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Stability of Abrikosov vortex lattices of the Ginzburg-Landau equations of superconductivity

Timmy Tzaneteas (TU Braunschweig)
Fredag, 24 februar, 2012, at 15:15-16:15, in Koll. D (1531-211)
Abrikosov lattice solutions of the Ginzburg-Landau equations of superconductivity (in $\R^2$) consist of vortices that arrange themselves in a lattice at the points where an applied magnetic field penetrates the material. Such solutions have been shown to exist for any lattice shape parameter $\tau$ (a complex number determining the shape of the lattice, for example, $\tau = i$ for square lattices and $\tau = e^{\frac{i\pi}{3}}$ for triangular lattices), and for constant applied magnetic fields with strength close to but less than a critical field. These lattices have been observed experimentally in Type II superconductors, for which the Ginzburg-Landau parameter $\kappa$, a constant depending on the material properties of the superconductor, is greater than $1/\sqrt{2}$.

We study the stability of such lattice solutions within the context of a gradient-flow time-dependent version of the Ginzburg-Landau equations for applied fields close to the critical field. We show that under gauge-periodic perturbations, which preserve the lattice structure of the solution, Abrikosov lattice solutions are asymptotically stable if $\kappa^2 > \frac{1}{2}(1 - \frac{1}{\beta(\tau)})$, where $\beta(\tau)$ is the Abrikosov constant depending only on the lattice shape parameter $\tau$, and unstable if $\kappa^2 < \frac{1}{2}(1 - \frac{1}{\beta(\tau)})$. In the case of more general finite-energy perturbations, we show in that if $\kappa^2 > \frac{1}{2}$, the solution is orbitally stable for all $\tau$.
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