Numerical Method for Computation of Sliding Velocities for Vortices in Nonlocal Josephson Electrodynamics

In this paper, a model of infinite Josephson layered structure is considered. The structure consists of alternating superconducting and tunnel layers and it is assumed that (i) the electrodynamics of the structure is nonlocal and (ii) the current-phase relation is presented by sum of Fourier harmonics instead of one sinusoidal harmonic for the case of the sine-Gordon equation. The governing equation is a nonlocal generalization of the nonlinear Klein-Gordon equation with periodic nonlinearity that depends on external parameter of nonlocality λ. The velocity of vortices (2 π-kinks) in models of such kind are not arbitrary, but belong to some discrete set. The paper presents a method for computation of these velocities (called also “sliding velocities”) and the shapes of kinks. The estimation of error of the method is given. The results of computations are the families of 2 π-kinks parametrized by λ. It is observed that the 2 π-kinks corresponding to different families for the same λ have nearly the same central part but differ in asymptotics of the tails. The numerical algorithm has been incorporated into a program complex “Kink solutions” in MatLab environment. The complex enables to compute the shapes and velocities of 2 π-kinks for nonlinearities represented by sums of up to ten Fourier harmonics, as well as to model the propagation of these kinks.

Josephson junction, nonlocal Josephson electrodynamics, embedded solitons, sliding velocities, nonsinusoidal nonlinearity