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 equatio...

In 1957 Abrikosov published his work on periodic solutions to the linearized Ginzburg-Landau equations. Abrikosov’s analysis assumes periodic boundary conditions, which are very different from the natural boundary conditions the minimizer of the Ginzburg-Landau energy functional should satisfy. In the present work we prove that the global minimizer of the fully non-linear functional can be appr...

The authors consider the Ginzburg-Landau model for superconductivity. First some well-known features of superconducting materials are reviewed and then various results concerning the model, the resultant differential equations, and their solution on bounded domains are derived. Then, finite element approximations of the solutions of the Ginzburg-Landau equations are considered and error estimat...

We study a gradient-flow version of the Ginzburg-Landau equations with an addition of a compactly supported potential term. We consider initial data close to a magnetic vortex solution of the Ginzburg-Landau equations and find the dynamical law governing the motion of the vortex center in the presence of the potential.

Using a recent construction of Bezrukavnikov and Etingof, [BE], we prove that there is a factorization of the Etingof-Ginzburg sheaf on the generalized Calogero-Moser space associated to a complex reflection group. In the case W = Sn, this confirms a conjecture of Etingof and Ginzburg, [EG].

Abstract The hybrid model is the Landau–Ginzburg-type theory that expected, via Landau–Ginzburg/ Calabi–Yau correspondence, to match Gromov–Witten of a complete intersection in weighted projective space. We prove wall-crossing formula exhibiting dependence genus-zero on its stability parameter, generalizing work [21] for quantum singularity and paralleling Ciocan-Fontanine–Kim [7] quasimaps. Th...

The phenomenological Ginzburg-Landau model for lowtemperature superconductivity has received much attention. However, it is not applicable to physical contexts that do not take into account factors such as material defects or thermal fluctuations. Thus, existing studies of the stability, dynamics, interactions, and other properties of the vortex state do not necessarily carry over to situations...

The Ginzburg-Landau equation is essential for understanding the dynamics of patterns in a wide variety of physical contexts. It governs the evolution of small amplitude instabilities near criticality. It is well known that the (cubic) Ginzburg-Landau equation has various unstable solitary pulse solutions. However, such localized patterns have been observed in systems in which there are two comp...

Several commentaries have appeared recently on the status of ecology as a science, with some being quite pessimistic when comparing ecology to the hard sciences like physics. For example, some ecologists wonder if we will ever have general laws and theories (Roughgarden 1998, Lawton 1999) or become a predictive science (McIntosh 1985, Peters 1991). The reason for this, says Murray (1992), is th...

We show that there exists multi-vortex, non-radial, finite energy solutions to the magnetic Ginzburg-Landau equations on all of R2. We use Lyapunov-Schmidt reduction to construct solutions which are invariant under rotations by 2π k (but not by rotations in O(2) in general) and reflections in the x− axis for some k ≥ 7. 1 Ginzburg-Landau equations 1.