Vortex dynamics of the full time‐dependent Ginzburg‐Landau equations

In the Ginzburg-Landau model for superconductivity a large Ginzburg-Landau parameter κ corresponds to the formation of tight, stable vortices. These vortices are located exactly where an applied magnetic field pierces the superconducting bulk, and each vortex induces a quantized supercurrent about the vortex. The energy of large-κ 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 the renormalized energy (the free energy less the vortex self-induction energy). A rigorous study of the full time-dependent Ginzburg-Landau equations under the classical Lorentz gauge is done under the asymptotic limit κ ∞. Under slow times the vortices remain pinned to their initial configuration. Under a fast time of order κ the vortices move according to a steepest descent of the renormalized energy. © 2002 John Wiley & Sons, Inc.

[1]  Sylvia Serfaty,et al.  Global minimizers for the Ginzburg–Landau functional below the first critical magnetic field , 2000 .

[2]  F. Lin,et al.  Evolution of harmonic maps with Dirichlet boundary conditions , 1993 .

[3]  Jacob Rubinstein,et al.  Vortex dynamics in U(1) Ginzberg-Landau models , 1993 .

[4]  Sylvia Serfaty,et al.  Stable Configurations in Superconductivity: Uniqueness, Multiplicity, and Vortex‐Nucleation , 1999 .

[5]  Petru Mironescu,et al.  The behavior of a Ginzburg-Landau minimizer near its zeroes , 1996 .

[6]  Sylvia Serfaty,et al.  LOCAL MINIMIZERS FOR THE GINZBURG–LANDAU ENERGY NEAR CRITICAL MAGNETIC FIELD: PART II , 1999 .

[7]  Halil Mete Soner,et al.  Dynamics of Ginzburg‐Landau Vortices , 1998 .

[8]  Qiang Du,et al.  Global existence and uniqueness of solutions of the time-dependent ginzburg-landau model for superconductivity , 1994 .

[9]  F. Lin.  Static and moving vortices in Ginzburg-Landau theories , 1997 .

[10]  Mariano Giaquinta,et al.  Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105), Volume 105 , 1984 .

[11]  K. Kitazawa,et al.  Motion of vortices in superconductors , 1999, Nature.

[12]  E. Abrahams,et al.  Time variation of the Ginzburg--Landau order parameter , 1966 .

[13]  Paul A. Pearce,et al.  Yang-Baxter equations, conformal invariance and integrability in statistical mechanics and field theory : proceedings of a conference : Centre for Mathematical Analysis, Australian National University, Canberra, Australia, July 10-14, 1989 , 1990 .

[14]  E Weinan,et al.  Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity , 1994 .

[15]  K. Maki,et al.  FLUCTUATIONS OF THE ORDER PARAMETER IN TYPE-II SUPERCONDUCTORS. , 1967 .

[16]  Qiang Du,et al.  Ginzburg-Landau vortices: dynamics, pinning, and hysteresis , 1997 .

[17]  L. Landau,et al.  On the theory of superconductivity , 1955 .

[18]  Hans G. Kaper,et al.  Dynamics of the Ginzburg-Landau equations of superconductivity , 1998 .

[19]  F. Lin.  Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds , 1998 .

[20]  F. Béthuel,et al.  Vortices for a variational problem related to superconductivity , 1995 .

[21]  John C. Neu,et al.  Vortices in complex scalar fields , 1990 .

[22]  Fanghua Lin,et al.  Some Dynamical Properties of Ginzburg-Landau Vortices , 1996 .

[23]  M. J. Stephen.  Transport Equations for Superconductors , 1965 .

[24]  Bascom S. Deaver,et al.  Experimental Evidence for Quantized Flux in Superconducting Cylinders , 1961 .

[25]  Shouhong Wang,et al.  Time dependent Ginzburg-Landau equations of superconductivity , 1995 .

[26]  Ginzburg-Landau dynamics with a time-dependent magnetic field , 1998 .

[27]  P. D. Gennes,et al.  Superconductivity of metals and alloys , 1966 .

[28]  Machida,et al.  Direct simulation of the time-dependent Ginzburg-Landau equation for type-II superconducting thin film: Vortex dynamics and V-I characteristics. , 1993, Physical review letters.

[29]  M. Nabauer,et al.  Experimental Proof of Magnetic Flux Quantization in a Superconducting Ring , 1961 .

[30]  L. Cooper,et al.  Microscopic theory of superconductivity , 1957 .

[31]  Qiang Du,et al.  High-Kappa Limits of the Time-Dependent Ginzburg-Landau Model , 1996, SIAM J. Appl. Math..

[32]  Michael Struwe,et al.  On the asymptotic behavior of minimizers of the Ginzburg-Landau model in $2$ dimensions , 1994, Differential and Integral Equations.