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Stability of Gaussian-type soliton in the cubic–quintic nonlinear media with fourth-order diffraction and \(\mathcal {PT}\)-symmetric potentials

  • Camus Gaston Latchio TiofackEmail author
  • Nathan Nkouessi Tchepemen
  • Alidou Mohamadou
  • Timoléon Crépin Kofané
Original paper
  • 14 Downloads

Abstract

We report on the existence and stability of Gaussian-type soliton in the nonlinear Schrödinger (NLS) equation with interplay of cubic–quintic nonlinearity, fourth-order diffraction (FOD) and novel quartic anharmonic parity-time (\(\mathcal {PT}\))-symmetric Gaussian potential. We study numerically the impact of the FOD coefficient on the regions of unbroken/broken linear \(\mathcal {PT}\)-symmetric phases. In the nonlinear domain, we derive exact soliton solutions of the one-dimensional and two-dimensional cubic–quintic NLS equation with \(\mathcal {PT}\)-symmetric Gaussian potential and FOD coefficients. Moreover, the stability of the constructed soliton solution is investigated. The results of linear stability analysis are validated by comparison with numerical simulations. Furthermore, we also show that the relative strength of the FOD coefficient influences the direction of the power flow.

Keywords

Fourth-order diffraction Cubic–quintic nonlinearities \(\mathcal {PT}\)-symmetric quartic Gaussian potential Gaussian soliton Stability 

Notes

Acknowledgements

C. G. L. Tiofack acknowledges the support of the “Laboratoire d’Excellence CEMPI: Centre Européen pour les Mathématiques, la Physique et leurs Interactions.” The authors would like to thank the anonymous reviewers for their useful and valuable comments and suggestions.

References

  1. 1.
    Bender, C.M., Boettcher, S.: Real spectra in non-hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bender, C.M., Brody, D.C., Jones, H.F.: Complex extension of quantum mechanics. Phys. Rev. Lett. 89, 270401 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chong, Y.D., Ge, L., Cao, H., Stone, A.D.: Coherent perfect absorbers: time-reversed lasers. Phys. Rev. Lett. 105, 053901 (2011)CrossRefGoogle Scholar
  4. 4.
    Wan, W., Chong, Y., Ge, L., Noh, H., Stone, A.G., Cao, H.: Time-reversed lasing and interferometric control of absorption. Science 331, 889–892 (2011)CrossRefGoogle Scholar
  5. 5.
    Sun, Y., Tan, W., Li, H.Q., Li, J., Chen, H.: Experimental demonstration of a coherent perfect absorber with PT phase transition. Phys. Rev. Lett. 112, 143903 (2014)CrossRefGoogle Scholar
  6. 6.
    Guo, A., Salamo, G.J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G.A., Christodoulides, D.N.: Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009)CrossRefGoogle Scholar
  7. 7.
    Longhi, S.: Bloch oscillations in complex crystals with PT symmetry. Phys. Rev. Lett. 103, 123601 (2009)CrossRefGoogle Scholar
  8. 8.
    Lin, Z., Ramezani, H., Eichelkraut, T., Kottos, T., Cao, H., Christodoulides, D.N.: Unidirectional invisibility induced by PT-symmetric periodic structures. Phys. Rev. Lett. 106, 213901 (2011)CrossRefGoogle Scholar
  9. 9.
    Castaldi, G., Savoia, S., Galdi, V., Alu, A., Engheta, N.: PT metamaterials via complex-coordinate transformation optics. Phys. Rev. Lett. 110, 173901 (2013)CrossRefGoogle Scholar
  10. 10.
    Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100, 030402 (2008)CrossRefGoogle Scholar
  11. 11.
    Shi, Z., Jiang, X., Zhu, X., Li, H.: Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials. Phys. Rev. A 84, 053855 (2011)CrossRefGoogle Scholar
  12. 12.
    Achilleos, V., Kevrekidis, P.G., Frantzeskakis, D.J., Carretero-González, R.: Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions. Phys. Rev. A 86, 013808 (2012)CrossRefGoogle Scholar
  13. 13.
    Kartashov, Y.V.: Vector solitons in parity-time-symmetric lattices. Opt. Lett. 38, 2600–2603 (2013)CrossRefGoogle Scholar
  14. 14.
    Abdullaev, K.K., Kartashov, Y.V., Konotop, V.V., Zezyulin, D.A.: Solitons in PT-symmetric nonlinear lattices. Phys. Rev. A 83, 041805(R) (2011)CrossRefGoogle Scholar
  15. 15.
    He, Y., Zhu, X., Mihalache, D., Liu, J., Chen, Z.: Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices. Phys. Rev. A 85, 013831 (2012)CrossRefGoogle Scholar
  16. 16.
    Jisha, C.P., Alberucci, A., Brazhnyi, V.A., Assanto, G.: Nonlocal gap solitons in PT-symmetric periodic potentials with defocusing nonlinearity. Phys. Rev. A 89, 013812 (2014)CrossRefGoogle Scholar
  17. 17.
    Driben, R., Malomed, B.A.: Stability of solitons in parity-time-symmetric couplers. Opt. Lett. 36, 4323–4325 (2011)CrossRefGoogle Scholar
  18. 18.
    Dai, C.Q., Wang, Y.Y.: Controllable combined Peregrine soliton and Kuznetsov-Ma soliton in PT-symmetric nonlinear couplers with gain and loss. Nonlinear Dyn. 80, 715–721 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Konotop, V.V., Yang, J., Zezyulin, D.A.: Nonlinear waves in PT-symmetric systems. Rev. Mod. Phys. 88, 035002 (2016)CrossRefGoogle Scholar
  20. 20.
    Suchkov, S.V., Sukhorukov, A.A., Huang, J., Dmitriev, S.V., Lee, C., Kivshar, Y.S.: Nonlinear switching and solitons in PT-symmetric photonic systems. Laser Photonics Rev. 10, 177–213 (2016)CrossRefGoogle Scholar
  21. 21.
    Desyatnikov, A., Maimistov, A., Malomed, B.A.: Three-dimensional spinning solitons in dispersive media with the cubic-quintic nonlinearity. Phys. Rev. E 61, 3107 (2000)CrossRefGoogle Scholar
  22. 22.
    Malomed, B.A., Crasovan, L.C., Mihalache, D.: Stability of vortex solitons in the cubic-quintic model. Physica D 161, 187 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Burlak, G., Malomed, B.A.: Stability boundary and collisions of two-dimensional solitons in PT-symmetric couplers with the cubic-quintic nonlinearity. Phys. Rev. E 88, 062904 (2013)CrossRefGoogle Scholar
  24. 24.
    Zhu, Y., Qin, W., Li, J.T., Han, J.Z., Wang, Y.Y., Dai, C.Q.: Recurrence behavior for controllable excitation of rogue waves in a two-dimensional PT-symmetric coupler. Nonlinear Dyn. 88, 1883–1889 (2017)CrossRefGoogle Scholar
  25. 25.
    Liu, S., Ma, C., Zhang, Y., Lu, K.: Bragg gap solitons in PT symmetric lattices with competing nonlinearity. Opt. Commun. 285(7), 1934–1939 (2012)CrossRefGoogle Scholar
  26. 26.
    Ge, L., Shen, M., Zang, T., Ma, C., Dai, L.: Stability of optical solitons in parity-time-symmetric optical lattices with competing cubic and quintic nonlinearities. Phys. Rev. E 91, 023203 (2015)CrossRefGoogle Scholar
  27. 27.
    Khare, A., Al-Marzoug, S.M., Bahlouli, H.: Solitons in PT-symmetric potential with competing nonlinearity. Phys. Lett. A 376, 2880–2886 (2012)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Chen, Y.X.: Sech-type and Gaussian-type light bullet solutions to the generalized (3+1)-dimensional cubic-quintic Schrödinger equation in PT-symmetric potentials. Nonlinear Dyn. 79, 427–436 (2015)CrossRefGoogle Scholar
  29. 29.
    Li, P., Li, L., Mihalache, D.: Optical solitons in PT-symmetric potential with competing cubic-quintic nonlinearity: existence, stability, and dynamics. Rom. Rep. Phys. 70, 408 (2018)Google Scholar
  30. 30.
    Chen, Y., Yan, Z.: Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials. Sci. Rep. 6, 23478 (2016)CrossRefGoogle Scholar
  31. 31.
    Ge, L., Shen, M., Ma, C., Zang, T., Dai, L.: Gap solitons in PT-symmetric optical lattices with higher-order diffraction. Opt. Express 22, 29435–29444 (2014)CrossRefGoogle Scholar
  32. 32.
    Zhu, X., Shi, Z., Li, H.: Gap solitons in parity-time-symmetric mixed linear-nonlinear optical lattices with fourth-order diffraction. Opt. Commun. 382, 455–461 (2017)CrossRefGoogle Scholar
  33. 33.
    Tiofack, C.G.L., Ndzana, F.I.I., Mohamadou, A., Kofane, T.C.: Spatial solitons and stability in the one-dimensional and the two-dimensional generalized nonlinear Schrödinger equation with fourth-order diffraction and parity-time-symmetric potentials. Phys. Rev. E 90, 032204 (2018)CrossRefGoogle Scholar
  34. 34.
    Cole, J.T., Musslimani, Z.H.: Band gaps and lattice solitons for the higher-order nonlinear Schrödinger equation with a periodic potential. Phys. Rev. A 90, 013815 (2014)CrossRefGoogle Scholar
  35. 35.
    Simon, B., Dicke, A.: Coupling constant analyticity for the anharmonic oscillator original. Ann. Phys. (NY) 58, 76–136 (1970)CrossRefGoogle Scholar
  36. 36.
    Weniger, E.J.: Construction of the strong coupling expansion for the ground state energy of the quartic, sextic, and octic anharmonic oscillator via a renormalized strong coupling expansion. Phys. Rev. Lett. 77, 2859–2862 (1996)CrossRefzbMATHGoogle Scholar
  37. 37.
    Midya, B.: Analytical stable Gaussian soliton supported by a parity-time-symmetric potential with power-law nonlinearity. Nonlinear Dyn. 79, 409–415 (2015)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Ashkin, A., Dziedzic, J.M., Bjorkholm, J.E., Chu, S.: Observation of a single-beam gradient force optical trap for dielectric particles. Opt. Lett. 11(1986), 288–290 (1986)CrossRefGoogle Scholar
  39. 39.
    Richardson, A.C., Reihani, S.N.S., Oddershede, L.B.: Non-harmonic potential of a single beam optical trap. Opt. Express 16, 15709–15717 (2008)CrossRefGoogle Scholar
  40. 40.
    Ahmed, Z.: Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential. Phys. Lett. A 282, 343–348 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  42. 42.
    Zezyulin, D.A., Konotop, V.V.: Nonlinear modes in the harmonic PT-symmetric potential. Phys. Rev. A 85, 043840 (2012)CrossRefGoogle Scholar
  43. 43.
    Nixon, S., Ge, L., Yang, J.: Stability analysis for solitons in PT-symmetric optical lattices. Phys. Rev. A 85, 023822 (2012)CrossRefGoogle Scholar
  44. 44.
    Yang, J.: Iteration methods for stability spectra of solitary waves. J. Comput. Phys. 227, 6862–6876 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Yang, J.: Nonlinear Waves in Integrable and Nonintegrable Systems. SIAM, Philadelphia (2010)CrossRefzbMATHGoogle Scholar
  46. 46.
    Cole, J.T., Musslimani, Z.H.: Spectral transverse instabilities and soliton dynamics in the higher-order multidimensional nonlinear Schrödinger equation. Physica D 313, 26–36 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Laboratory of Mechanics, Department of Physics, Faculty of ScienceUniversity of Yaounde IYaoundéCameroon
  2. 2.Fundamental Physics Laboratory, Group of Nonlinear Physics and Complex Systems, Department of Physics, Faculty of ScienceUniversity of DoualaDoualaCameroon
  3. 3.Condensed Matter Laboratory, Department of Physics, Faculty of ScienceUniversity of MarouaMarouaCameroon
  4. 4.The Abdus Salam International Centre for Theoretical PhysicsTriesteItaly

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