Ginzburg-Landau-Schrodinger equation

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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.