Qualitative Aspects of Ginzburg-Landau Equations

  • Frank Pacard
  • Tristan Rivière
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 39)

Abstract

The Ginzburg-Landau functional was introduced by V.L. Ginzburg and L.D. Landau in [27] as a model for superconductivity. If Ω is a domain of ℝ n which is diffeomorphic to the unit ball B 1 ⊃ ℝ n , the functional has the following form:
$$ {{G}_{K}}(u,A): = \int_{\Omega } {|\nabla u - iAu{{|}^{2}} + \frac{{{{\kappa }^{2}}}}{2}} \int_{\Omega } {{{{(1 - |u{{|}^{2}})}}^{2}}} + \int_{\Omega } {|dA - {{h}_{{ext}}}{{|}^{2}}} . $$
(1.1)
Here the condensate wave function u is defined from Ω into ℂ, and A is a 1-form defined in Ω which represents the potential associated to the induced magnetic field h = dA in the material. The quantity |u|2 is nothing but the density of cooper pairs of electrons that produce the superconductivity. Finally, h ext denotes the external magnetic field which is applied and then appears in the problem. The parameter κ > 0 is usually called the Ginzburg-Landau parameter.

Keywords

Vortex Manifold Vorticity Eter Dition 

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Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Frank Pacard
    • 1
  • Tristan Rivière
    • 2
    • 3
  1. 1.Département de MathématiquesUniversité de Paris XIICreteil CedexFrance
  2. 2.Department of MathematicsCourant Institute of Mathematical SciencesNew YorkUSA
  3. 3.CMLA, ENS-CACHANCentre National de la Recherche Scientifique 61CachanFrance

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