Nonlinear Dynamics

, Volume 89, Issue 3, pp 1745–1752 | Cite as

Stability of Gaussian-type light bullets in the cubic-quintic-septimal nonlinear media with different diffractions under \({\mathcal {PT}}\)-symmetric potentials

  • Hai-Ping Zhu
  • Zhen-Huan Pan
Original Paper


From the governing equation \(-(3+1)\)-dimensional nonlinear Schrödinger equation with cubic-quintic-septimal nonlinearities, different diffractions and \({\mathcal {PT}}\)-symmetric potentials, we obtain two kinds of analytical Gaussian-type light bullet solutions. The septimal nonlinear term has a strong impact on the formation of light bullets. The eigenvalue method and direct numerical simulation to analytical solutions imply that stable and unstable evolution of light bullets against white noise attributes to the coaction of cubic-quintic-septimal nonlinearities, dispersion, different diffractions and \({\mathcal {PT}}\)-symmetric potential.


Cubic-quintic-septimal nonlinearity Different diffractions \({\mathcal {PT}}\)-symmetric potential Light bullet Stability 



This work was supported by the National Natural Science Foundation of China (Grant No. 11375079).


  1. 1.
    Kong, L.Q., Dai, C.Q.: Some discussions about variable separation of nonlinear models using Riccati equation expansion method. Nonlinear Dyn. 81, 1553–1561 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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
  3. 3.
    Zhang, B., Zhang, X.L., Dai, C.Q.: Discussions on localized structures based on equivalent solution with different forms of breaking soliton model. Nonlinear Dyn. 87, 2385–2393 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Wang, Y.Y., Dai, C.Q., Zhou, G.Q., Fan, Y., Chen, L.: Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium. Nonlinear Dyn. 87, 67–73 (2017)CrossRefGoogle Scholar
  5. 5.
    Dai, C.Q., Wang, Y., Liu, J.: Spatiotemporal Hermite–Gaussian solitons of a (3 + 1)-dimensional partially nonlocal nonlinear Schrodinger equation. Nonlinear Dyn. 84, 1157–1161 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dai, C.Q., Fan, Y., Zhou, G.Q., Zheng, J., Cheng, L.: Vector spatiotemporal localized structures in (3 + 1)-dimensional strongly nonlocal nonlinear media. Nonlinear Dyn. 86, 999–1005 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kong, L.Q., Liu, J., Jin, D.Q., Ding, D.J., Dai, C.Q.: Soliton dynamics in the three-spine \(\alpha \)-helical protein with inhomogeneous effect. Nonlinear Dyn. 87, 83–92 (2017)CrossRefGoogle Scholar
  8. 8.
    Raju, T.S.: Spatiotemporal optical similaritons in dual core waveguide with an external source. Commun. Nonlinear Sci. Numer. Simul. 45, 75–80 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Aitchison, J.S., Weiner, A.M., Silberberg, Y., Oliver, M.K., Jackel, J.L., Leaird, D.E., Vogel, E.M., Smith, P.W.E.: Observation of spatial optical solitons in a nonlinear glass waveguide. Opt. Lett. 15, 471–474 (1990)CrossRefGoogle Scholar
  10. 10.
    Dai., C.Q., Zhou, G.Q., Chen, R.P., Lai, X.J., Zheng, J.: Vector multipole and vortex solitons in two dimensional Kerr media. Nonlinear Dyn. (2017). doi: 10.1007/s11071-017-3399-z
  11. 11.
    Skarka, V., Berezhiani, V.I., Miklaszewski, R.: Spatiotemporal soliton propagation in saturating nonlinear optical media. Phys. Rev. E 56, 1080–1087 (1997)CrossRefGoogle Scholar
  12. 12.
    Bang, O., Rasmussen, J.J., Christiansen, P.L.: Subcritical localization in the discrete nonlinear Schrodinger equation with arbitrary power nonlinearity. Nonlinearity 7, 205–218 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rasmussen, J.J., Rypdal, K.: Blow-up in nonlinear Schrodinger equations-I, A general review. Phys. Scr. 33, 481 (1986)CrossRefzbMATHGoogle Scholar
  14. 14.
    Falcao-Filho, E.L., de Araujo, C.B., Boudebs, G., Leblond, H., Skarka, V.: Phys. Rev. Lett. 110, 013901 (2013)CrossRefGoogle Scholar
  15. 15.
    Reyna, A.S., Jorge, K.C., de Araujo, C.B.: Two-dimensional solitons in a quintic-septimal medium. Phys. Rev. A 90, 063835 (2014)CrossRefGoogle Scholar
  16. 16.
    Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT-symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wang, Y.Y., Dai, C.Q., Wang, X.G.: Stable localized spatial solitons in PT-symmetric potentials with power-law nonlinearity. Nonlinear Dyn. 77, 1323–1330 (2014)CrossRefGoogle Scholar
  18. 18.
    Dai, C.Q., Wang, X.G.: Light bullet in parity-time symmetric potential. Nonlinear Dyn. 77, 1133–1139 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dai, C.Q., Chen, R.P., Wang, Y.Y., Fan, Y.: Dynamics of light bullets in inhomogeneous cubic-quintic-septimal nonlinear media with PT-symmetric potentials. Nonlinear Dyn. 87, 1675–1683 (2017)CrossRefGoogle Scholar
  20. 20.
    Chen, Y.X.: Sech-type and Gaussian-type light bullet solutions to the generalized (3 + 1)-dimensional cubic-quintic Schrdinger equation in PT-symmetric potentials. Nonlinear Dyn. 79, 427–436 (2015)CrossRefGoogle Scholar
  21. 21.
    Li, J.T., Zhang, X.T., Meng, M., Liu, Q.T., Wang, Y.Y., Dai, C.Q.: Control and management of the combined Peregrine soliton and Akhmediev breathers in PT-symmetric coupled waveguides. Nonlinear Dyn. 84, 473–479 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Li, J.T., Zhu, Y., Liu, Q.T., Han, J.Z., Wang, Y.Y., Dai, C.Q.: Vector combined and crossing Kuznetsov–Ma solitons in PT-symmetric coupled waveguides. Nonlinear Dyn. 85, 973–980 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    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
  24. 24.
    Reyna, A.S., Malomed, B.A., de Araújo, C.B.: Stability conditions for one-dimensional optical solitons in cubic-quintic-septimal media. Phys. Rev. A 92, 033810 (2015)CrossRefGoogle Scholar
  25. 25.
    Ultanir, E.A., Stegeman, G.I., Michaelis, D., Lange, C.H., Lederer, F.: Stable dissipative solitons in semiconductor optical amplifiers. Phys. Rev. Lett. 90, 253903 (2003)CrossRefzbMATHGoogle Scholar
  26. 26.
    Dai, C.Q., Wang, Y.: Higher-dimens ional locali zed mode families in parity-time-symmetric potentials with competing nonlinearities. J. Opt. Soc. Am. B 31, 2286–2294 (2014)CrossRefGoogle Scholar
  27. 27.
    Abramowitz, M., Stegun, I.A.: Chapter 15, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1965)zbMATHGoogle Scholar
  28. 28.
    Zhao, L., Sui, Z., Zhu, Q.H., Zhang, Y., Zuo, Y.L.: Improvement and precision analysis of the split-step Fourier method in solving the general nonlinear Schrödinger equation. Acta Phys. Sin. 58, 4731–4737 (2009)Google Scholar
  29. 29.
    Dai, C.Q., Zhang, X.F., Fan, Y., Chen, L.: Localized modes of the (n + 1)-dimensional Schrödinger equation with power-law nonlinearities in PT-symmetric potentials. Commun. Nonlinear Sci. Numer. Simul. 43, 239–250 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.College of EcologyLishui UniversityLishuiPeople’s Republic of China
  2. 2.College of Engineering and DesignLishui UniversityLishuiPeople’s Republic of China

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