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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

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: The existence and uniqueness of vortex solutions is proved for Ginzburg-Landau equations with external potentials in $\mathbb{R} ^2$. These equations describe the equilibrium states of superconductors and the stationary states of the $U(1)$-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 $\mathbb{R} ^2$. For smooth and sufficiently small external potentials, it is shown that for each critical point $z_0$ of the potential there exists a perturbed vortex solution centered near $z_0$, 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
PII: S 1061-0022(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