The resistive state in a superconducting wire: Bifurcation from the normal state

TL;DR

This paper rigorously analyzes bifurcations in a superconducting wire model, revealing how steady and periodic states emerge from the normal state through explicit formulas and a detailed dynamical systems approach.

Contribution

It provides a rigorous mathematical framework for understanding bifurcations in superconducting wires, including explicit formulas and a center manifold construction.

Findings

Explicit asymptotic formulas for bifurcating solutions

Identification of phase slip centers in the bifurcation process

Complete description of finite-dimensional dynamics near bifurcation

Abstract

We study formally and rigorously the bifurcation to steady and time-periodic states in a model for a thin superconducting wire in the presence of an imposed current. Exploiting the PT-symmetry of the equations at both the linearized and nonlinear levels, and taking advantage of the collision of real eigenvalues leading to complex spectrum, we obtain explicit asymptotic formulas for the stationary solutions, for the amplitude and period of the bifurcating periodic solutions and for the location of their zeros or "phase slip centers" as they are known in the physics literature. In so doing, we construct a center manifold for the flow and give a complete description of the associated finite-dimensional dynamics.