Skip to main content
SpringerLink
Log in

Emergence and Decay of π-Kinks in the Sine-Gordon Model with High-Frequency Pumping

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this paper, we consider the sine-Gordon equation with a high-frequency parametric pumping and a weak dissipative force. We examine the class of π-kink-type solutions that are soliton solutions of the nonperturbed sine-Gordon equation. In contrast to stable 2π-kinks, these solutions are unstable. We prove that the time of decaying of π-kinks due to small perturbations is proportional to the cube of the inverse period of fast oscillations of the parametric pumping. We derive a two-time asymptotic expansion of a solution of the boundary-value problem and analyze evolution of a wave packet whose leading term has the form of a π-kink. Numerical simulations of solutions confirm the good qualitative agreement with asymptotic expansions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. N. Bogolyubov and Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Fizmatlit, Moscow (1958).

  2. L. N. Bulaevskii and A. E. Koshelev, “Radiation due to Josephson oscillations in layered superconductors,” Phys. Rev. Lett., 99, 057002 (2007).

    Article  Google Scholar 

  3. E. I. Butikov, “An improved criterion for Kapitzas pendulum stability,” J. Phys. A: Math. Theor., 44, 295202 (2011).

    Article  MathSciNet  Google Scholar 

  4. B. V. Chirikov, Nonlinear Resonance [in Russian], Novosibirsk (1977).

  5. S. G. Glebov, O. M. Kiselev, and N. Tarkhanov, Nonlinear Equations with Small Parameter, De Gruyter, Berlin–New York (2017).

    MATH  Google Scholar 

  6. X. Hu and S. Lin, “Three-dimensional phase-kink state in a thick stack of Josephson junctions and terahertz radiation,” Phys. Rev. B, 78, No. 13, 134510 (2008).

    Article  Google Scholar 

  7. P. L. Kapitsa, “Dynamic stability of a pendulum with an oscillating suspension point,” Zh. Eksper. Teor. Fiz., 21, 588–597 (1951).

    MathSciNet  Google Scholar 

  8. O. M. Kiselev, “Oscillations around a separatrix in the Duffing equation,” Tr. Inst. Mat. Mekh. Ufim. Nauch. Tsenr. Ross. Akad. Nauk, 18, No. 2, 141–153 (2012).

    Google Scholar 

  9. O. M. Kiselev and V. Yu. Novokshenov, “Autoresonance in a model of the generator of terahertz waves,” Tr. Inst. Mat. Mekh. Ufim. Nauch. Tsenr. Ross. Akad. Nauk, 23, No. 2, 117–132 (2017).

    Google Scholar 

  10. Yu. S. Kivshar, N. Gronbech-Jensen, M. R. Samuelsen, “Pi-kinks in a parametrically driven sine-Gordon chain,” Phys. Rev. B, 45, 7789–7794 (1992).

    Article  Google Scholar 

  11. N. M. Krylov and N. N. Bogolyubov, Introduction to Nonlinear Mechanics [in Russian], Kiev (1937).

  12. G. A. Kuzmak, “Asymptotic solutions of second-order nonlinear differential equations with variable coefficients,” Prik. Mat. Mekh., 23, 515–526 (1959).

    MathSciNet  MATH  Google Scholar 

  13. L. D. Landau and E. M. Lifshits, Mechanics [in Russian], Nauka, Moscow (1988).

  14. J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Dunod, Paris (1968).

    MATH  Google Scholar 

  15. L. S. Revin and A. L. Pankratov, “Spectral and power properties of inline long Josephson junctions,” Phys. Rev. B, 86, 054501 (2012).

    Article  Google Scholar 

  16. A. M. Samoylenko and I. V. Polesya, “Existence of separatrix surfaces of systems in the standard form,” Differ. Uravn., 11, No. 10, 1827-1831 (1975).

    MathSciNet  Google Scholar 

  17. A. S. Sobolev, A. L. Pankratov, and J. Mygindc, “Numerical simulation of the self-pumped long Josephson junction using a modified sine-Gordon model,” Phys. C, 435, 112–113 (2006).

    Article  Google Scholar 

  18. V. F. Zaitsev and A. D. Polyanin, Reference Book on First-Order Partial Differential Equations [in Russian], Fizmatlit, Moscow (2003).

  19. V. Zharnitsky, I. Mitkov, and N. Gronbech-Jensen, “π-Kinks in strongly ac driven sine-Gordon systems,” Phys. Rev. E, 58, R52–R55 (1998).

    Article  Google Scholar 

  20. V. Zharnitsky, I. Mitkov, and M. Levi, “Parametrically forced sine-Gordon equation and domain walls dynamics in ferromagnets,” Phys. Rev. B, 57, 5033 (1998).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics with Computing Center, Ufa Federal Research Center of the Russian Academy of Sciences, Ufa, Russia

    O. M. Kiselev & V. Yu. Novokshenov

Authors
  1. O. M. Kiselev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Yu. Novokshenov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. M. Kiselev.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 152, Mathematical Physics, 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kiselev, O.M., Novokshenov, V.Y. Emergence and Decay of π-Kinks in the Sine-Gordon Model with High-Frequency Pumping. J Math Sci 252, 175–189 (2021). https://doi.org/10.1007/s10958-020-05152-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-020-05152-x

Keywords and phrases

AMS Subject Classification

Search

Navigation