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.     

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.     

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.     

Well-posedness of fractional Ginzburg–Landau equation in Sobolev spaces

This article is concerned with real fractional Ginzburg–Landau equation. Existence and uniqueness of local and global mild solution for both whole space case and flat torus case are obtained by contraction semigroup method, and Gevrey regularity of mild solution for flat torus case is discussed.     

Analysis of a Ginzburg–Landau type energy model for smectic C* liquid crystals with defects

This work investigates properties of a smectic C* liquid crystal film containing defects that cause distinctive spiral patterns in the film?s texture. The phenomena are described by a Ginzburg–Landau type model and the investigation provides a detailed analysis of minimal energy configurations for the film?s director field. The study demonstrates the existence of a limiting location for the defects (vortices) so as to minimize a renormalized energy. It is shown that if the degree of the boundary data is positive then the vortices each have degree +1 and that they are located away from the boundary. It is proved that the limit of the energies for a sequence of minimizers minus the sum of the energies around their vortices, as the G–L parameter ε tends to zero, is equal to the renormalized energy for the limiting state.     

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).     

Random attractors for a Ginzburg–Landau equation with additive noise

The existence of a compact random attractor for the random dynamical system generated by the complex Ginzburg–Landau equation with additive white noise has been proved. And a precise estimate of the upper bound of the Hausdorff dimension of the random attractor is obtained.     

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.     

Gamma-convergence and the emergence of vortices for Ginzburg–Landau on thin shells and manifolds

We analyze the Ginzburg–Landau energy in the presence of an applied magnetic field when the superconducting sample occupies a thin neighborhood of a bounded, closed manifold in ${\mathbb R^3}$ . We establish Γ-convergence to a reduced Ginzburg–Landau model posed on the manifold in which the magnetic potential is replaced in the limit by the tangential component of the applied magnetic potential. We then study the limiting problem, constructing two-vortex critical points when the manifold ${\mathcal{M}}$ is a simply connected surface of revolution and the applied field is constant and vertical. Finally, we calculate that the exact asymptotic value of the first critical field H _c1 is simply (4π/(area of ${\mathcal{M}}$ )) ln κ for large values of the Ginzburg–Landau parameter κ. Merging this with the Γ-convergence result, we also obtain the same asymptotic value for H _c1 in 3d valid for large κ and sufficiently thin shells.     

Finite-time blowup for a complex Ginzburg–Landau equation with linear driving

In this paper, we consider the complex Ginzburg–Landau equation ${u_t = e^{i\theta} [\Delta u + |u|^\alpha u] + \gamma u}$ on ${\mathbb{R}^N}$ , where ${\alpha > 0,\,\gamma \in \mathbb{R}}$ and ${-\pi /2 < \theta < \pi /2}$ . By convexity arguments, we prove that, under certain conditions on ${\alpha,\theta,\gamma}$ , a class of solutions with negative initial energy blows up in finite time.     

Large deviations for the Ginzburg–Landau ∇φ interface model

Hydrodynamic large scale limit for the Ginzburg-Landau ∇φ interface model was established in [6]. As its next stage this paper studies the corresponding large deviation problem. The dynamic rate functional is given by

A Ginzburg–Landau model for the phase transition in Helium II

The paper deals with the second-order phase transition in Helium II by a Ginzburg–Landau model, in which any particle has simultaneously a normal and superfluid velocity. This pattern is able to describe the classical effects of Helium II as the phase diagram, the vortices, the second sound and the thermomechanical effect. Finally, the vorticities and turbulence are described by an extension of the model in which the material time derivative is used.     

Lattice Solutions in a Ginzburg–Landau Model for a Chiral Magnet

Journal of Nonlinear Science - We examine micromagnetic pattern formation in chiral magnets, driven by the competition of Heisenberg exchange, Dzyaloshinskii–Moriya interaction, easy-plane...