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Broken and unbroken \(\varvec{\mathcal {PT}}\)-symmetric solutions of semi-discrete nonlocal nonlinear Schrödinger equation

  • Y. HanifEmail author
  • U. Saleem
Original paper
  • 10 Downloads

Abstract

In this letter, we obtain multi-soliton solutions in terms of ratio of ordinary determinants for semi-discrete nonlocal nonlinear Schrödinger (sd-NNLS) equation by employing the Darboux transformation. We construct explicit expressions of single and double soliton solutions in zero background. We obtain symmetry-broken and symmetry-unbroken soliton solutions of sd-NNLS equation by using appropriate eigenfunctions. We notice that for symmetry non-preserving case, the potential term exhibits stable structure whereas individual fields display instability. We also obtain blowup or oscillating singular-type soliton solutions for symmetry-preserving case.

Keywords

Reverse space \(\mathcal {PT}\)-symmetry Semi-discrete nonlinear Schrödinger equation Darboux transformation Multi-soliton solutions Exceptional points 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no potential conflict of interest.

References

  1. 1.
    Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)CrossRefGoogle Scholar
  2. 2.
    Sarma, A.K., Musslimani, M.A., Christodoulides, D.N.: Continuous and discrete Schrödinger systems with parity-time-symmetric nonlinearities. Phys. Rev. E 89, 052918 (2014)CrossRefGoogle Scholar
  3. 3.
    Valchev, T., Slavova, A.: Mathematics in Industry, vol. 36. Cambridge Scholars Publishing, Cambridge (2014)Google Scholar
  4. 4.
    Khare, A., Saxena, A.: Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations. J. Math. Phys. 56, 032104 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Wen, X.Y., Yan, Z., Yang, Y.: Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. Chaos 26, 063123 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Priya, N.V., Senthivelan, M., Rangarajan, G., Lakshmanan, M.: On symmetry preserving and symmetry broken bright and dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation. Phys. Lett. A 383, 15 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bender, C.M., Boettcher, S.: Real Spectra in Non-Hermitian Hamiltonians Having \(\cal{PT}\) Symmetry. Phys. Rev. Lett. 80, 5243 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Optical solitons in \(\cal{PT}\) periodic potentials. Phys. Rev. Lett. 100, 030402 (2008)CrossRefGoogle Scholar
  9. 9.
    Longhi, S.: Bloch oscillations in complex crystals with \(\cal{PT}\) symmetry. Phys. Rev. Lett. 103, 123601 (2009)CrossRefGoogle Scholar
  10. 10.
    Markum, H., Pullirsch, R., Wettig, T.: Non-Hermitian random matrix theory and lattice QCD with chemical potential. Phys. Rev. Lett. 83, 484 (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Cartarius, H., Wunner, G.: Model of a \(\cal{PT}\)-symmetric Bose–Einstein condensate in a \(\delta \)-function double-well potential. Phys. Rev. A 86, 013612 (2012)CrossRefGoogle Scholar
  12. 12.
    Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: Beam dynamics in \(\cal{PT}\) symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008)CrossRefGoogle Scholar
  13. 13.
    Guo, A., et al.: Observation of \(\cal{PT}\)-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009)CrossRefGoogle Scholar
  14. 14.
    Ruter, C.E., et al.: Observation of parity-time symmetry in optics. Nat. Phys. 6, 192 (2010)CrossRefGoogle Scholar
  15. 15.
    Kottos, T.: Broken symmetry makes light work. Nat. Phys. 6, 166 (2010)CrossRefGoogle Scholar
  16. 16.
    Giorgi, G.L.: Spontaneous \(\cal{PT}\) symmetry breaking and quantum phase transitions in dimerized spin chains. Phys. Rev. B 82, 052404 (2010)CrossRefGoogle Scholar
  17. 17.
    Regensburger, A., et al.: Parity-time synthetic photonic lattices. Nature (London) 488, 167 (2012)CrossRefGoogle Scholar
  18. 18.
    Joglekar, Y.N., Thompson, C., Scott, D.D., Vemuri, G.: Optical waveguide arrays: quantum effect and \(\cal{PT}\) symmetry breaking. Eur. Phys. J. Appl. Phys. 63, 30001 (2013)CrossRefGoogle Scholar
  19. 19.
    El-Ganainy, R., Makris, K.G., Khajavikhan, M., Musslimani, Z.H., Rotter, S., Christodoulides, D.N.: Non-Hermitian physics and \(\cal{PT}\) symmetry. Nat. Phys. 14, 11 (2018)CrossRefGoogle Scholar
  20. 20.
    Ablowitz, M.J., Ladik, J.F.: Nonlinear differential-difference equation. J. Math. Phys. 16, 598 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Davydov, A.S.: The theory of contraction of proteins under their excitation. J. Theor. Biol. 38, 559 (1973)CrossRefGoogle Scholar
  22. 22.
    Su, W.P., Schrieffer, J.R., Heeger, A.J.: Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698 (1979)CrossRefGoogle Scholar
  23. 23.
    Kenkre, V.M., Campbell, D.K.: Self-trapping on a dimer: time-dependent solutions of a discrete nonlinear Schrödinger equation. Phys. Rev. B 34, 4959 (1986)CrossRefGoogle Scholar
  24. 24.
    Papaanicoulau, N.: Complete integrability for a discrete Heisenberg chain. J. Phys. A: Math. Gen. 20, 3637 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Abdullaev, F.K., Kartashov, Y.V., Zezyulin, D.A.: Solitons in \(\cal{PT}\)-symmetric nonlinear lattices. Phys. Rev. A 83, 041805 (2011)CrossRefGoogle Scholar
  26. 26.
    Fring, A.: \(\cal{PT}\)-Symmetric deformations of integrable models. Philos. Trans. R. Soc. A 371, 20120064 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zyablovsky, A.A., Vinorgradow, A.P., Pukhov, A.A., Dorofeenko, A.V., Lisyansky, A.A.: \(\cal{PT}\)-symmetry in optics. Phys. Uspekhi 57, 1063 (2014)CrossRefGoogle Scholar
  28. 28.
    Ablowitz, M.J., Musslimani, Z.H.: Integrable discrete \(\cal{PT}\) symmetric model. Phys. Rev. Lett. E 90, 032912 (2014)CrossRefGoogle Scholar
  29. 29.
    Grahovski, G.G., Mohammed, A.J., Susanto, H.: Nonlocal reductions of the Ablowitz–Ladik equation. arXiv:1711.08419 [nlin.SI]
  30. 30.
    Mitchell, M., Segev, M., Coskun, T.H., Christodoulides, D.N.: Theory of self-trapped spatially incoherent light beams. Phys. Rev. Lett. 79, 4990 (1997)CrossRefGoogle Scholar
  31. 31.
    Ma, L.Y., Zhu, Z.N.: N-soliton solution for an integrable nonlocal focusing nonlinear Schrödinger equation. Appl. Math. Lett. 59, 115 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ablowitz, M.J., Prinari, B., Trubatch, A.D.: Discrete and Continuous Nonlinear Schrödinger Systems. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  33. 33.
    Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Germany (1991)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of the PunjabLahorePakistan

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