed in the study of topological defects to distinct but related problems. The most important and promising such problems involve vortex filaments in classical fluids such as (idealized) air or water. For example, a phenomenon known as vortex leapfrogging has been predicted in classical fluids since foundational work of Helmholtz in the 1850s. This has been studied in great detail by physicists and applied mathematicians, and it can even be seen in videos on youtube. But a mathematical understanding has been elusive. Leapfrogging was first proved to occur only in recent work of myself and my collaborator D Smets, and only for quantized vortices in superfluids. The proposed research will investigate whether techniques that we developed can be applied to study leapfrogging in ideal classical fluids.3. the study of effective laws that govern the behaviour of topological defects. For example, the above-mentioned BCF has remarkable but poorly-understood propertes that we will investigate.In addressing these problems, we will develop new mathematical techniques to provide a bridge between certain analytic and geometric questions.