We want to describe the dynamics of magnetic vortices in type-II superconductors using the Gor’kov-Eliashberg equations. To solve this system is very difficult so we want to use an approximation, called Slow-Motion Approximation. This approximation is used quite a lot in physics and for our system of nonlinear partial differential equations we want to show, using rigorous mathematical arguments, that it is in fact an approximation to the exact solution. For the Abelian-Higgs model which shares the same time independent equations with the Gor’kov- Eliashberg system, such a mathematically rigorous proof was given by Stuart (1994).
The mathematical discussion starts with an ansatz for the solution that involves the exact solution of the static problem and a small correction. It is well-known that the static solution is a 2N real-parameter family. Let us denote the parameters by q. In the Slow-Motion Approximation we assume that the parameters are time dependent. In our case we want to find the trajectory in the space of static solutions which is the closest, in some sense, to the exact solution.
As in many approximation techniques we need a small parameter such that the approximation gets better and better the smaller the parameter becomes. The small parameter, denoted by e, is given by the Higgs self-coupling constant k2 = (1 + e ) /2. Guided by Stuart’s proof we assume that the time derivative of the parameters q is 0 (e).
So the problem of proving the validity of the approximation is now turned into proving the existence and smallness of the corrections, which are the solutions of a parabolic linear partial differential equations system on M2. In order to prove this we try to imitate the techniques for finding solutions of the same class of equations in a bounded domain. We need also an iterative method that provide us with certain estimates in suitable Sobolev spaces. We get a system of equations for the parameters q(t) that is a Cauchy problem as soon as we fix initial conditions for q. Imposing initial conditions as well on the corrections of the static solution we simplify the equations for q and solve them.
Substituting these q(tys into the static solution we obtain a good approximation for the exact solution of Gor’kov-Eliashberg equations.