Ginzburg-Landau-Schrodinger equation

The Ginzburg-Landau-Schrodinger equation is

$i u^\epsilon_t + \Delta u^\epsilon = \frac{1}{\epsilon^2} (|u^\epsilon|^2 - 1)u^\epsilon.$

The main focus of study for this equation is the formation of vortices and their dynamics in the limit $\epsilon \to 0$.

The Ginzburg-Landau theory is briefly surveyed on Wikipedia.

Perturbative Approach

The limit $\epsilon\to 0$ can be treated with the same methods given in Perturbation theory. To see this we note that an exact solution can be written as

$u^\epsilon = \sqrt{n_0}e^{-i(n_0-1)\frac{t}{\epsilon^2}}$

n0 being a real constant. Then, if we rescale time as τ = t / ε2 and take the solution series

$u^\epsilon = u_0+\epsilon^2 u_1+\epsilon^4 u_2+\ldots$

one has the non trivial set of equations

$i\dot u_0=u_0(|u_0|^2-1)$

$i\dot u_1+\Delta u_0=u_1(|u_0|^2-1)+(u_1^*u_0+u_0^*u_1)u_0$

$i\dot u_2+\Delta u_1=u_2(|u_0|^2-1)+(u_1^*u_0+u_0^*u_1)u_1+(|u_1|^2+u_2^*u_0+u_0^*u_2)u_0$

$\ldots$.

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

$u_0 = \sqrt{n_0(x)}e^{-i[n_0(x)-1]\tau}$.

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

$u_1 = \phi(x,\tau)e^{-i[2n_0(x)-1]\tau}$

$i\dot\phi=n_0(x)\phi^*e^{-i[2n_0(x)-1]\tau}-(\Delta u_0)e^{-i[2n_0(x)-1]\tau}$

$-i\dot\phi^*=n_0(x)\phi e^{i[2n_0(x)-1]\tau}-(\Delta u_0^*)e^{i[2n_0(x)-1]\tau}$.

This set is easy to solve. The most important point to notice is the limit surface n0(x) = 1 / 2 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 τ and τ2. These terms can be treated with several known techniques.