On the dynamical law of the Ginzburg‐Landau vortices on the plane

We study the Ginzburg‐Landau equation on the plane with initial data being the product of n well‐separated +1 vortices and spatially decaying perturbations. If the separation distances are O(ϵ⁻¹), ϵ ≪ l, we prove that the n vortices do not move on the time scale $O(\varepsilon^{‐2}\lambda_{\varepsilon}), \lambda_{\varepsilon} = o(\log {1\over \varepsilon})$; instead, they move on the time scale according to the law ẋ_j = − ∇_xj W, W = − Σ_l≠j log|x_l −x_j|, x_j = (ξ_j, η_j) ∈ ℝ², the location of the j^th vortex. The main ingredients of our proof consist of estimating the large space behavior of solutions, a monotonicity inequality for the energy density of solutions, and energy comparisons. Combining these, we overcome the infinite energy difficulty of the planar vortices to establish the dynamical law. © John & Wiley Sons, Inc.     

A quantization property for static Ginzburg‐Landau vortices

For any critical point of the complex Ginzburg‐Landau functional in dimension 3, we prove that, for large coupling constants, ; if the energy of this critical point on a ball of a given radius r is relatively small compared to r log , then the ball of half‐radius contains no vortex (the modulus of the solution is larger than ½). We then show how this property can be applied to describe limiting vortices as ε → 0. © 2001 John Wiley & Sons, Inc.     

Local minimizers with vortices to the Ginzburg‐Landau system in three dimensions

We construct local minimizers to the Ginzburg‐Landau energy in certain three‐dimensional domains based on the asymptotic connection between the energy and the total length of vortices using the theory of weak Jacobians. Whenever there exists a collection of locally minimal line segments spanning the domain, we can find local minimizers with arbitrarily assigned degrees with respect to each segment. © 2003 Wiley Periodicals, Inc.     

On the Ginzburg‐Landau critical field in three dimensions

We study the three‐dimensional Ginzburg‐Landau model of superconductivity. Several “natural” definitions of the (third) critical field, H, governing the transition from the superconducting state to the normal state, are considered. We analyze the relation between these fields and give conditions as to when they coincide. An interesting part of the analysis is the study of the monotonicity of the ground state energy of the Laplacian with constant magnetic field and with Neumann (magnetic) boundary condition in a domain Ω. It is proved that the ground state energy is a strictly increasing function of the field strength for sufficiently large fields. As a consequence of our analysis, we give an affirmative answer to a conjecture by Pan. © 2008 Wiley Periodicals, Inc.     

On a non-stationary Ginzburg–Landau superconductivity model

In this paper, we study a non-stationary superconductivity model derived from Ginzburg–Landau macroscopic theory. By using gauge invariance and studying a linear problem with curl boundary conditions, we obtain the existence of solutions. The solution is unique in the sense of gauge equivalence.     

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.     

On the Validity of the degenerate Ginzburg—Landau equation

The Ginzburg–Landau equation which describes nonlinear modulation of the amplitude of the basic pattern does not give a good approximation when the Landau constant (which describes the influence of the nonlinearity) is small. In this paper a derivation of the so-called degenerate (or generalized) Ginzburg–Landau (dGL)-equation is given. It turns out that one can understand the dGL-equation as an example of a normal form of a co-dimension two bifurcation for parabolic PDEs. The main body of the paper is devoted to the proof of the validity of the dGL as an equation whose solution approximate the solution of the original problem. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On an initial‐boundary value problem for the p‐Ginzburg–Landau system

This paper is concerned with the asymptotic behavior of the decreasing energy solution u_ε to a p‐Ginzburg–Landau system with the initial‐boundary data for p > 4/3. It is proved that the zeros of u_ε in the parabolic domain G × (0,T] are located near finite lines {a_i}×(0,T]. In particular, all the zeros converge to these lines when the parameter ε goes to zero. In addition, the author also considers the uniform energy estimation on a domain far away from the zeros. At last, the Hölder convergence of u_ε to a heat flow of p‐harmonic map on this domain is proved when p > 2. Copyright © 2014 John Wiley & Sons, Ltd.     

On the global existence and small dispersion limit for a class of complex Ginzburg–Landau equations

In this paper we consider a class of complex Ginzburg–Landau equations. We obtain sufficient conditions for the existence and uniqueness of global solutions for the initial‐value problem in d‐dimensional torus ??^d, and that solutions are initially approximated by solutions of the corresponding small dispersion limit equation for a period of time that goes to infinity as dispersive coefficient goes to zero. Copyright © 2008 John Wiley & Sons, Ltd.     

Gamma‐convergence of gradient flows with applications to Ginzburg‐Landau

We present a method to prove convergence of gradient flows of families of energies that Γ‐converge to a limiting energy. It provides lower‐bound criteria to obtain the convergence that correspond to a sort of C¹‐order Γ‐convergence of functionals. We then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the Ginzburg‐Landau energy in dimension 2, retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field. © 2004 Wiley Periodicals, Inc.     

Recurrent solutions of the linearly coupled complex cubic‐quintic Ginzburg‐Landau equations

In this paper, we will establish the bounded solutions, periodic solutions, quasiperiodic solutions, almost periodic solutions, and almost automorphic solutions for linearly coupled complex cubic‐quintic Ginzburg‐Landau equations, under suitable conditions. The main difficulty is the nonlinear terms in the equations that are not Lipschitz‐continuity, traditional methods cannot deal with the difficulty in our problem. We overcome this difficulty by the Galerkin approach, energy estimate method, and refined inequality technique.     

Difference methods for computing the Ginzburg‐Landau equation in two dimensions

In this article, three difference schemes of the Ginzburg‐Landau Equation in two dimensions are presented. In the three schemes, the nonlinear term is discretized such that nonlinear iteration is not needed in computation. The plane wave solution of the equation is studied and the truncation errors of the three schemes are obtained. The three schemes are unconditionally stable. The stability of the two difference schemes is proved by induction method and the time‐splitting method is analysized by linearized analysis. The algebraic multigrid method is used to solve the three large linear systems of the schemes. At last, we compute the plane wave solution and some dynamics of the equation. The numerical results demonstrate that our schemes are reliable and efficient. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 507–528, 2011py; 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 507–528, 2011     

Ginzburg‐Landau vortex dynamics driven by an applied boundary current

In this paper we study the time‐dependent Ginzburg‐Landau equations on a smooth, bounded domain Ω ? ?², subject to an electrical current applied on the boundary. The dynamics with an applied current are nondissipative, but via the identification of a special structure in an interaction energy, we are able to derive a precise upper bound for the energy growth. We then turn to the study of the dynamics of the vortices of the solutions in the limit ε → 0. We first consider the original time scale in which the vortices do not move and the solutions undergo a “phase relaxation.” Then we study an accelerated time scale in which the vortices move according to a derived dynamical law. In the dynamical law, we identify a novel Lorentz force term induced by the applied boundary current. © 2010 Wiley Periodicals, Inc.     

Asymptotic behaviour of time‐dependent Ginzburg–Landau equations of superconductivity

In this paper, we establish the global fast dynamics for the time‐dependent Ginzburg–Landau equations of superconductivity. We show the squeezing property and the existence of finite‐dimensional exponential attractors for the system. In addition we prove the existence of the global attractor in L² × L² for the Ginzburg–Landau equations in two spatial dimensions. Copyright © 1999 John Wiley & Sons, Ltd.     

Vortex analysis of the periodic Ginzburg–Landau model

We study the vortices of energy minimizers in the London limit for the Ginzburg–Landau model with periodic boundary conditions. For applied fields well below the second critical field we are able to describe the location and number of vortices. Many of the results presented appeared in [H. Aydi, Doctoral Dissertation, Université Paris-XII, 2004], others are new.     

Analysis of some finite difference schemes for two‐dimensional Ginzburg‐Landau equation

We study the rate of convergence of some finite difference schemes to solve the two‐dimensional Ginzburg‐Landau equation. Avoiding the difficulty in estimating the numerical solutions in uniform norm, we prove that all the schemes are of the second‐order convergence in L₂ norm by an induction argument. The unique solvability, stability, and an iterative algorithm are also discussed. A numerical example shows the correction of the theoretical analysis.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1340‐1363, 2011     

On the equilibrium position of Ginzburg Landau vortices

We study a few problems related to superconducting vortices. Our main concern is the stable equilibrium distribution of them. Our starting point is the asymptotic form of the Ginzburg Landau energy functional with a large Ginzburg Landau parameter. We consider in particular the interaction of the vortices with an applied magnetic field, and the effects of impurities on the vortex locations.