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Advances in Mathematical Physics
Volume 2016, Article ID 9514230, 12 pages
http://dx.doi.org/10.1155/2016/9514230
Research Article

Bright Solitons in a -Symmetric Chain of Dimers

1Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, UK
2Department of Mathematics, University of Indonesia, Depok, Indonesia

Received 17 July 2016; Revised 17 September 2016; Accepted 17 October 2016

Academic Editor: Yao-Zhong Zhang

Copyright © 2016 Omar B. Kirikchi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. N. Moiseyev, Non-Hermitian Quantum Mechanics, Cambridge University Press, Cambridge, UK, 2011.
  2. C. M. Bender and S. Boettcher, “Real spectra in non-hermitian hamiltonians having PT symmetry,” Physical Review Letters, vol. 80, no. 24, p. 5243, 1998. View at Publisher · View at Google Scholar
  3. C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” Journal of Mathematical Physics, vol. 40, no. 5, pp. 2201–2229, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Reports on Progress in Physics, vol. 70, no. 6, p. 947, 2007. View at Publisher · View at Google Scholar
  5. A. Guo, G. J. Salamo, D. Duchesne et al., “Observation of PT-symmetry breaking in complex optical potentials,” Physical Review Letters, vol. 103, no. 9, Article ID 093902, 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Physics, vol. 6, no. 3, pp. 192–195, 2010. View at Publisher · View at Google Scholar · View at Scopus
  7. S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Yu. S. Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser & Photonics Reviews, vol. 10, no. 2, pp. 177–213, 2016. View at Publisher · View at Google Scholar
  8. V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Reviews of Modern Physics, vol. 88, no. 3, Article ID 035002, 2016. View at Publisher · View at Google Scholar
  9. J. Pickton and H. Susanto, “Integrability of PT-symmetric dimers,” Physical Review A, vol. 88, no. 6, Article ID 063840, 2013. View at Publisher · View at Google Scholar
  10. P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, “Nonlinear dynamics in PT-symmetric lattices,” Journal of Physics. A. Mathematical and Theoretical, vol. 46, no. 36, Article ID 365201, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. I. V. Barashenkov, G. S. Jackson, and S. Flach, “Blow-up regimes in the PT -symmetric coupler and the actively coupled dimer,” Physical Review A, vol. 88, no. 5, Article ID 053817, 2013. View at Publisher · View at Google Scholar
  12. S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, “Solitons in a chain of parity-time-invariant dimers,” Physical Review E, vol. 84, no. 4, Article ID 046609, 2011. View at Publisher · View at Google Scholar
  13. P. L. Chu, B. A. Malomed, and G. D. Peng, “Soliton switching and propagation in nonlinear fiber couplers: analytical results,” Journal of the Optical Society of America B, vol. 10, no. 8, p. 1379, 1993. View at Publisher · View at Google Scholar
  14. E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers,” Physical Review A, vol. 40, no. 8, pp. 4455–4466, 1989. View at Publisher · View at Google Scholar · View at Scopus
  15. N. Akhmediev and J. M. Soto-Crespo, “Propagation dynamics of ultrashort pulses in nonlinear fiber couplers,” Physical Review E, vol. 49, no. 5, pp. 4519–4529, 1994. View at Publisher · View at Google Scholar · View at Scopus
  16. B. A. Malomed, I. M. Skinner, P. L. Chu, and G. D. Peng, “Symmetric and asymmetric solitons in twin-core nonlinear optical fibers,” Physical Review E, vol. 53, no. 4, pp. 4084–4091, 1996. View at Google Scholar · View at Scopus
  17. R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Optics Letters, vol. 36, no. 22, pp. 4323–4325, 2011. View at Publisher · View at Google Scholar · View at Scopus
  18. F. K. Abdullaev, V. V. Konotop, M. Ögren, and M. P. Sørensen, “Zeno effect and switching of solitons in nonlinear couplers,” Optics Letters, vol. 36, no. 23, pp. 4566–4568, 2011. View at Publisher · View at Google Scholar · View at Scopus
  19. R. Driben and B. A. Malomed, “Stabilization of solitons in PT models with supersymmetry by periodic management,” EPL (Europhysics Letters), vol. 96, no. 5, Article ID 51001, 2011. View at Publisher · View at Google Scholar
  20. N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Physical Review A, vol. 85, no. 6, Article ID 063837, 2012. View at Publisher · View at Google Scholar
  21. I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, “Breathers in PT-symmetric optical couplers,” Physical Review A, vol. 86, no. 5, Article ID 053809, 2012. View at Publisher · View at Google Scholar
  22. P. Li, L. Li, and B. A. Malomed, “Multisoliton Newton's cradles and supersolitons in regular and parity-time-symmetric nonlinear couplers,” Physical Review E, vol. 89, Article ID 062926, 2014. View at Publisher · View at Google Scholar
  23. J. D'Ambroise, P. G. Kevrekidis, and B. A. Malomed, “Staggered parity-time-symmetric ladders with cubic nonlinearity,” Physical Review E, vol. 91, no. 3, Article ID 033207, 11 pages, 2015. View at Publisher · View at Google Scholar
  24. G. Herring, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Frantzeskakis, “Symmetry breaking in linearly coupled dynamical lattices,” Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, vol. 76, no. 6, Article ID 066606, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. A. Chernyavsky and D. Pelinovsky, “Breathers in hamiltonian PT -symmetric chains of coupled pendula under a resonant periodic force,” Symmetry, vol. 8, no. 7, p. 59, 2016. View at Publisher · View at Google Scholar
  26. C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel, “Observation of PT phase transition in a simple mechanical system,” American Journal of Physics, vol. 81, no. 3, pp. 173–179, 2013. View at Publisher · View at Google Scholar · View at Scopus
  27. A. Chernyavsky and D. E. Pelinovsky, “Long-time stability of breathers in Hamiltonian PT-symmetric lattices,” https://arxiv.org/abs/1606.02333.
  28. R. S. MacKay and S. Aubry, “Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators,” Nonlinearity, vol. 7, no. 6, pp. 1623–1643, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. V. V. Konotop, D. E. Pelinovsky, and D. A. Zezyulin, “Discrete solitons in PT-symmetric lattices,” EPL, vol. 100, no. 5, Article ID 56006, 2012. View at Publisher · View at Google Scholar · View at Scopus
  30. D. E. Pelinovsky, D. A. Zezyulin, and V. V. Konotop, “Nonlinear modes in a generalized PT-symmetric discrete nonlinear Schrödinger equation,” Journal of Physics A: Mathematical and Theoretical, vol. 47, no. 8, Article ID 085204, 2014. View at Publisher · View at Google Scholar
  31. K. Li and P. G. Kevrekidis, “PT-symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics-symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics,” Physical Review E, vol. 83, no. 6, Article ID 066608, 2011. View at Publisher · View at Google Scholar
  32. Z. Chen, J. Liu, S. Fu, Y. Li, and B. A. Malomed, “Discrete solitons and vortices on two-dimensional lattices of PT—symmetric couplers,” Optics Express, vol. 22, no. 24, pp. 29679–29692, 2014. View at Publisher · View at Google Scholar · View at Scopus