zbMATH — the first resource for mathematics

Stable nucleation for the Ginzburg-Landau system with an applied magnetic field. (English) Zbl 0922.35157
The authors deal with the boundary value problem \[ \Biggl({i\over\kappa}+ A\Biggr)^2\psi- \psi+| \psi|^2\psi= 0,\;(\text{curl})^2A+{i\over 2\kappa} (\psi^*\nabla\psi- \psi\nabla\psi^*)+ |\psi|^2 A=0\quad\text{in }\Omega, \] \[ n\Biggl({i\over\kappa} \nabla+ A\Biggr)\psi= 0,\quad \text{curl }A= he_3\quad \text{on }\partial\Omega \] describing a superconducting material placed in a vacuum subject to an applied magnetic field. The domain \(\Omega\) is a cylinder and the applied field \(he_3\) is parallel to the axis. Then \(|\psi|^2\) represents the density of the superconducting electrons and \(\text{curl }A\) (where \(A\) is a two-dimensional vector field) represents the induced magnetic field.
If \(h\) is large, then the normal (nonsuperconducting) state (with \(\psi= 0\)) is stable. As \(h\) decreases, the normal state becomes unstable. However, if the constant \(\kappa\) is large, then, excepting a discrete set of radii of \(\Omega\), there exists precisely one superconducing solution which is stable. Then \(|\psi|\) becomes uniformly positive in a strip near \(\partial\Omega\) and rapidly decreases toward the interior of \(\Omega\). This is the surface nucleation phenomenon thoroughly analyzed in the article.

35Q55 NLS equations (nonlinear Schrödinger equations)
82D55 Statistical mechanical studies of superconductors
Full Text: DOI