Numerical Solution of a Class of Boundary Value Problems Arising in the Physics of Josephson Junctions

In this paper we propose a method of numerical solution of non-linear boundary value problems for systems of ODEs given on the embedded intervals. The algorithm is based on the continuous analog of Newton method coupled with spline-collocation scheme of fourth order of accuracy. Demonstrative examples of similar problems take place in physics of stacked Jo-sephson junctions with different layers lengths. As a concrete example we consider the problem of calculating the possible distributions of magnetic ux in a system of two magnetically coupled long Josephson junctions. The innuence of length ratio on the main physical properties of basic bound states is investigated numerically. The existence of bifurcations by change of the lengths of the layers for some couples of solutions has been proved.