Here we study the asymptotic behavior of solutions to the complex Ginzburg-Landau equations and their associated heat flows in arbitrary dimensions when the Ginzburg-Landau parameter 1/ε tends to infinity. We prove that the energies of solutions in the flow are concentrated at vortices in two dimensions, filaments in three dimensions, and codimension-2 submanifolds in higher dimensions. Moreover, we show the dynamical laws for the motion of these vortices, filaments, and codimension-2 submanifolds. Away from the energy concentration sets, we use some measure-theoretic arguments to show the strong convergence of solutions in both static and heat flow cases.
|Original language||English (US)|
|Number of pages||57|
|Journal||Communications on Pure and Applied Mathematics|
|State||Published - Apr 1998|
ASJC Scopus subject areas
- Applied Mathematics