Ginzburg-Landau equations and their generalizations

TL;DR

This paper reviews the Ginzburg-Landau equations, their extensions to complex geometries, and related scattering theories, highlighting their significance in superconductivity and mathematical physics with open problems.

Contribution

It surveys recent generalizations of Ginzburg-Landau equations to Riemann surfaces and 4-manifolds, and discusses vortex scattering theory and unresolved issues.

Findings

Extensions to Riemann surfaces and 4-manifolds

Analysis of vortex scattering and hyperbolic equations

Identification of open problems in the field

Abstract

The Ginzburg-Landau equations were proposed in the superconductivity theory to describe mathematically the intermediate state of superconductors in which the normal conductivity is mixed with the superconductivity. It was understood later on that these equations play an important role also in various problems of mathematical physics. We mention here the extension of these equations to compact Riemann surfaces and Riemannian 4-manifolds. A separate interesting topic is the scattering theory of vortices reducing to the study of hyperbolic Ginzburg-Landau equations. In this review we tried to touch these interesting topics with some unsolved problems.