Multiple-spiral-wave solutions of the general cubic complex Ginzburg-Landau equation in bounded domains are considered. We investigate the effect of the boundaries on spiral motion under homogeneous Neumann boundary conditions, for small values of the twist parameter $q$. We derive explicit laws of motion for rectangular domains and we show that the motion of spirals becomes exponentially slow when the twist parameter exceeds a critical value depending on the size of the domain. The oscillation frequency of multiple-spiral patterns is also analytically obtained.
Building on the earlier work of Odeh, Barany, Golubitsky, Turski and Lasher we give a proof of existence of Abrikosov vortex lattices in the Ginzburg-Landau model of superconductivity.
For a reduced Ginzburg-Landau model in which the magnetic field is neglected, we prove, for weak electric currents, the existence of a steady-state solution in a vicinity of the purely superconducting state. We further show that this solution is linearly stable.
The Ginzburg-Landau equations were proposed in the superconductivity theory to describe mathematically the intermediate state of superconductors in which the normal conductivity is mixed with the superconductivity. It was understood later on that these equations play an important role also in various problems of mathematical physics. We mention here the extension of these equations to compact Riemann surfaces and Riemannian 4-manifolds. A separate interesting topic is the scattering theory of vortices reducing to the study of hyperbolic Ginzburg-Landau equations. In this review we tried to tou…
Eigenvalue problem for two coupled Ginzburg-Landau equations is numerically investigated. The fixed points of corresponding equations system are found. The classification of these points is made. The phase portraits of corresponding ordinary differential equations and the dependence of some parameters of the equations system and the total energy on the initial values are given.
In this paper we study the problem of ergodicity for the complex Ginzburg--Landau equation perturbed by an unbounded random kick-force. Randomness is introduced both through the kicks and through the times between the kicks. We show that the Markov process associated with the equation in question possesses a unique stationary distribution and satisfies a property of polynomial mixing.
We consider the Aharonov-Bohm magnetic potential and study the transition from normal to superconducting solutions within the Ginzburg-Landau model of superconductivity. We obtain oscillations consistent with the Little-Parks effect. We study the same problem but for a regularization of the Aharonov-Bohm potential, which leads to an interesting Aharonov-Bohm like magnetic field, and we prove that the transition between superconducting and normal solutions is not monotone too. Our results show a mechanism to derive the Aharonov-Bohm magnetic potential starting from a step magnetic field thereby…
We introduce a duality construction for toric Landau-Ginzburg models, applicable to complete intersections in toric varieties via the sigma model / Landau-Ginzburg model correspondence. This construction is shown to reconstruct those of Batyrev-Borisov, Berglund-H"ubsch, Givental, and Hori-Vafa. It can be done in more general situations, and provides partial resolutions when the above constructions give a singular mirror. An extended example is given: the Landau-Ginzburg models dual to elliptic curves in (P^1)^2 .
We study vortex nucleation for minimizers of a Ginzburg-Landau energy with discontinuous constraint. For applied magnetic fields comparable with the first critical field of vortex nucleation, we determine the limiting vorticities.
We prove existence of Abrikosov vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, with multiple magnetic flux quanta per a fundamental cell. We also revisit the existence proof for the Abrikosov vortex lattices, streamlining some arguments and providing some essential details missing in earlier proofs for a single magnetic flux quantum per a fundamental cell.
In this paper, an extension of the random field Ginzburg-Landau model on the hypercubic lattice is considered by adding $p$-spin ($p\geqslant 2$) interactions coupled to general disorders. This new model is called the random field mixed-spin Ginzburg-Landau model. We proved that, in the infinite volume limit of this model, the variance of spin overlap vanishes.
We reconsider the elliptic estimates for magnetic operators in two and three dimensions used in connection with Ginzburg-Landau theory. Furthermore we discuss the so-called blow-up technique in order to obtain optimal estimates in the limiting cases.
It is proved that the vortices are attracted by impurities or inhomogeities in the superconducting materials. The strong H^1-convergence for the corresponding Ginzburg-Landau system is also proved.
We study the asymptotic behavior of solutions for $n$-component Ginzburg-Landau equations as $\ve \to 0$. We prove that the minimizers converges locally in any $C^k$-norm to a solution of a system of generalized harmonic map equations.
We consider the Ginzburg-Landau functional defined over a bounded and smooth three dimensional domain. Supposing that the strength of the applied magnetic field varies between the first and second critical fields, in such a way that $H_{C_1}\ll H\ll H_{C_2}$, we estimate the ground state energy to leading order as the Ginzburg-Landau parameter tends to infinity.
This is a review about a series of results on vortices in the Ginzburg-Landau model of superconductivity on the one hand, and point patterns in Coulomb gases on the other hand, as well as the connections between the two topics.
In Ginzburg-Landau Theory of superconductivity, the density and location of the superconducting electrons are measured by a complex-valued wave function, the order parameter. In this paper, when the intensity of the applied magnetic field is close to the second critical field, and when the order parameter minimizes the Ginzburg-Landau functional defined over a two dimensional domain, the leading order approximation of its $L^2$-norm in `small' squares is given as the Ginzburg-Landau parameter tends to infinity.
The time evolution of several interacting Ginzburg-Landau vortices according to an equation of Schroedinger type is approximated by motion on a finite-dimensional manifold. That manifold is defined as an unstable manifold of an auxiliary dynamical system, namely the gradient flow of the Ginzburg-Landau energy functional. For two vortices the relevant unstable manifold is constructed numerically and the induced dynamics is computed. The resulting model provides a complete picture of the vortex motion for arbitrary vortex separation, including well-separated and nearly coincident vortices.
We establish a new relationship (the MLK correspondence) between twisted FJRW theory and local Gromov-Witten theory in all genera. As a consequence, we show that the Landau-Ginzburg/Calabi-Yau correspondence is implied by the crepant transformation conjecture for Fermat type in genus zero. We use this to then prove the Landau-Ginzburg/Calabi-Yau correspondence for Fermat type, generalizing the results of A. Chiodo and Y. Ruan.
We study Ginzburg-Landau equations for a complex vector order parameter. We consider the Dirichlet problem in the disk in the plane 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 both the Ginzburg-Landau parameter as well as the sign of the interaction term in the energy.
Ginzburg-Landau fields are the solutions of the Ginzburg-Landau equations which depend on two positive parameters, $α$ and $β$. We give conditions on $α$ and $β$ for the existence of irreducible solutions of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded domains in $\mathbb{R}^2$, spheres, tori, etc.) with de Gennes-Neumann boundary conditions. We also prove that, for each such manifold and all positive $α$ and $β$, the Ginzburg-Landau free energy is a Palais-Smale function on the space of gauge equivalence classes, Ginzburg-Land…
We study the Ginzburg-Landau model of type-I superconductors in the regime of small external magnetic fields. We show that, in an appropriate asymptotic regime, flux patterns are described by a simplified branched transportation functional. We derive the simplified functional from the full Ginzburg-Landau model rigorously via $Γ$-convergence. The detailed analysis of the limiting procedure and the study of the limiting functional lead to a precise understanding of the multiple scales contained in the model.
We analyze 2-dimensional Ginzburg-Landau vortices at critical coupling, and establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. We then compute the corresponding Berry curvature and holonomy in the large volume limit.
We consider a superconducting/normal/superconducting Josephson junction modeled through the Ginzburg-Landau theory. When the normal material is sufficiently thin and the applied magnetic field is below the critical field of vortex nucleation, we prove to leading order that jump boundary conditions of the type predicted by de Gennes are satisfied across the junction.
In a recent paper we give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Here we present our results in the simplified case of a one-dimensional system of particles interacting via a delta-potential.
We provide a quantitative three-dimensional vortex approximation construction for the Ginzburg-Landau functional. This construction gives an approximation of vortex lines coupled to a lower bound for the energy, optimal to leading order, analogous to the 2D ones, and valid for the first time at the $\varepsilon$-level. These tools allow for a new approach to analyze the behavior of global minimizers for the Ginzburg-Landau functional below and near the first critical field in 3D, followed in two forthcoming papers. In addition, they allow to obtain an $\varepsilon$-quantitative product estimat…
Group classification of the generalized complex Ginzburg-Landau equations is presented. An approach to group classification of systems of reaction-diffusion equations with general diffusion matrix is developed.
For certain nonlinear elliptic PDE problems in two dimensions, the classical isoperimetric inequality produces a sharp inequality that violates a Pohozaev identity except for radial symmetric, decreasing solutions. A generalized version of this technique is used here to prove radial symmetry of curl-free Ginzburg-Landau vortices.
We formulate a spectral problem related to the onset of superconductivity for a generalized Ginzburg-Landau model, where the order parameter and the magnetic potential are defined in the whole space. This model is devoted to the `proximity effect' for a superconducting sample surrounded by a normal material. In the regime when the Ginzburg-Landau parameter (of the superconducting material) is large, we estimate the critical applied magnetic field for which the normal state will lose its stability, a result that has some roots in the physical literature. In some asymptotic situations, we recove…
Using the improved lower bound on the sum of the eigenvalues of the Dirichlet Laplacian proved by A. D. Melas (Proc. Amer. Math. Soc. \textbf{131} (2003) 631-636), we report a new and sharp estimate for the dimension of the global attractor associated to the complex Ginzburg-Landau equation supplemented with Dirichlet boundary conditions.
We describe rules for computing a homology theory of knots and links in $\mathbb{R}^3$. It is derived from the theory of framed BPS states bound to domain walls separating two-dimensional Landau-Ginzburg models with (2,2) supersymmetry. We illustrate the rules with some sample computations, obtaining results consistent with Khovanov homology. We show that of the two Landau-Ginzburg models discussed in this context by Gaiotto and Witten one, (the so-called Yang-Yang-Landau-Ginzburg model) does not lead to topological invariants of links while the other, based on a model with target space equal …
Given an algebra with group $G$-action, we construct brace structures for its $G$-twisted Hochschild cochains. An an application, we construct $G$-Frobenius algebras for orbifold Landau-Ginzburg B-models and present explicit orbifold cup product formula for all invertible polynomials.
Ginzburg-Landau energy models arise as autonomous sto-chastic dynamics for the energies in coupled systems after a weak coupling limit (cf. [3, 6]). We prove here that, under certain conditions, the energy fluctuations of these stochastic dynamics are driven by the heat equation, under a diffusive space time scaling.
We prove a version of the Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic. In particular we calculate the genus-zero FJRW theory for the pair (W, G) where W is the Fermat quintic polynomial and G = SL(W). We identify it with the Gromov-Witten theory of the mirror quintic three-fold via an explicit analytic continuation and symplectic transformation. In the process we prove a mirror theorem for the corresponding Landau-Ginzburg model (W,G).
These are the lecture notes of a "Nachdiplomvorlesung" course taught at ETH Zurich in the Spring of 2013. They appeared in the EMS series Zurich Lectures in Advanced Mathematics.
This is the author's PhD thesis, which focuses on the so-called Landau-Ginzburg/conformal field theory correspondence. This correspondence dates from the late 80s and early 90s in the physics literature and in particular, it predicts a relation between defects in Landau-Ginzburg models and defects in conformal field theories. This relation is supported by examples, but not understood in general, nor up to date is there a clear mathematical conjecture of the Landau-Ginzburg/conformal field theory correspondence. The pursuit of a precise mathematical statement of this conjecture continues to gen…
We present a class of three-dimensional integrable structures associated with the Darboux-Egoroff metric and classical Euler equations of free rotations of a rigid body. They are obtained as canonical structures of rational Landau-Ginzburg potentials and provide solutions to the Painleve VI equation.
We prove the existence of novel, nonminimal and irreducible solutions to the (self-dual) Ginzburg-Landau equations on closed surfaces. To our knowledge these are the first such examples on nontrivial line bundles, that is, with nonzero total magnetic flux. Our method works with the 2-dimensional, critically coupled Ginzburg-Landau theory and uses the topology of the moduli space. The method is nonconstructive, but works for all values of the remaining coupling constant. We also prove the instability of these solutions.
This article analyzes well-definedness and regularity of renormalized powers of Ornstein-Uhlenbeck processes and uses this analysis to establish local existence, uniqueness and regularity of strong solutions of stochastic Ginzburg-Landau equations with polynomial nonlinearities in two space dimensions and with quadratic nonlinearities in three space dimensions.
In this paper we prove an analogue of the McKay correspondence for Landau-Ginzburg models. Our proof is based on the ideas introduced by T. Bridgeland, A. King and M. Reid, which reformulate and generalize the McKay correspondence in the language of derived categories, along with the techniques introduced by J.-C. Chen.
We derive the fractional generalization of the Ginzburg-Landau equation from the variational Euler-Lagrange equation for fractal media. To describe fractal media we use the fractional integrals considered as approximations of integrals on fractals. Some simple solutions of the Ginzburg-Landau equation for fractal media are considered and different forms of the fractional Ginzburg-Landau equation or nonlinear Schrodinger equation with fractional derivatives are presented. The Agrawal variational principle and its generalization have been applied.
The post Gaussian effective potential in D=3 dimensions and the Gaussian effective potential in D=2+epsilon are evaluated for the Ginzburg-Landau theory of superconductivity. It is shown that, the next order correction to the Gaussian approximation of the Ginzburg-Landau parameter (kappa) is significant, whereas contribution from the two dimensionality is rather small. This strongly indicates that strong correlation plays a more dominant role than the two dimensionality does in high T_c superconductivity.
