\documentclass[12pt]{article} \usepackage{amstex} \pagestyle{empty} \textwidth=156mm \textheight=239mm \oddsidemargin=5mm \topmargin=-10mm %\title{Numerical Investigation of Bifurcations in Inhomogeneous Josephson Junctions} \title{A Free Boundary Value Problem for Long Inhomogeneous Josephson Junctions} \bigskip \author{ M. D. Todorov\\ \small\it Institute of Applied Mathematics and Informatics\\[-1.mm] \small\it Technical University of Sofia \\ \\ T. L. Boyadjiev\\ \small\it Faculty of Mathematics and Informatics\\[-1.mm] \small\it Sofia University 'St.Kl.Ohridski' } \date{ } \begin{document} \maketitle \bigskip A method for calculating the bifurcations of a magnetic field in long inhomogeneous Josephson junctions is proposed. Since the length of the junctions is a variable quantity the nonlinear spectral problem as a problem with free boundary is interpreted. We solve the following differential system consisting of the perturbed Sin--Gordon equation, stability condition (Sturm--Liouville problem) both with boundary conditions of Neumann type and norm condition for the eigenfunctions, namely: \begin{eqnarray*} &&-\varphi^{''}+j_D(x) \sin \varphi + \gamma =0\\ &&\varphi^{\prime}(-R)=\varphi^{\prime}(R)=h_B\\ &&-\psi^{''}+\left(j_D(x) \cos \varphi -\lambda\right) \psi =0\\ &&\psi^{\prime}(-R)=\psi^{\prime}(R)=0\\ &&\int_{-R}^R \psi^2 dx = 1\>, \end{eqnarray*} \noindent where $\varphi$ is magnetic flux, $j_D$ - the amplitude of the Josephson junction, $\gamma$ - the external current, $h_B$ - the boundary magnitude of magnetic field. The above system is linearized appropriately after that a continuous analogue of the Newton method with optimal iterative step and a spline scheme of fourth order of approximation on an uniform mesh is used. \end{document} 