Chapter 2
Magnetic Vortices, Abrikosov Lattices, and Automorphic Functions
Israel Michael Sigal,
Israel Michael Sigal
Department of Mathematics, University of Toronto, Ontario, Canada
Search for more papers by this authorIsrael Michael Sigal,
Israel Michael Sigal
Department of Mathematics, University of Toronto, Ontario, Canada
Search for more papers by this authorBook Editor(s):Roderick Melnik,
Roderick Melnik
Wilfrid Laurier University, Waterloo, Ontario, Canada
Search for more papers by this authorSummary
This chapter presents some recent results on the Ginzburg-Landau equations of superconductivity, and reviews appropriate background. The Ginzburg–Landau equations describe the key mesoscopic and macroscopic properties of superconductors and form the basis of the phenomenological theory of superconductivity. One of the most interesting mathematical and physical phenomena connected with Ginzburg–Landau equations is the presence of vortices in their solutions. Vortex represents a localized defect where the normal state intrudes and magnetic flux penetrates. For vortex lattices the energy is infinite, but the flux quantization still holds for each lattice cell because of gauge-periodic boundary conditions. The chapter describes the existence of Abrikosov solutions at low magnetic fields near the first critical magnetic field. Configurations containing several vortices are not static solutions. Heuristically, this is due to an effective inter-vortex interaction, which causes the vortex centers to move.
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