Bifurcations and Vortex Formation in the Ginzburg-Landau Equations

• Published in: Zeitschrift für angewandte Mathematik und Mechanik Vol. 81; no. 8; pp. 523 - 539
• Main Author: TAKAC, P
• Format: Journal Article
• Language: English
• Published: Berlin WILEY-VCH Verlag Berlin GmbH 01.08.2001; WILEY‐VCH Verlag Berlin GmbH; Wiley-VCH
• Subjects: Classical and quantum physics: mechanics and fields; Exact sciences and technology; Functional analytical methods; Functions of a complex variable; Ginzburg-Landau equations; inhomogeneous Cauchy-Riemann equations; Mathematical analysis; Mathematics; nucleation of vortices; Partial differential equations; Physics; Quantum mechanics; quasi-periodic boundary conditions; Sciences and techniques of general use; singular perturbation analysis; superconductivity; symmetry; Nucleation; Singular perturbation; Superconductivity; Cauchy Riemann equation; Bifurcation; Boundary condition; Fourier coefficient; Coulomb gauge; Ginzburg Landau equation; Periodic boundary condition; Vorticity; Analytical function; Periodic system; Symmetry
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/1521-4001(200108)81:8<523::AID-ZAMM523>3.0.CO;2-9

Bifurcations from the normal to superconducting state are investigated for the two‐dimensional Ginzburg‐Landau system modelling a superconductor near the critical temperature Tc. Nucleation of vortices is shown under periodic boundary conditions imposed on three observables, the local density of superconducting electrons, the local supercurrent density, and the microscopic magnetic field. This is demonstrated by rigorous analysis showing that the zeros of the eigenfunction involved in the bifurcation process (the so‐called vortices) remain preserved in the solution during the bifurcation process for all values of the applied magnetic field H (used as the bifurcation parameter) near a critical value Hc2 (H < Hc2). The method is based on a factorization of the order parameter as the product of the eigenfunction and a bounded function. In particular, the solution describes the Abrikosov vortex state. Classical elliptic functions of Weierstraß are used to perform a singular perturbation analysis which employs well‐known complex‐analytic tools for the inhomogeneous Cauchy‐Riemann equations.