TY - JOUR

T1 - Vortex motion law for the Schrödinger-Ginzburg-Landau equations

AU - Spirn, Daniel

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

KW - Ginzburg-Landau theory

KW - Superconductivity

KW - Vortex dynamics

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U2 - 10.1137/S0036141001396667

DO - 10.1137/S0036141001396667

M3 - Article

AN - SCOPUS:0242511184

VL - 34

SP - 1435

EP - 1476

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 6

ER -