The Ginzburg - Landau equation is a nonlinear partial differential equation which was proposed around 1950 in the modelling of superconductors. Since then, it has become an extremely popular tool in many other areas of physics where vortices and/or topological defects appear, e.g. superfluids; new problems in physics with a similar flavour arise constantly e.g. ferromagnetism, Bose-Einstein condensates etc.... Starting around 1990, there has been remarkable progress in the mathematical understanding of such equations. This involves a combined use of techniques from many areas of mathematics:
- Nonlinear Partial Differential Equations.
- Geometric measure theory.
- Sobolev maps with values into manifolds.
- Harmonic maps in Differential Geometry.
- Renormalized Energies. Concentration effects. Vortices.
- Geometry and Topology of singularities.
- Questions of stability and symmetry for the Ginzburg-Landau system.
This subject is a very active current area of research, in particular in France and in China.