Dynamics of the Ginzburg-Landau equations of superconductivity

Abstract

Abstract. This article is concerned with the dynamical properties of solutions of the time-dependent Ginzburg-Landau (TDGL) equations of superconductivity. It is shown that the TDGL equations define a dynamical process when the applied magnetic field varies with time, and a dynamical system when the applied magnetic field is stationary. The dynamical system describes the large-time asymptotic behavior: Every solution of the TDGL equations is attracted to a set of stationary solutions, which are .divergence free. These results are obtained in the ''4 = -w(V A)" gauge, which reduces to the standard %qj = -V -A" gauge if w = 1 and to the zero-electric potential gauge if w = 0; the treatment captures both in a unified framework. This gauge forces the London gauge, V -A = 0, for any stationary solution of the TDGL equations.

Discrete gauge invariant approximations of a time dependent Ginzburg-Landau model of superconductivity

“…Questions related to the well-posedness of the above equations and the fixing of gauges have been studied in [4], [11], [13], [24] and the references cited therein. The gauge invariance has been a very desirable property for physicists who have proposed and who have been using these models.…”

“…Using similar ideas as in [5] for the continuous case (see also [13] for related discussion), the construction of the gauge transformation T h f is given by (3.19) and f 0 = 0. The case λ = 1 is often referred to as the Lorentz gauge.…”

“…At t n = n∆t, let the components of ϕ n be defined by ϕ n j := ψ(x j , t n ) (4.21a) and the components of A n be defined by a n jk := where c is a positive constant depending only on the appropriate norms of the exact solution (ψ, A). One may consult [5], [13], [24] for the regularity estimates for the weak and strong solutions of the time dependent GL equations in various gauges.…”

“…The well-posedness of the initial boundary value problems for such a model, the long time asymptotics of solutions and the gauge invariance properties have all been studied in, for example, [5], [11], [13], [24] and the references cited therein. The finite element approximations have also been discussed by various authors [2], [6], [7], [8].…”

Time-Dependent Ginzburg–Landau Simulations of the Critical Current in Superconducting Films and Junctions in Magnetic Fields

“…Eqs. (1) - (4) along with Maxwell's equations define the state of the system up to a gauge transformation [6].…”