Vortex motion law for the Schrödinger-Ginzburg-Landau equations

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In the Ginzburg-Landau model for superconductivity a large Ginzburg-Landau parameter k corresponds to the formation of tight, stable vortices. These vortices are located where an applied magnetic field pierces the superconducting bulk, and each vortex induces a quantized supercurrent about the vortex. The energy of large-k solutions blows up near each vortex, which brings about difficulties in analysis. Rigorous asymptotic static theory has previously established the existence of a finite number of the vortices, and these vortices are located precisely at the critical points of a renormalized energy. We consider the motion of such vortices in a dynamic model for superconductivity that couples a U(1) gauge-invariant Schrödinger-type Ginzburg-Landau equation to a Maxwell-type equation under the limit of large Ginzburg-Landau parameter k. It is shown that under an almost-energy-minimizing condition each vortex moves in the direction of the net supercurrent located at the vortex position, and these vortices behave like point vortices in the classical two-dimensional Euler equations.

Original languageEnglish (US)
Pages (from-to)1435-1476
Number of pages42
JournalSIAM Journal on Mathematical Analysis
Issue number6
StatePublished - 2003


  • Ginzburg-Landau theory
  • Superconductivity
  • Vortex dynamics


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