Ginzburg-Landau-Schrodinger equation

The Ginzburg-Landau-Schrodinger equation is

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

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

$u^{\epsilon }={\sqrt {n_{0}}}e^{-i(n_{0}-1){\frac {t}{\epsilon ^{2}}}}$

$n_{0}$ being a real constant. Then, if we rescale time as $\tau =t/\epsilon ^{2}$ 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 $\tau$. 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 $n_{0}(x)=1/2$ 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 $\tau$ and $\tau ^{2}$. These terms can be treated with several known techniques.