Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds

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Abstract

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 languageEnglish (US)
Pages (from-to)385-441
Number of pages57
JournalCommunications on Pure and Applied Mathematics
Volume51
Issue number4
StatePublished - Apr 1998

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Vortex Filament
Complex Ginzburg-Landau Equation
Heat Flow
Submanifolds
Codimension
Vortex flow
Heat transfer
Convergence of Solutions
Ginzburg-Landau
Filament
Asymptotic Behavior of Solutions
Energy
Strong Convergence
Higher Dimensions
Three-dimension
Vortex
Two Dimensions
Infinity
Tend
Motion

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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