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Kink movement on a periodic background

Tomasz Dobrowolski1, Jacek Gatlik2, and Panayotis G. Kevrekidis3

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Phys. Rev. E 111, 024203 – Published 7 February, 2025

DOI: https://doi.org/10.1103/PhysRevE.111.024203

Abstract

The behavior of the kink in the sine-Gordon (sG) model in the presence of periodic inhomogeneity is studied. An ansatz is proposed that allows for the construction of a reliable effective model with two degrees of freedom. Effective models featuring very good agreement with the original field-theoretic partial differential equation are constructed, including in the nonperturbative region and for relativistic velocities. The numerical solutions of the sG model describing the evolution of the kink in the presence of a barrier as well as in the case of a periodic heterogeneity under the potential additional influence of a switched bias current and/or dissipation were obtained. The results of the field equation and the effective models were favorably compared. The effect of the choice of initial conditions in the field model on the agreement of the results with the effective model is also discussed.

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References (43)

  1. N. J. Zabusky and M. D. Kruskal, Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15, 240 (1965).
  2. E. Fermi, J. Pasta, and S. Ulam, Studies of Nonlinear Problems, Technical Report LA-1940, (Los Alamos Scientific Laboratory, 1955).
  3. J. Cuevas-Maraver, P. Kevrekidis, and F. Williams, The sine-Gordon Model and its Applications, Nonlinear Systems and Complexity (Springer, Cham, 2014).
  4. J. Cuevas, L. Q. English, P. G. Kevrekidis, and M. Anderson, Discrete breathers in a forced-damped array of coupled pendula: Modeling, computation, and experiment, Phys. Rev. Lett. 102, 224101 (2009).
  5. L. Lam, Solitons in Liquid Crystals (Springer, New York, 1992), pp. 9–50.
  6. V. Zharnitsky, I. Mitkov, and M. Levi, Parametrically forced sine-Gordon equation and domain wall dynamics in ferromagnets, Phys. Rev. B 57, 5033 (1998).
  7. S. Bugaychuk, L. Kovács, G. Mandula, K. Polgár, and R. A. Rupp, Wave-mixing solitons in ferroelectric crystals, Radiat. Eff. Defects Solids 157, 995 (2002).
  8. B. A. Malomed, The sine-Gordon Model: General Background, Physical Motivations, Inverse Scattering, and Solitons, in The Sine-Gordon Model and its Applications: From Pendula and Josephson Junctions to Gravity and High-Energy Physics, edited by J. Cuevas-Maraver, P. G. Kevrekidis, and F. Williams, (Springer International Publishing, Cham, 2014), pp. 1–30.
  9. A. I. Braginski, Superconductor electronics: Status and outlook, J. Supercond. Novel Magn. 32, 23 (2019).
  10. R. Monaco, Engineering double-well potentials with variable-width annular Josephson tunnel junctions, J. Phys.: Condens. Matter 28, 445702 (2016).
  11. A. Benabdallah, J. G. Caputo, and A. C. Scott, Exponentially tapered Josephson flux-flow oscillator, Phys. Rev. B 54, 16139 (1996).
  12. A. Kemp, A. Wallraff, and A. Ustinov, Josephson vortex qubit: Design, preparation and read-out, Phys. Status Solidi B 233, 472 (2002).
  13. D. R. Gulevich and F. V. Kusmartsev, Flux cloning in Josephson transmission lines, Phys. Rev. Lett. 97, 017004 (2006).
  14. T. Dobrowolski, Curved Josephson junction, Ann. Phys. 327, 1336 (2012).
  15. J. Gatlik and T. Dobrowolski, Modeling kink dynamics in the sine–Gordon model with position dependent dispersive term, Physica D 428, 133061 (2021).
  16. E. Ekomasov, R. Murtazin, and V. Nazarov, Excitation of magnetic inhomogeneities in three-layer ferromagnetic structure with different parameters of the magnetic anisotropy and exchange, J. Magn. Magn. Mater. 385, 217 (2015).
  17. E. Ekomasov, R. Murtazin, O. Bogomazova, and V. Nazarov, Excitation and dynamics of domain walls in three-layer ferromagnetic structure with different parameters of the magnetic anisotropy and exchange, in Phase Transitions, Critical and Nonlinear Phenomena in Condensed Matter, Materials Science Forum Vol. 845 (Trans Tech Publications Ltd., Switzerland, 2016), pp. 195–198.
  18. A. M. Gumerov, E. G. Ekomasov, and R. V. Kudryavtsev, One-dimensional dynamics of magnetic inhomogeneities in a three- and five-layer ferromagnetic structure with different values of the magnetic parameters, J. Phys.: Conf. Ser. 1389, 012004 (2019).
  19. J. Gatlik and T. Dobrowolski, The impact of thermal noise on kink propagation through a heterogeneous system, Physica D 445, 133649 (2023).
  20. D. R. Gulevich and F. V. Kusmartsev, New phenomena in long Josephson junctions, Supercond. Sci. Technol. 20, S60 (2007).
  21. C. Gorria, Y. B. Gaididei, M. P. Soerensen, P. L. Christiansen, and J. G. Caputo, Kink propagation and trapping in a two-dimensional curved Josephson junction, Phys. Rev. B 69, 134506 (2004).
  22. E. Goldobin, A. Sterck, and D. Koelle, Josephson vortex in a ratchet potential: Theory, Phys. Rev. E 63, 031111 (2001).
  23. G. Carapella, G. Costabile, N. Martucciello, M. Cirillo, R. Latempa, A. Polcari, and G. Filatrella, Experimental realization of a relativistic fluxon ratchet, Physica C: Supercond. 382, 337 (2002).
  24. A. Kemp, A. Wallraff, and A. Ustinov, Testing a state preparation and read-out protocol for the vortex qubit, Physica C: Supercond. 368, 324 (2002).
  25. D. Gulevich, M. Gaifullin, O. Kusmartseva, F. Kusmartsev, and K. Hirata, Josephson fluxon pump: Theoretical aspects and experimental implementation of elementary flux quanta generator with bscco, Physica C: Superconductivity 468, 1903 (2008).
  26. A. Sánchez, R. Scharf, A. R. Bishop, and L. Vázquez, Sine-Gordon breathers on spatially periodic potentials, Phys. Rev. A 45, 6031 (1992).
  27. R. Scharf, Y. S. Kivshar, A. Sánchez, and A. R. Bishop, Sine-Gordon kink-antikink generation on spatially periodic potentials, Phys. Rev. A 45, R5369 (1992).
  28. D. W. McLaughlin and A. C. Scott, Perturbation analysis of fluxon dynamics, Phys. Rev. A 18, 1652 (1978).
  29. B. A. Malomed, Dynamics of a fluxon in a long Josephson junction with a periodic lattice of inhomogeneities, Phys. Rev. B 38, 9242 (1988).
  30. G. Mkrtychyan and V. Shmidt, On the radiation from inhomogeneous Josephson junction, Solid State Commun. 30, 783 (1979).
  31. I. Serpuchenko and A. Ustinov, Experimental-observation of fine-structure on the current-voltage characteristics of long Josephson junctions with a lattice of inhomogeneities, JETP Lett. 46, 549 (1987).
  32. Y. S. Kivshar and B. A. Malomed, Interaction of a fluxon with a local inhomogeneity in a long Josephson junction, Phys. Lett. A 129, 443 (1988).
  33. C. R. Galley, Classical mechanics of nonconservative systems, Phys. Rev. Lett. 110, 174301 (2013).
  34. P. G. Kevrekidis, Variational method for nonconservative field theories: Formulation and two PT-symmetric case examples, Phys. Rev. A 89, 010102(R) (2014).
  35. E. W. Weisstein, Adams' method.” From MathWorld–A Wolfram Web Resource. In our study, we employed Mathematica's advanced numerical solvers to integrate the field equations. For the numerical method, we selected the Adams algorithm, while also constraining the maximum step size (to 0.05) to maintain precise results throughout the computation, https://mathworld.wolfram.com/AdamsMethod.html.
  36. T. Dobrowolski and A. Jarmoliński, Perturbation scheme for a fluxon in a curved Josephson junction, Phys. Rev. E 96, 012214 (2017).
  37. J. Gatlik, T. Dobrowolski, and P. G. Kevrekidis, Kink-inhomogeneity interaction in the sine-Gordon model, Phys. Rev. E 108, 034203 (2023).
  38. N. Balmforth, R. Craster, and P. Kevrekidis, Being stable and discrete, Physica D 135, 212 (2000).
  39. B. A. Malomed and M. I. Tribelsky, Emission of waves by moving kinks in a spatially modulated sine-Gordon system, Phys. Rev. B 41, 11271 (1990).
  40. J. Gatlik, T. Dobrowolski, and P. G. Kevrekidis, Effective description of the impact of inhomogeneities on the movement of the kink front in 2+1 dimensions, Phys. Rev. E 109, 024205 (2024).
  41. J. Cuevas-Maraver A Dynamical Perspective on the ϕ4 Model, 1st, ed., Nonlinear Systems and Complexity, edited by P. G. Kevrekidis and J. Cuevas-Maraver (Springer International Publishing, Cham, Switzerland, 2019).
  42. M. Haataja, D. J. Srolovitz, and I. G. Kevrekidis, Apparent hysteresis in a driven system with self-organized drag, Phys. Rev. Lett. 92, 160603 (2004).
  43. M. D. Angelis and G. Fiore, Existence and uniqueness of solutions of a class of third order dissipative problems with various boundary conditions describing the josephson effect, J. Math. Anal. Appl. 404, 477 (2013).

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Phys. Rev. E 111, 024203– Published 7 February, 2025
  • Received 20 September 2024
  • Accepted 16 January 2025
Article part of CHORUS
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