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

The aim of this work is a mathematical modeling of the static magnetic flux distributions in long Josephson junctions (JJ) taking into account the second harmonic in the Fourier-decomposition of the Josephson current. Stability analysis is based on numerical solution of a spectral Sturm–Liouville problem formulated for each distribution. In this approach the nullification of the minimal eigenvalue of this problem indicates a bifurcation point in one of parameters. At each step of numerical continuation in parameters of the model, the corresponding nonlinear boundary problem is solved on the basis of the continuous analog of Newton's method with the spline-collocation discretization of linearized problems at Newtonian iterations. Main solutions of the double sine-Gordon equation have been found. Stability of magnetic flux distributions has been investigated. Numerical results are compared with the results of the standard JJ model.