Bifurcation diagram and pattern formation in superconducting wires with electric currents

TL;DR

This paper investigates the complex behavior and pattern formation in superconducting wires under electric currents using spectral analysis and numerical methods, revealing bifurcation structures and oscillatory states.

Contribution

It provides a detailed bifurcation analysis of the Ginzburg-Landau model in superconducting wires, highlighting the role of PT-symmetric eigenvalue collisions in pattern emergence.

Findings

Identification of parameter regions with different stable states

Explanation of oscillatory state emergence via eigenvalue collisions

Connection between spectrum and pattern formation

Abstract

We examine the behavior of a one-dimensional superconducting wire exposed to an applied electric current. We use the time-dependent Ginzburg-Landau model to describe the system and retain temperature and applied current as parameters. Through a combination of spectral analysis, asymptotics and canonical numerical computation, we divide this two-dimensional parameter space into a number of regions. In some of them only the normal state or a stationary state or an oscillatory state are stable, while in some of them two states are stable. One of the most interesting features of the analysis is the evident collision of real eigenvalues of the associated PT-symmetric linearization, leading as it does to the emergence of complex elements of the spectrum. In particular this provides an explanation to the emergence of a stable oscillatory state. We show that part of the bifurcation diagram and many of the emerging patterns are directly controlled by this spectrum, while other patterns arise due to nonlinear interaction of the leading eigenfunctions.