Well‐posedness of time‐space fractional stochastic evolution equations driven by α‐stable noise

This paper is devoted to the well‐posedness for time‐space fractional Ginzburg‐Landau equation and time‐space fractional Navier‐Stokes equations by α‐stable noise. The spatial regularity and the temporal regularity of the nonlocal stochastic convolution are firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag‐Leffler functions, respectively. Numerical simulations for time‐space fractional Ginzburg‐Landau equation are provided to verify the analysis results.     

Vortex dynamics of the full time‐dependent Ginzburg‐Landau equations

In the Ginzburg‐Landau model for superconductivity a large Ginzburg‐Landau parameter κ corresponds to the formation of tight, stable vortices. These vortices are located exactly where an applied magnetic field pierces the superconducting bulk, and each vortex induces a quantized supercurrent about the vortex. The energy of large‐κ solutions blows up near each vortex which brings about difficulties in analysis. Rigorous asymptotic static theory has previously established the existence of a finite number of the vortices, and these vortices are located precisely at the critical points of the renormalized energy (the free energy less the vortex self‐induction energy). A rigorous study of the full time‐dependent Ginzburg‐Landau equations under the classical Lorentz gauge is done under the asymptotic limit κ → ∞. Under slow times the vortices remain pinned to their initial configuration. Under a fast time of order κ the vortices move according to a steepest descent of the renormalized energy. © 2002 John Wiley & Sons, Inc.     

On the equilibrium position of Ginzburg Landau vortices

We study a few problems related to superconducting vortices. Our main concern is the stable equilibrium distribution of them. Our starting point is the asymptotic form of the Ginzburg Landau energy functional with a large Ginzburg Landau parameter. We consider in particular the interaction of the vortices with an applied magnetic field, and the effects of impurities on the vortex locations.     

The mixed boundary condition for the Ginzburg Landau model in thin films

The mixed boundary condition for the Ginzburg Landau model of superconductivity is considered in thin films. A simplified model is derived in the asymptotic limit of very small thickness. We also show that under certain conditions there is no nucleation of superconductivity at all.     

Existence and uniqueness for a non-isothermal dynamical Ginzburg–Landau model of superconductivity

In this paper, we present a time-dependent Ginzburg–Landau model which describes the phenomenon of superconductivity taking into account thermal effects. We modify the classical time-dependent Ginzburg–Landau equations by including temperature dependence. We prove that this model is compatible with the laws of thermodynamics. Moreover it allows us to express the critical magnetic field, which distinguishes the superconductive phase from the normal state, as a function of the absolute temperature. The theoretical temperature dependence of the threshold magnetic field agrees with the experimental results. Finally, we prove the existence and the uniqueness of the solutions of the non-isothemal Ginzburg–Landau equations.     

Size of Planar Domains and Existence of Minimizers of the Ginzburg–Landau Energy with Semistiff Boundary Conditions

The Ginzburg–Landau energy with semistiff boundary conditions is an intermediate model between the full Ginzburg–Landau equations, which leads to the appearance of both a condensate wave function and a magnetic potential, and the simplified Ginzburg–Landau model, coupling the condensate wave function to a Dirichlet boundary condition. In the semistiff model, there is no magnetic potential. The boundary data are not fixed, but circulation is prescribed on the boundary. Mathematically, this leads to prescribing the degrees on the components of the boundary. The corresponding problem is variational, but noncompact: in general, energy minimizers do not exist. Existence of minimizers is governed by the topology and the size of the underlying domain. We propose here various notions of domain size related to existence of minimizers, and discuss existence of minimizers or critical points, as well as their uniqueness and asymptotic behavior. We also present the state of the art in the study of this model, accounting for results obtained during the last decade by Berlyand, Dos Santos, Farina, Golovaty, Rybalko, Sandier, and the author.     

Global existence of strong solutions to a time‐dependent 3D Ginzburg‐Landau model for superconductivity with partial viscous terms

We study an initial boundary value problem for a time‐dependent 3D Ginzburg‐Landau model of superconductivity with partial viscous terms. We prove the global existence of strong solutions.     

A Numerical Method for Computing Time‐Periodic Solutions in Dissipative Wave Systems

A numerical method is proposed for computing time‐periodic and relative time‐periodic solutions in dissipative wave systems. In such solutions, the temporal period, and possibly other additional internal parameters such as the propagation constant, are unknown priori and need to be determined along with the solution itself. The main idea of the method is to first express those unknown parameters in terms of the solution through quasi‐Rayleigh quotients, so that the resulting integrodifferential equation is for the time‐periodic solution only. Then this equation is computed in the combined spatiotemporal domain as a boundary value problem by Newton‐conjugate‐gradient iterations. The proposed method applies to both stable and unstable time‐periodic solutions; its numerical accuracy is spectral; it is fast‐converging; its memory use is minimal; and its coding is short and simple. As numerical examples, this method is applied to the Kuramoto–Sivashinsky equation and the cubic‐quintic Ginzburg–Landau equation, whose time‐periodic or relative time‐periodic solutions with spatially periodic or spatially localized profiles are computed. This method also applies to systems of ordinary differential equations, as is illustrated by its simple computation of periodic orbits in the Lorenz equations. MATLAB codes for all numerical examples are provided in the Appendices to illustrate the simple implementation of the proposed method.     

Lines of vortices for solutions of the Ginzburg–Landau equations

For disc domains and for periodic models, we construct solutions of the Ginzburg–Landau equations which verify in the limit of a large Ginzburg–Landau parameter specified qualitative properties: the limit density of the vortices concentrates on lines.     

On the hydrodynamic limit of Ginzburg‐Landau wave vortices

In this paper, we study the hydrodynamic limit of the finite Ginzburg‐Landau wave vortices, which was established in [16]. Unlike the classical vortex method for incompressible Euler equations, we prove here that the densities approximated by the vortex blob method associated with the Ginzburg‐Landau wave vortices tend to the solutions of the pressure‐less compressible Euler‐Poisson equations. The convergence of such approximation is proved before the formation of singularities in the limit system as the blob sizes and the grid sizes tend to zero in appropriate rates. © 2002 John Wiley & Sons, Inc.     

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.     

Existence and uniqueness for a mathematical model in superfluidity

In this paper we propose a model to study superfluidity by considering as state variables the order parameter, describing the concentration of the superfluid phase, the velocity of the superfluid and the absolute temperature. We assume that the order parameter satisfies a Ginzburg–Landau equation and that the velocity is decomposed as the sum of a normal and a superfluid component. The heat equation provides the evolution equation for the temperature. We prove that this model is consistent with the principles of thermodynamics. Well‐posedness of the resulting initial and boundary value problem is shown. Copyright © 2008 John Wiley & Sons, Ltd.     

Hysteresis loops for para–ferromagnetic phase transitions

In this paper, by an extension of the Ginzburg–Landau theory, we propose a mathematical model describing hard magnets within which we are able to explore the para–ferromagnetic transition and by using the Landau–Lifshitz–Gilbert equation, to study the 3D evolution of magnetic field. Finally, the hysteresis loops are obtained and represented by numerical implementations.     

Vortices and pinning effects for the Ginzburg‐Landau model in multiply connected domains

We consider the two‐dimensional Ginzburg‐Landau model with magnetic field for a superconductor with a multiply connected cross section. We study energy minimizers in the London limit as the Ginzburg‐Landau parameter κ = 1/? → ∞ to determine the number and asymptotic location of vortices. We show that the holes act as pinning sites, acquiring nonzero winding for bounded fields and attracting all vortices away from the interior for fields up to a critical value h_ex = O(|1n?|). At the critical level the pinning effect breaks down, and vortices appear in the interior of the superconductor at locations that we identify explicitly in terms of the solutions of an elliptic boundary value problem. The method involves sharp upper and lower energy estimates, and a careful analysis of the limiting problem that captures the interaction between the vortices and the holes. © 2005 Wiley Periodicals, Inc.     

Amplitude equation for the stochastic reaction‐diffusion equations with random Neumann boundary conditions

In this paper, we consider a quite general class of reaction‐diffusion equations with cubic nonlinearities and with random Neumann boundary conditions. We derive rigorously amplitude equations, using the natural separation of time‐scales near a change of stability and investigate whether additive degenerate noise and random boundary conditions can lead to stabilization of the solution of the stochastic partial differential equation or not. The nonlinear heat equation (Ginzburg–Landau equation) is used to illustrate our result. Copyright © 2015 John Wiley & Sons, Ltd.     

Triggered Fronts in the Complex Ginzburg Landau Equation

We study patterns that arise in the wake of an externally triggered, spatially propagating instability in the complex Ginzburg–Landau equation. We model the trigger by a spatial inhomogeneity moving with constant speed. In the comoving frame, the trivial state is unstable to the left of the trigger and stable to the right. At the trigger location, spatio-temporally periodic wave trains nucleate. Our results show existence of coherent, “heteroclinic” profiles when the speed of the trigger is slightly below the speed of a free front in the unstable medium. Our results also give expansions for the wavenumber of wave trains selected by these coherent fronts. A numerical comparison yields very good agreement with observations, even for moderate trigger speeds. Technically, our results provide a heteroclinic bifurcation study involving an equilibrium with an algebraically double pair of complex eigenvalues. We use geometric desingularization and invariant foliations to describe the unfolding. Leading-order terms are determined by a condition of oscillations in a projectivized flow, which can be found by intersecting absolute spectra with the imaginary axis.     

Ginzburg–Landau equations and their generalizations

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 turned out that these equations have interesting and non-trivial generalizations. First of all, they can be extended to arbitrary compact Riemann surfaces. Next, they can be generalized to dimension 3 as dynamical (or hyperbolic) Ginzburg–Landau equations. They also have a 4-dimensional extension provided by Seiberg–Witten equations. In this review we describe all these interesting topics together with some unsolved problems.     

Asymptotic behaviour of time‐dependent Ginzburg–Landau equations of superconductivity

In this paper, we establish the global fast dynamics for the time‐dependent Ginzburg–Landau equations of superconductivity. We show the squeezing property and the existence of finite‐dimensional exponential attractors for the system. In addition we prove the existence of the global attractor in L² × L² for the Ginzburg–Landau equations in two spatial dimensions. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

A three‐level linearized compact difference scheme for the Ginzburg–Landau equation

A high‐order finite difference method for the two‐dimensional complex Ginzburg–Landau equation is considered. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergent order in maximum norm is two in temporal direction and four in spatial direction. In addition, an efficient alternating direction implicit scheme is proposed. Some numerical examples are given to confirm the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 876–899, 2015