The Adiabatic Limit of Fu–Yau Equations

In this paper, we consider the adiabatic limit of Fu–Yau equations on a product of two Calabi–Yau manifolds. We prove that the adiabatic limit of Fu–Yau equations are quasilinear equations.     

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.     

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.     

Symmetric vortices for two-component Ginzburg–Landau systems

We study Ginzburg–Landau equations for a complex vector order parameter Ψ=(_ψ+,_ψ−)∈C²$\Psi =({\psi }_{+},{\psi }_{-})\in {\mathbb{C}}^2$. We consider symmetric vortex solutions in the plane R²${\mathbb{R}}^2$, ψ(x)=_f±(r)e^i_n±θ$\psi (x)=f_{\pm }(r)e^{i{n}_{\pm }\theta }$, with given degrees _n±∈Z$n_{\pm }\in \mathbb{Z}$, and prove the existence, uniqueness, and asymptotic behavior of solutions as r→∞$r\to \infty$. We also consider the monotonicity properties of solutions, and exhibit parameter ranges in which both vortex profiles _f+$f_{+}$, _f−$f_{-}$ are monotone, as well as parameter regimes where one component is non-monotone. The qualitative results are obtained by means of a sub- and super-solution construction and a comparison theorem for elliptic systems.     

Nehari method for the generalized Ginzburg–Landau system

The Dirichlet problem for the generalized Ginzburg–Landau system is considered. The existence of positive vector solutions is proved in the following three cases: (1) the cross term has weak growth; (2) the interaction constant is large enough; and (3) the cross term has strong growth and the interaction constant is positive and close to zero.     

Exact chirped solitary-wave solutions for Ginzburg–Landau equation

In this short letter, by applying specially envelope transform and direct ansatz approach to (1 + 1)D Ginzburg–Landau equation the authors obtain a new type of exact solitary wave solution including chirped bright solitary-wave and chirped dark solitary-wave solutions.     

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.     

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. (Received 30 June 2000; in revised form 30 December 2000)     

Homoclinic Orbit for the Cubic–Quintic Ginzburg–Landau Equation

The Ginzburg–Landau equation with small complex coefficients is discussed in this paper. A transformation is introduced to change the equation into a three order, ordinary differential equation and the existence of the homoclinic orbit for this system has been proved by analytical methods.     

A geometric Ginzburg–Landau problem

For surfaces embedded in a three-dimensional Euclidean space, consider a functional consisting of two terms: a version of the Willmore energy and an anisotropic area penalising the first component of the normal vector, the latter weighted with the factor ${1/\epsilon^2}$ . The asymptotic behaviour of such functionals as ${\epsilon}$ tends to 0 is studied in this paper. The results include a lower and an upper bound on the minimal energy subject to suitable constraints. Moreover, for embedded spheres, a compactness result is obtained under appropriate energy bounds.     

Adiabatic limit in Ginzburg—Landau and Seiberg—Witten equations

Theoretical and Mathematical Physics - We present the concept of an adiabatic limit of Ginzburg—Landau dynamical equations on ?1+2 and Seiberg—Witten equations on four-dimensional...     

Stability of symmetric vortices for two-component Ginzburg–Landau systems

We study Ginzburg–Landau equations for a complex vector order parameter Ψ=(_ψ+,_ψ−)∈C²$\Psi =({\psi }_{+},{\psi }_{-})\in {\mathbb{C}}^2$. We consider the Dirichlet problem in the disk in R²${\mathbb{R}}^2$ with a symmetric, degree-one boundary condition, and study its stability, in the sense of the spectrum of the second variation of the energy. We find that the stability of the degree-one equivariant solution depends on the Ginzburg–Landau parameter as well as the sign of the interaction term in the energy.     

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     

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).