Bifurcation Diagram and Pattern Formation of Phase Slip Centers in Superconducting Wires Driven with Electric Currents

Abstract

We provide here new insights into the classical problem of a one-dimensional superconducting wire exposed to an applied electric current using the time-dependent Ginzburg-Landau model. The most striking feature of this system is the well-known appearance of oscillatory solutions exhibiting phase slip centers (PSC's) where the order parameter vanishes. Retaining temperature and applied current as parameters, we present a simple yet definitive explanation of the mechanism within this nonlinear model that leads to the PSC phenomenon and we establish where in parameter space these oscillatory solutions can be found. 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.

An exactly solvable $\mathcal {PT}$-symmetric dimer from a Hamiltonian system of nonlinear oscillators with gain and loss

“…The transition between the two phases has been observed in a variety of experimental environments including optics [2,3], superconductivity [10], microwave cavities [11], atomic diffusion [12], and nuclear magnetic resonance [13]. In particular, the PT -symmetric phase transition is easily recognised in structures consisting of coupled oscillators with gain and loss [9,[14][15][16].…”

“…The existence of two integrals of motion establishes the complete integrability of the system (13) and the dimer (10). We can treat r as an independent variable and add the evolution equation (15) to the system (13).…”

“…[24].) The coupler described by (10) consists of a waveguide with loss and a guide with an equal amount of optical gain. The variables ψ 1 and ψ 2 represent the corresponding complex beam amplitudes, γ > 0 is their common gain-loss rate, and τ measures the distance along the parallel cores.…”

Non-Hermitian Swanson model with a time-dependent metric

“…Reinforcing the practical features, there are optical structures described by PTS concepts that enable unprecedented control of light [8]. At a classical level, PTS properties have also been observed in a variety of experimental set-ups, ranging from quantum optics [9] to NMR [10] and superconductivity [11].…”

Non‐Hermitian potentials and real eigenvalues

“…These fundamental results have in turn lead to a plethora of studies in all areas of physics including: a resolution of the problem of ghosts in conformal gravity theory [10], a large variety of studies extending classical mechanical systems and chaotic systems into the complex plane [11][12][13][14], as well as explorations of the deformations of wave-equations [15]. In the last four years, a P T -symmetric systems have become the objects of experimental work in optics such that P T -symmetric effects have now been observed in the lab [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].…”