Numerical modeling of long Josephson junctions in the frame of double sine-gordon equation

The aim of this work is the mathematical modeling of static distributions of the magnetic flux in long Josephson junctions (JJ), taking into account the second harmonic in the decomposition of the Josephson current and the sequential comparison of the results with the conventional model. For the analysis of stability, each concrete distribution of magnetic current in the junction is put into relationship with the Sturm-Liouville spectral problem; the nullification of its minimal eigenvalue indicates the bifurcation distribution by one of the problem parameters. The corresponding nonlinear boundary problem is numerically solved by a continuous analog of the Newton’s method with the spline-collocation scheme for linearized problems at each Newtonian iteration. The main distributions of the magnetic flux have been found and their stability against changes of model parameters investigated. The obtained results were compared with the results of the JJ conventional model of the superconductor-insulator-superconductor type.