Long-Time Stabilization of Solutions to the Ginzburg–Landau Equations of Superconductivity

 The long-time dynamical properties of solutions (φ,A) to the time-dependent Ginzburg–Landau (TDGL) equations of superconductivity are investigated. The applied magnetic field varies with time, but it is assumed to approach a long-time asymptotic limit. Sufficient conditions (in terms of the time rate of change of the applied magnetic field) are given which guarantee that the dynamical process defined by the TDGL equations is asymptotically autonomous, i.e., it approaches a dynamical system as time goes to infinity. Analyticity of an energy functional is used to show that every solution of the TDGL equations asymptotically approaches a (single) stationary solution of the (time-independent) Ginzburg–Landau equations. The standard “φ = − ∇ · A” gauge is chosen.     

Inviscid Limits of the Complex Generalized Ginzburg—Landau Equations

1 IntroductionDerivative Ginzburg-Landau equation appeared in many physical problem. It was derivedfor instability waves in hydrodynamic such as the nonlinear growth of Rayleigh-Benard convectiverolls, the appearance of Taylor Vortices in the couette flow between counter-rotating cylinders.This paper is concerning with the generalized derivative Ginzburg-Landau equations given by     

Adiabatic Limit for Hyperbolic Ginzburg–Landau Equations

We study the adiabatic limit in hyperbolic Ginzburg–Landau equations which are Euler–Lagrange equations for the Abelian Higgs model. Solutions of Ginzburg–Landau equations in this limit converge to geodesics on the moduli space of static solutions in the metric determined by the kinetic energy of the system. According to heuristic adiabatic principle, every solution of Ginzburg–Landau equations with sufficiently small kinetic energy can be obtained as a perturbation of some geodesic. A rigorous proof of this result was proposed recently by Palvelev.     

A Model for Vortex Nucleation in the Ginzburg–Landau Equations

This paper studies questions related to the dynamic transition between local and global minimizers in the Ginzburg–Landau theory of superconductivity. We derive a heuristic equation governing the dynamics of vortices that are close to the boundary, and of dipoles with small inter-vortex separation. We consider a small random perturbation of this equation and study the asymptotic regime under which vortices nucleate.     

Well-posedness of the fractional Ginzburg–Landau equation

In this paper, we investigate the well-posedness of the real fractional Ginzburg–Landau equation in several different function spaces, which have been used to deal with the Burgers’ equation, the semilinear heat equation, the Navier–Stokes equations, etc. The long time asymptotic behavior of the nonnegative global solutions is also studied in details.     

Periodic minimizers of the anisotropic Ginzburg–Landau model

We consider the anisotropic Ginzburg–Landau model in a three-dimensional periodic setting, in the London limit as the Ginzburg–Landau parameter \({\kappa=1/{\epsilon}\to\infty}\) . By means of matching upper and lower bounds on the energy of minimizers, we derive an expression for a limiting energy in the spirit of Γ-convergence. We show that, to highest order as \({\epsilon\to0}\) , the normalized induced magnetic field approaches a constant vector. We obtain a formula for the lower critical field H _c1 as a function of the orientation of the external field \({h^\epsilon_{ex}}\) with respect to the principal axes of the anisotropy, and determine the direction of the limiting induced field as a minimizer of a convex geometrical problem.     

A Finite-difference Solution of the Ginzburg–Landau Equation

Two finite-difference methods, which differ only in the way that they approximate the derivative boundary conditions, are developed for solving a particular form of the complex Ginzburg–Landau equation of superconductivity. The non-linear term in this equation is linearized in a way familiar to readers of Professor Mickens' work, and the numerical solution is obtained at each time step by solving a linear algebraic system. Consistency and stability are discussed and some numerical results are reported.     

Null controllability of the complex Ginzburg–Landau equation

The paper investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg–Landau equation. Zero-controllability results are derived from a new Carleman estimate and an analysis based upon the theory of sectorial operators.     

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Ginzburg–Landau Equations

This paper considers the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg–Landau equations driven by additive noise with α∈(0, 1). First, we give some conditions for bounding the fractal dimension of a random invariant set of non-autonomous random dynamical system. Second, we derive uniform estimates of solutions and establish the existence and uniqueness of tempered pullback random attractors for the equation in H. At last, we prove the finiteness of fractal dimension of random attractors.     

Homogenization of a Ginzburg–Landau functional

We consider a nonlinear homogenization problem for a Ginzburg–Landau functional with a (positive or negative) surface energy term describing a nematic liquid crystal with inclusions. Assuming that sizes and distances between inclusions are of the same order ?, we obtain a limiting functional as $?\to 0$. We generalize the method of mesocharacteristics to show that a corresponding homogenized problem for arbitrary, periodic or non-periodic geometries is described by an anisotropic Ginzburg–Landau functional. We give computational formulas for material characteristics of an effective medium. To cite this article: L. Berlyand et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Singularity analysis of Ginzburg–Landau energy related to p-wave superconductivity

The following Ginzburg–Landau energy in the absence of a magnetic field $$E_\varepsilon(\psi) = \int\limits_G\left[\frac{1}{2}|\nabla\psi|^2 + \frac{1}{4\varepsilon^2}(1-|\psi|^2)^2\right]{\rm d}x$$ was well studied during recent twenty years. Here, ${G \subset \mathbf{R}^2}$ is a bounded smooth domain, ${\psi}$ is an order parameter, ${\varepsilon >0 }$ . In particular, several global properties including the weighted energy estimation, the concentration compactness properties and the quantization effect of the energy had been established. This paper is concerned with another Ginzburg–Landau type free energy associated with p-wave superconductivity $$E_\varepsilon (\psi, u; G) = \frac{1}{2} \int\limits_G(|\nabla \psi|^2 + |\nabla u|^2 - |\nabla|\psi||^2){\rm d}x + \frac{1}{4\varepsilon^2} \int\limits_G(1-|\psi|^2)^2{\rm d}x.$$ Here, u is also an order parameter. We will prove that those global properties still hold for this more complicated energy functional. Such global properties describe the locations of the regular and the singular domains, and also show the convergence relation between the Ginzburg–Landau minimizers and the harmonic maps.     

Stabilization of Solutions of the Nonlinear Fokker–Planck Equation

One considers the equation $$ \mathrm{div}\left( {{u^{\sigma }}Du} \right)+b(x)Du-{u_t}=f(x)g(u),\quad x\in {{\mathbb{R}}^n},\quad t\in \left( {0,\infty } \right), $$ where $ b:{{\mathbb{R}}^n}\to {{\mathbb{R}}^n} $ and $ f:{{\mathbb{R}}^n}\to [0,\infty ) $ are locally bounded measurable functions, g: (0,∞)?→?(0,∞) is continuous and nondecreasing, One obtains the conditions ensuring that its positive solutions stabilize to zero as t?→?∞.     

Singular Solutions to the Quantum Yang–Baxter Equations

Let X and A be weak Hopf algebras in the sense of Li (1998 Li , F. ( 1998 ). Weak Hopf algebras and some new solutions of the quantum Yang–Baxter equation . J. Algebra 208 ( 1 ): 72 – 100 .[Crossref], [Web of Science ®] , [Google Scholar]). As in the case of Hopf algebras (Majid, 1990 Majid , S. ( 1990 ). Quasitriangular Hopf algebras and Yang–Baxter equations . Internat. J. Modern Phys. A 5 : 1 – 91 . [Google Scholar]), a weak bicrossed coproduct X∞ _R A is constructed by means of good regular R-matrices of the weak Hopf algebras X and A. Using this, we provide a new framework of obtaining singular solutions of the quantum Yang–Baxter equation by constructing weak quasitriangular structures over X∞ _R A when both X and A admit a weak quasitriangular structure. Finally, two explicit examples are given.     

Existence of standing waves for the complex Ginzburg–Landau equation

We study the existence of standing wave solutions of the complex Ginzburg–Landau equation

equation(GL)

_φt−e^iθ(ρI−Δ)φ−e^iγ|φ|^αφ=0${\phi }_t-e^{i\theta }(\rho I-\mathrm{\Delta })\phi -e^{i\gamma }{|\phi |}^{\alpha }\phi =0$

in R^N${\mathbb{R}}^N$, where α>0$\alpha >0$, (N−2)α<4$(N-2)\alpha <4$, ρ>0$\rho >0$ and θ,γ∈R$\theta ,\gamma \in \mathbb{R}$. We show that for any θ∈(−π/2,π/2)$\theta \in (-\pi /2,\pi /2)$ there exists ε>0$\epsilon >0$ such that (GL) has a non-trivial standing wave solution if |γ−θ|<ε$|\gamma -\theta |<\epsilon$. Analogous result is obtained in a ball Ω∈R^N$\Omega \in {\mathbb{R}}^N$ for ρ>−_λ1$\rho >-{\lambda }_1$, where _λ1${\lambda }_1$ is the first eigenvalue of the Laplace operator with Dirichlet boundary conditions.