On the solutions of the one dimensional Ginzburg-Landau equations for superconductivity

Abstract

This paper gives a complete description of the solutions of the one dimensional Ginzburg-Landau equations which model superconductivity phenomena in infinite slabs. We investigate this problem over the entire range of physically important parameters: $a$ the size of the slab, $κ$ the Ginzburg-Landau parameter, and $h_0$, the exterior magnetic field. We do extensive numerical computations using the software AUTO, and determine the number, symmetry and stability of solutions for all values of the parameters. In particular, our experiments reveal the existence of two key-points in parameter space which play a central role in the formation of the complicated patterns by means of bifurcation phenomena. Our global description also allows us to separate the various physically important regimes, to classify previous results in each regime according to the values of the parameters and to derive new open problems. In addition, our investigation provides new insight into the problem of differentiating between the types of superconductors in terms of the parameters.