Numerical Investigation of a Bifurcation Problem with free Boundaries Arising from the Physics of Josephson Junctions

TL;DR

This paper addresses the computation of the minimal half-length $R_{\min}$ of a long Josephson junction that still admits a nontrivial stable magnetic-flux distribution, formulated as a nonlinear boundary-value problem with a free boundary. The authors recast the coupled BVP/SLP stability problem into a nonlinear eigenvalue problem for the half-length $R$ using a Landau-type transformation, and solve it via the continuous analogue of Newton's method, decomposing into four linear two-point BVPs solved with a spline-difference scheme of second-order accuracy. They demonstrate numerical convergence and provide quantitative results for $R_{\min}$ in both homogeneous and inhomogeneous junctions, showing that increasing the boundary magnetic field $h_B$ lowers the critical length required for a stable main fluxon. The method yields direct bifurcation curves in parameter space and is presented as a versatile tool that can be extended to other nonlinear free-boundary problems in applied physics. Overall, the work offers a rigorous numerical framework to identify stability boundaries and practical limits for device dimensions in Josephson-junction systems.

Abstract

A direct method for calculating the minimal length of ``one-dimensional'' Josephson junctions is proposed, in which the specific distribution of the magnetic flux retains its stability. Since the length of the junctions is a variable quantity, the corresponding nonlinear spectral problem as a problem with free boundaries is interpreted. The obtained results give us warranty to consider as ``long'', every Josephson junction in which there exists at least one nontrivial stable distribution of the magnetic flux for fixed values of all other parameters.