Pinning of magnetic vortices by an external potential

Authors:
I. M. Sigal and F. Ting

Original publication:
Algebra i Analiz, tom **16** (2004), nomer 1.

Journal:
St. Petersburg Math. J. **16** (2005), 211-236

MSC (2000):
Primary 58E50; Secondary 35B20, 82D55, 35Q55, 35Q60

Published electronically:
December 17, 2004

MathSciNet review:
2069485

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Abstract | References | Similar Articles | Additional Information

Abstract: The existence and uniqueness of vortex solutions is proved for Ginzburg-Landau equations with external potentials in . These equations describe the equilibrium states of superconductors and the stationary states of the -Higgs model of particle physics. In the former case, the external potentials are due to impurities and defects. Without the external potentials, the equations are translationally (as well as gauge) invariant, and they have gauge equivalent families of vortex (equivariant) solutions called magnetic or Abrikosov vortices, centered at arbitrary points of . For smooth and sufficiently small external potentials, it is shown that for each critical point of the potential there exists a perturbed vortex solution centered near , and that there are no other single vortex solutions. This result confirms the ``pinning'' phenomena observed and described in physics, whereby magnetic vortices are pinned down to impurities or defects in the superconductor.

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Additional Information

**I. M. Sigal**

Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada

**F. Ting**

Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618

Address at time of publication:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON P7B 5E1, Canada

Email:
fridolin.ting@lakeheadu.ca

DOI:
http://dx.doi.org/10.1090/S1061-0022-04-00848-9

Keywords:
Superconductivity,
Ginzburg--Landau equations,
pinning,
magnetic vortices,
external potential,
existence

Received by editor(s):
November 20, 2003

Published electronically:
December 17, 2004

Additional Notes:
Supported by NSERC (grant N7901).

Dedicated:
Dedicated to M. Sh. Birman with admiration and friendship

Article copyright:
© Copyright 2004
American Mathematical Society