Nonlinear Dynamics

, Volume 79, Issue 1, pp 409–415 | Cite as

Analytical stable Gaussian soliton supported by a parity-time symmetric potential with power-law nonlinearity

Original Paper
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Abstract

We address the existence and stability of spatial localized modes supported by a parity-time symmetric complex potential in the presence of power-law nonlinearity. The analytical expressions of the localized modes, which are Gaussian in nature, are obtained in both (1 + 1) and (2 + 1) dimensions. A linear stability analysis corroborated by the direct numerical simulations reveals that these analytical localized modes can propagate stably for a wide range of the potential parameters and for various order nonlinearities. Some dynamical characteristics of these solutions, such as the power and the transverse power-flow density, are also examined.

Keywords

Parity-time symmetry Gain and loss Nonlinear Schrödinger equation Optical soliton 
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Notes

Acknowledgments

The author thanks Prof. Rajkumar Roychoudhury for discussions. He also acknowledges the postdoctoral grant from the Belgian Federal Science Policy Office co-funded by the Marie-Curie Actions (FP7) from the European Commission.

References

  1. 1.
    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
  2. 2.
    Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: Beam dynamics in PT-symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008)CrossRefGoogle Scholar
  3. 3.
    Achilleos, V., Kevrekidis, P.G., Frantzeskakis, D.J., Carretero-Gonzales, 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
  4. 4.
    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
  5. 5.
    Abdullaev, FKh, Kartashov, Y.V., Konotop, V.V., Zezyulin, D.A.: Solitons in PT-symmetric nonlinear lattice. Phys. Rev. A 83, 041805(R) (2011)CrossRefGoogle Scholar
  6. 6.
    Nixon, S., Ge, L., Yang, J.: Stability analysis for solitons in PT-symmetric optical lattices. Phys. Rev. A 85, 023822 (2012)CrossRefGoogle Scholar
  7. 7.
    Alexeeva, N.V., Barashenkov, I.V., Sukhorukov, A.A., Kivshar, Y.S.: Optical solitons in PT-symmetric nonlinear couplers with gain and loss. Phys. Rev. A 85, 063837 (2012)CrossRefGoogle Scholar
  8. 8.
    Mayteevarunyoo, T., Malomed, B.A., Reoksabutr, A.: Solvable model for solitons pinned to a parity-time-symmetric dipole. Phys. Rev. E 88, 022919 (2013)CrossRefGoogle Scholar
  9. 9.
    Midya, B., Roychoudhury, R.: Nonlinear localized modes in PT-symmetric Rosen-Morse potential wells. Phys. Rev. A 87, 045803 (2013)CrossRefGoogle Scholar
  10. 10.
    Midya, B., Roychoudhury, R.: Nonlinear localized modes in PT-symmetric optical media with competing gain and loss. Ann. Phys. 341, 12 (2014)CrossRefGoogle Scholar
  11. 11.
    Khare, A., Al-Marzoug, S.M., Bahlouli, H.: Solitons in PT-symmetric potential with competing nonlinearity. Phys. Lett. A 376, 2880 (2012)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Li, K., Kevrekidis, P.G., Malomed, B.A., Gunther, U.: Nonlinear PT-symmetric plaquettes. J. Phys. A 45, 444021 (2012)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Zezyulin, D.A., Konotop, V.V.: Nonlinear modes in the harmonic PT-symmetric potential. Phys. Rev. A 85, 043840 (2012)CrossRefGoogle Scholar
  14. 14.
    Li, H., Shi, Z., Jiang, X., Zhu, X.: Gray solitons in parity-time symmetric potentials. Opt. Lett. 36, 3290 (2011)CrossRefGoogle Scholar
  15. 15.
    Xu, X.J., Dai, C.Q.: Nonlinear tunneling of spatial solitons in PT-symmetric potential. Opt. Commun. 318, 112 (2014)CrossRefGoogle Scholar
  16. 16.
    Dai, C.Q., Wang, X.G., Zhou, G.Q.: Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials. Phys. Rev. A 89, 013834 (2014)CrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    Hu, S., Hu, W.: Optical solitons in the parity-time-symmetric Bessel complex potential. J. Phys. B 45, 225401 (2012)CrossRefGoogle Scholar
  19. 19.
    Chen, Y., Dai, C., Wang, X.: Two-dimensional nonautonomous solitons in parity-time symmetric optical media. Opt. Commun. 324, 10 (2014)CrossRefGoogle Scholar
  20. 20.
    Dai, C., Huang, W.: Multi-rogue wave and multi-breather solutions in PTPT-symmetric coupled waveguides. Appl. Math. Lett. 32, 35 (2014)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Bludov, Y.V., Driben, R., Konotop, V.V., Malomed, B.A.: Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides. J. Opt. 15, 064010 (2013)CrossRefGoogle Scholar
  22. 22.
    Benderr, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947 (2007)CrossRefGoogle Scholar
  23. 23.
    Ruter, C.E., Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Segev, M., Kip, D.: Observation of parity-time symmetry in optics. Nat. Phys. 6, 192 (2010)CrossRefGoogle Scholar
  24. 24.
    Guo, A., et al.: Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009)CrossRefGoogle Scholar
  25. 25.
    Klaiman, S., Gunther, U., Moiseyev, N.: Visualization of branch points in PT-symmetric waveguides. Phys. Rev. Lett. 101, 080402 (2008)Google Scholar
  26. 26.
    Zhong, W., Belic, M.R., Huang, T.: Two-dimensional accessible solitons in PT-symmetric potentials. Nonlinear Dyn. 70, 2027 (2012) Google Scholar
  27. 27.
    Dai, C.Q., Wang, Y.Y.: A bright 2D spatial soliton in inhomogeneous Kerr media with PT-symmetric potentials. Laser Phys. 24, 035401 (2014)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Driben, R., Malomed, B.: Stability of solitons in parity-time-symmetric couplers. Opt. Lett. 36, 4323 (2011)CrossRefGoogle Scholar
  29. 29.
    Driben, R., Malomed, B.A.: Dynamics of higher-order solitons in regular and PT-symmetric nonlinear couplers. Europhys. Lett. 99, 54001 (2012)CrossRefGoogle Scholar
  30. 30.
    Driben, R., Malomed, B.A.: Stabilization of solitons in PT models with supersymmetry by periodic management. Europhys. Lett. 96, 51001 (2011)CrossRefGoogle Scholar
  31. 31.
    Wang, Y., Dai, C., Wan, X.: Stable localized spatial solitons in PT-symmetric potentials with power-law nonlinearity. Nonlinear Dyn. 77, 1323 (2014)CrossRefGoogle Scholar
  32. 32.
    Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Analytical solutions to a class of nonlinear Schr\(\ddot{d}\)inger equations with PT-like potentials. J. Phys. A 41, 244019 (2008)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Xu, H., Kevrikidis, P.G., Zhou, Q., Frantzeskakis, D.J., Achilleos, V., Carretero-Gonzalez, R.: Nonlinear PT-Symmetric models bearing exact solutions. Rom. J. Phys. 59, 185 (2014)Google Scholar
  34. 34.
    Zhu, P.: Nonlinear tunneling for controllable rogue waves in two dimensional graded-index waveguides. Nonlinear Dyn. 72, 873 (2013)CrossRefGoogle Scholar
  35. 35.
    Zhu, H.P., Pan, Z.H.: Combined akhmediev breather and kuznetsov—ma solitons in a two-dimensional graded-index waveguide. Laser Phys. 24, 045406 (2014)Google Scholar
  36. 36.
    Liu, W.J., Tian, B., Lei, M.: Elastic and inelastic interactions between optical spatial solitons in nonlinear optics. Laser Phys. 23, 095401 (2013)CrossRefGoogle Scholar
  37. 37.
    Liu, W.J., et al.: Breathers in a hollow-core photonic crystal fiber. Laser Phys. Lett. 11, 045402 (2014)CrossRefGoogle Scholar
  38. 38.
    Lü, X., Peng, M.: Painlevé-integrability and explicit solutions of the general two-coupled nonlinear Schrodinger system in the optical fiber communications. Nonlinear Dyn. 73, 405 (2013)CrossRefMATHGoogle Scholar
  39. 39.
    Yang, J.: Iteration methods for stability spectra of solitary waves. J. Comp. Phys. 227, 6862 (2008)CrossRefMATHGoogle Scholar
  40. 40.
    Cartarius, H., Wunner, G.: Model of a PT-symmetric Bose-Einstein condensate in a \(\delta \)-function double-well potential. Phys. Rev. A 86, 013612 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Physique Nucleare et Physique QuantiqueUniversite Libre de BruxellesBrusselsBelgium

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