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.     

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.     

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.     

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

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.     

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.     

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.     

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.     

Generalized Ginzburg–Landau equations in high dimensions

In this work, we study critical points of the generalized Ginzburg–Landau equations in dimensions \(n\ge 3\) which satisfy a suitable energy bound, but are not necessarily energy-minimizers. When the parameter in the equations tend to zero, such solutions are shown to converge to singular n-harmonic maps into spheres, and the convergence is strong away from a finite set consisting (1) of the infinite energy singularities of the limiting map, and (2) of points where bubbling off of finite energy n-harmonic maps could take place. The latter case is specific to dimensions \({>}2\). We also exhibit a criticality condition satisfied by the limiting n-harmonic maps which constrains the location of the infinite energy singularities. Finally we construct an example of non-minimizing solutions to the generalized Ginzburg–Landau equations satisfying our assumptions.     

A product estimate for Ginzburg–Landau and application to the gradient-flow

We prove a new inequality for the Jacobian (or vorticity) associated to the Ginzburg–Landau energy in any dimension, and give static and dynamical corollaries. We then present a method to prove convergence of gradient-flows of families of energies which Gamma-converge to a limiting energy, which we apply to establish, thanks to the previous dynamical estimate, the limiting dynamical law of a finite number of vortices for the heat-flow of Ginzburg–Landau in dimension 2, with and without magnetic field. To cite this article: E. Sandier, S. Serfaty, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Well-posedness for a mean field model of Ginzburg–Landau vortices with opposite degrees

In this paper we analyze the hydrodynamic equations for Ginzburg–Landau vortices as derived by E (Phys. Rev. B. 50(3):1126–1135, 1994). In particular, we are interested in the mean field model describing the evolution of two patches of vortices with equal and opposite degrees. Many results are already available for the case of a single density of vortices with uniform degree. This model does not take into account the vortex annihilation, hence it can also be seen as a particular instance of the signed measures system obtained in Ambrosio et al. (Ann. Inst. H. Poincaré Anal. Non Linéaire 28(2):217–246, 2011) and related to the Chapman et al. (Eur. J. Appl. Math. 7(2):97–111, 1996) formulation. We establish global existence of L ^p solutions, exploiting some optimal transport techniques introduced in this context in Ambrosio and Serfaty (Commun. Pure Appl. Math. LXI(11):1495–1539, 2008). We prove uniqueness for L ^∞ solutions, as expected by analogy with the incompressible Euler equations in fluidodynamics. We also consider the corresponding Dirichlet problem in a bounded domain. Moreover, we show some simple examples of 1-dimensional dynamic.     

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.     

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...