# Ginzburg-Landau-Schrodinger equation

The **Ginzburg-Landau-Schrodinger** equation is

The main focus of study for this equation is the formation of vortices and their dynamics in the limit .

The Ginzburg-Landau theory is briefly surveyed on Wikipedia.

## Perturbative Approach

The limit can be treated with the same methods given in Perturbation theory. To see this we note that an exact solution can be written as

being a real constant. Then, if we rescale time as and take the solution series

one has the non trivial set of equations

.

where dot means derivation with respect to . The leading order solution is easily written down as

.

With this expression we can write down the next order correction as

.

This set is easy to solve. The most important point to notice is the limit surface that denotes a change into the stability of the solution of GL equation. It should also be pointed out the appearence at this order of secular terms going like and . These terms can be treated with several known techniques.