Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8567Research ArticleNumerical Method for Computation of Sliding Velocities for Vortices in Nonlocal Josephson ElectrodynamicsMedvedevaE VDepartment of Higher Mathematics-1elinamedvedeva87@gmail.comNational Research University of Electronic Technology150120151455208092016Copyright © 2015,2015In 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 junctionnonlocal Josephson electrodynamicsembedded solitonssliding velocitiesnonsinusoidal nonlinearityджозефсоновский переходнелокальная джозефсоновская электродинамикавложенные солитоныскорости скольжениянесинусоидальная нелинейность[Abdumalikov A.A., Alfimov G.L., Malishevskii A.S. Nonlocal Electrodynamics of Josephson Vortices in Superconducting Circuits, Superconductor Science and Technology 22 (2), art 023001.][Alfimov G.L., Eleonsky V.M., Lerman L.M. Solitary Wave Solutions of Nonlocal Sine-Gordon Equations, Chaos 8 (1) (1998) 257-271.][Aliev Y.M., Ovchinnikov K.N., Silin V.P. Nonlocal Josephson Electrodynamics of Layered Structures, Journal of Experimental and Theoretical Physics 80 (3) (1995) 551-559.][Savel’ev S., Yampol’skii V.A., Rakhmanov A.L., Nori F. Terahertz Josephson Plasma Waves in Layered Superconductors: Spectrum, Generation, Nonlinear and Quantum Phenomena, Reports on Progress in Physics 73 (2), art 026501.][Alfimov G.L., Malishevskii A.V., Medvedeva E.V. Discrete Set of Kink Velocities in Josephson Structures: the Nonlocal Double Sine-Gordon Model, Physica D 282 (2014) 16-26.][Golubov A.A., Kupriyanov M.Y., Il’ichev E. The Current-Phase Relation in Josephson Junctions, Reviews of Modern Physics 76 (2) (2004) 411-469.][Atanasova P.K., Boyadjiev T.L., Shukrinov Y.M., Zemlyanaya E.V. Numerical Modeling of Long Josephson Junctions in the Frame of Double Sine-Gordon Equation, Mathematical Models and Computer Simulations 3 (2011) 389-398.][Alfimov G.L., Medvedeva E.V. Moving Nonradiating Kinks in Nonlocal f4 and f4 - f6 Models, Physical Review E 84 (5), art 056606.][Kalitkin N.N., Koryakin P.V. Numerical Methods, Book 2, Methods of Mathematical Physics, Academia, Moscow, 2013, in Russian.]