Two problems in the theory of Ginzburg-Landau vortices

Specialeforsvar: Jingxuan Zhang

Titel: Two problems in the theory of Ginzburg-Landau vortices

Abstract: In this thesis, we make two contributions to the theory of Ginzburg-Landau vortices. Firstly, we construct non-trivial equivariant solutions to the (magnetic) Ginzburg-Landau equations on non-compact Riemann surfaces. These solutions are the non-Abelian generalizations of the Abrikosov vortex lattices. Our existence result does not require self-duality (i.e. Bogomolnyi regime). We obtain precise asymptotics for the solution, and show that the energy of our solution is strictly less than that
of the constant curvature (magnetic field) one. This is a joint work with N.M. Ercolani and I.M. Sigal. Secondly, we study the motion of superfluid vortex filaments in a three dimensional cylindrical domain. We show that at the leading order, the full dynamics of the vortex filaments under the dispersive Ginzburg-Landau equation (without magnetic field) can be reduced to an effective dynamics, namely the binormal curvature flow, of some one-dimensional concentration sets. This gives an affirmative answer to a long standing conjecture of R. L. Jerrard. This is an independent work of the Author. This thesis contains a survey of key concepts in the Ginzburg-Landau theory, and two papers dealing with the aforementioned problems about Ginzburg-Landau vortices respectively.

Vejleder: Niels Martin Møller
Censor: Søren Fournais, Aarhus Universitet