Nonlinear nonlocal diffusion of magnetic flux in thin type-II superconductors and Josephson junction arrays: Exact solutions

TL;DR

This paper presents an exact solution to a nonlinear nonlocal diffusion problem modeling magnetic flux evolution in thin type-II superconductors and Josephson junction arrays, revealing self-similar flux distributions with sharp fronts.

Contribution

It provides the first exact analytical solution for the nonlinear nonlocal diffusion of magnetic flux in these superconducting systems, including novel flux distribution profiles.

Findings

Self-similar flux distributions with square-root fronts

Fronts expand with power law in time

Sharp peak appears in hard superconductor flux profiles

Abstract

An exact solution of the nonlinear nonlocal diffusion problem is obtained that describes the evolution of the magnetic flux injected into a soft or hard type-II superconductor film or a two-dimensional Josephson junction array. (The magnetic field in vortices is assumed to be perpendicular to the film; the electric field induced by the vortex motion is proportional to the local magnetic induction; flux creep in the hard superconductors under consideration is described by the logarithmic U(j) dependence.) Self-similar flux distributions with sharp square-root fronts are found. The fronts are shown to expand with power law time-dependence. A sharp peak in the middle of the distribution appears in the hard superconductor case.