Bifurcations in Long Josephson Junctions with Second Harmonic in the Current-Phase Relation: Numerical Study

  • Pavlina Atanasova
  • Elena Zemlyanaya
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)


Critical regimes in the long Josephson junction (LJJ) are studied within the frame of a model accounting the second harmonic in the current-phase relation (CPR). Numerical approach is shown to provide a good agreement with analytic results. Numerical results are presented to demonstrate the availabilities and advantages of the numerical scheme for investigation of bifurcations and properties of the magnetic flux distributions in dependence on the sign and value of the second harmonic in CPR.


Long Josephson junction double sine-Gordon equation continuous analogue of Newton’s method numerical continuation stability bifurcations 


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  1. 1.
    Atanasova, P., Kh., B.T.L., Dimova, S.N.: Numerical Modeling of Critical Dependence for Symmetric Two-Layer Josephson Junctions. Comput. Maths. and Math. Phys. 46(4), 666–679 (2006)CrossRefGoogle Scholar
  2. 2.
    Atanasova, P.K., Boyadjiev, T.L., Shukrinov, Y.M., Zemlyanaya, E.V., Seidel, P.: Influence of Josephson current second harmonic on stability of magnetic flux in long junctions. J. Phys. Conf. Ser. 248, 012044 (2010), arXiv:1007.4778v1Google Scholar
  3. 3.
    Atanasova, P.K., Boyadjiev, T.L., Zemlyanaya, E.V., Shukrinov, Y.M.: Numerical study of magnetic flux in the LJJ model with double sine-gordon equation. In: Dimov, I., Dimova, S., Kolkovska, N. (eds.) NMA 2010. LNCS, vol. 6046, pp. 347–352. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Atanasova, P.K., Boyadjiev, T.L., Shukrinov, Y.M., Zemlyanaya, E.V.: Numerical investigation of the second harmonic effects in the LJJ. In: Koleva, M., Vulkov, L. (eds.) Proc. the FDM 2010 Conf., Lozenetz, Bulgaria, pp. 1–8 (July 2010), arXiv:1005.5691v1Google Scholar
  5. 5.
    Atanasova, P.K., Zemlyanaya, E., Shukrinov, Y.: Numerical study of fluxon solutions of sine-gordon equation under the influence of the boundary conditions. In: Adam, G., Buša, J., Hnatič, M. (eds.) MMCP 2011. LNCS, vol. 7125, pp. 201–206. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Atanasova, P.K., Zemlyanaya, E.V., Boyadjiev, T.L., Shukrinov, Y.M.: Numerical modeling of long Josephson junctions in the frame of double sine-Gordon equation. Mathematical Models and Computer Simulations 3(3), 388–397 (2011)CrossRefGoogle Scholar
  7. 7.
    Buzdin, A., Koshelev, A.E.: Periodic alternating 0-and π-junction structures as realization of ϕ-Josephson junctions. Phys. Rev. B. 67, 220504(R) (2003)Google Scholar
  8. 8.
    Galpern, Y.S., Filippov, A.T.: Joint solution states in inhomogeneous Josephson junctions. Sov. Phys. JETP 59, 894 (1984) (Russian)Google Scholar
  9. 9.
    Goldobin, E., Koelle, D., Kleiner, R., Buzdin, A.: Josephson junctions with second harmonic in the current-phase relation: Properties of junctions. Phys. Rev. B. 76, 224523 (2007)CrossRefGoogle Scholar
  10. 10.
    Goldobin, E., Koelle, D., Kleiner, R., Mints, R.G.: Josephson junction with magnetic-field tunable ground state. Phys. Rev. Lett. 107, 227001 (2011)CrossRefGoogle Scholar
  11. 11.
    Hatakenaka, N., Takayanag, H., Kasai, Y., Tanda, S.: Double sine-Gordon fluxons in isolated long Josephson junction. Physica B. 284-288, 563–564 (2000)CrossRefGoogle Scholar
  12. 12.
    Likharev, K.K.: Introduction in Josephson junction dynamics, M. Nauka, GRFML (1985) (Russian)Google Scholar
  13. 13.
    Puzynin, I.V., et al.: Methods of Computational Physics for Investigation of Models of Complex Physical Systems. Physics of Particles and Nuclei 38(1), 70–116 (2007)CrossRefGoogle Scholar
  14. 14.
    Ryazanov, V.V., et al.: Coupling of two superconductors through a ferromagnet: evidence for a pi junction. Phys. Rev. Lett. 36, 2427–2430 (2001)CrossRefGoogle Scholar
  15. 15.
    Ermakov, V.V., Kalitkin, N.N.: The optimal step and regularisation for Newton’s method, Comput. Maths. and Math. Phys. 21(2), 235–242 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhanlav, T., Puzynin, I.V.: On iterations convergence on the base of continuous analog of Newton’s method. Comput. Maths. and Math. Phys. 32(6), 729–737 (1992)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Zemlyanaya, E.V., Barashenkov, I.V.: Numerical study of the multisoliton complexes in the damped-driven NLS. Math. Modelling 16(3), 3–14 (2004) (Russian)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Pavlina Atanasova
    • 1
  • Elena Zemlyanaya
    • 2
  1. 1.FMIUniversity of PlovdivPlovdivBulgaria
  2. 2.Laboratory of Information TechnologiesJoint Institute for Nuclear ResearchMoscow RegionRussia

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