Ginzburg-Landau-Schrodinger equation

The Ginzburg-Landau-Schrodinger equation is

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

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

The Ginzburg-Landau theory is briefly surveyed on Wikipedia.

Perturbative Approach

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

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

${\displaystyle n_{0}}$ being a real constant. Then, if we rescale time as ${\displaystyle \tau =t/\epsilon ^{2}}$ and take the solution series

${\displaystyle u^{\epsilon }=u_{0}+\epsilon ^{2}u_{1}+\epsilon ^{4}u_{2}+\ldots }$

one has the non trivial set of equations

${\displaystyle i{\dot {u}}_{0}=u_{0}(|u_{0}|^{2}-1)}$

${\displaystyle i{\dot {u}}_{1}+\Delta u_{0}=u_{1}(|u_{0}|^{2}-1)+(u_{1}^{*}u_{0}+u_{0}^{*}u_{1})u_{0}}$

${\displaystyle i{\dot {u}}_{2}+\Delta u_{1}=u_{2}(|u_{0}|^{2}-1)+(u_{1}^{*}u_{0}+u_{0}^{*}u_{1})u_{1}+(|u_{1}|^{2}+u_{2}^{*}u_{0}+u_{0}^{*}u_{2})u_{0}}$

${\displaystyle \ldots }$.

where dot means derivation with respect to ${\displaystyle \tau }$. The leading order solution is easily written down as

${\displaystyle u_{0}={\sqrt {n_{0}(x)}}e^{-i[n_{0}(x)-1]\tau }}$.

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

${\displaystyle u_{1}=\phi (x,\tau )e^{-i[2n_{0}(x)-1]\tau }}$

${\displaystyle i{\dot {\phi }}=n_{0}(x)\phi ^{*}e^{-i[2n_{0}(x)-1]\tau }-(\Delta u_{0})e^{-i[2n_{0}(x)-1]\tau }}$

${\displaystyle -i{\dot {\phi }}^{*}=n_{0}(x)\phi e^{i[2n_{0}(x)-1]\tau }-(\Delta u_{0}^{*})e^{i[2n_{0}(x)-1]\tau }}$.

This set is easy to solve. The most important point to notice is the limit surface ${\displaystyle n_{0}(x)=1/2}$ 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 ${\displaystyle \tau }$ and ${\displaystyle \tau ^{2}}$. These terms can be treated with several known techniques.