Nonlinear Dynamics

, Volume 93, Issue 4, pp 2389–2397 | Cite as

Spatiotemporal traveling and solitary wave solutions to the generalized nonlinear Schrödinger equation with single- and dual-power law nonlinearity

  • Nikola Z. PetrovićEmail author
Original Paper


We generalize previously obtained solutions to the generalized nonlinear Schrödinger equation (NLSE) with cubic-quintic nonlinearity and distributed coefficients to obtain spatiotemporal traveling and solitary wave solutions for the NLSE with a general p-2p dual-power law nonlinearity, where p is an arbitrary positive real number (the cubic-quintic model being a special case for \(p=2\)). In addition, it is possible to eliminate the lower exponent, producing spatiotemporal traveling and solitary wave solutions to the NLSE with a single power law nonlinearity of arbitrary positive real power, which models many important systems including superfluid Fermi gas.


Nonlinear Schrödinger Cubic-quintic Dual-power 



Work at the Institute of Physics is supported by Project OI 171006 of the Serbian Ministry of Education and Science.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.


  1. 1.
    Akhmediev, N., Ankiewicz, A.: Solitons. Chapman and Hall, London (1997)zbMATHGoogle Scholar
  2. 2.
    Kivshar, Y., Agrawal, G.: Optical Solitons, from Fibers to Photonic Crystals. Academic, New York (2003)Google Scholar
  3. 3.
    Hasegawa, A., Matsumoto, M.: Optical Solitons in Fibers. Springer, New York (2003)CrossRefGoogle Scholar
  4. 4.
    Malomed, B.: Soliton Management in Periodic Systems. Springer, New York (2006)zbMATHGoogle Scholar
  5. 5.
    Zhong, W.P., et al.: Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrödinger equation with distributed coefficients. Phys. Rev. A 78, 023821 (2008)CrossRefGoogle Scholar
  6. 6.
    Belić, M., et al.: Analytical light bullet Solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation. Phys. Rev. Lett. 101, 0123904 (2008)CrossRefGoogle Scholar
  7. 7.
    Petrović, N., et al.: Exact spatiotemporal wave and soliton solutions to the generalized (3+1)-dimensional Schrödinger equation for both normal and anomalous dispersion. Opt. Lett. 34, 1609 (2009)CrossRefGoogle Scholar
  8. 8.
    Petrović, N., et al.: Modulation stability analysis of exact multidimensional solutions to the generalized nonlinear Schrödinger equation and the Gross-Pitaevskii equation using a variational approach. Opt. Exp. 23, 10616 (2015)CrossRefGoogle Scholar
  9. 9.
    Petrović, N., et al.: Exact traveling-wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional Schrödinger equation with polynomial nonlinearity of arbitrary order. Phys. Rev. E 83, 026604 (2011)CrossRefGoogle Scholar
  10. 10.
    Hong-Yu, W., et al.: Self-similar solutions of variable-coefficient cubic-quintic nonlinear Schrdinger equation with an external potential. Commun. Theor. Phys. (Beijing, China) 54, 55 (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Towers, I., et al.: Stability of spinning ring solitons of the cubicquintic nonlinear Schrdinger equation. Phys. Lett. A 288, 292 (2001)CrossRefGoogle Scholar
  12. 12.
    Schürmann, H.W.: Traveling-wave solutions of the cubic-quintic nonlinear Schrdinger equation. Phys. Rev. E 54, 4313 (1996)CrossRefGoogle Scholar
  13. 13.
    Liu, X.B., et al.: Exact self-similar wave solutions for the generalized (3+1)-dimensional cubic-quintic nonlinear Schröinger [sic] equation with distributed coefficients. Opt. Commun. 285, 779 (2012)CrossRefGoogle Scholar
  14. 14.
    Dai, C., et al.: Chirped and chirp-free self-similar cnoidal and solitary wave solutions of the cubic-quintic nonlinear Schrödinger equation with distributed coefficients. Opt. Commun. 283, 1489 (2010)CrossRefGoogle Scholar
  15. 15.
    Belmonte-Beitia, J., Cuevas, J.: Solitons for the cubic-quintic nonlinear Schrödinger equation with time- and space-modulated coefficients. J. Phys. A. 42, 165201 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    He, J.R., Li, H.M.: Analytical solitary-wave solutions of the generalized nonautonomous cubic-quintic nonlinear Schrödinger equation with different external potentials. Phys. Rev. E 83, 066607 (2011)CrossRefGoogle Scholar
  17. 17.
    Hao, R., et al.: A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients. Opt. Commun. 236, 79 (2004)CrossRefGoogle Scholar
  18. 18.
    Zhou, Q., et al.: Optical solitons in media with time-modulated nonlinearities and spatiotemporal dispersion. Nonlinear Dyn. 80, 983 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Biswas, A.: Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients. Nonlinear Dyn. 58, 345 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Biswas, A., Khalique, C.M.: Stationary solutions for nonlinear dispersive Schrdingers equation. Nonlinear Dyn. 63, 623 (2011)CrossRefGoogle Scholar
  21. 21.
    Eslami, M., Mirzazadeh, M.: Optical solitons with Biswas-Milović equation for power law and dual-power law nonlinearities. Nonlinear Dyn. 83, 731 (2016)CrossRefzbMATHGoogle Scholar
  22. 22.
    Micallef, R., et al.: Optical solitons with power-law asymptotics. Phys. Rev. E 54, 2936 (1996)CrossRefGoogle Scholar
  23. 23.
    Biswas, A.: 1-soliton solution of (1+2)-dimensional nonlinear Schrödinger equation in dual-power law media. Phys. Lett. A 372, 5941 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Biswas, A.: Soliton-soliton interaction with dual-power law nonlinearity. Appl. Math. Comput. 198, 605 (2008)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Bouzida, A., et al.: Chirped optical solitons in nano optical fibers with dual-power law nonlinearity. Optik 142, 77 (2017)CrossRefGoogle Scholar
  26. 26.
    Mirzazadeh, M., et al.: Topological solitons of resonant nonlinear Schödinger’s equation with dual-power law nonlinearity by G/G-expansion technique. Optik 125, 5480 (2014)CrossRefGoogle Scholar
  27. 27.
    Ali, A., et al.: Soliton solutions of the nonlinear Schrödinger equation with the dual power law nonlinearity and resonant nonlinear Schrödinger equation and their modulation instability analysis. Optik 145, 79 (2017)CrossRefGoogle Scholar
  28. 28.
    Biswas, A.: Optical solitons with time-dependent dispertion, nonlinearity and attenuation in a power-law media. Commun. Nonlinear Sci. Numer. Simulat. 14, 1078 (2009)CrossRefzbMATHGoogle Scholar
  29. 29.
    Wazwaz, A.: Reliable analysis for nonlinear Schrödinger equations with a cubic nonlinearity and a power law nonlinearity. Math. Comput. Model. 43, 178 (2006)CrossRefzbMATHGoogle Scholar
  30. 30.
    Mirzazadeh, M., et al.: Soliton solutions to resonant nonlinear Schrödinger’s equation with time-dependent coefficients by trial solution approach. Nonlinear Dyn. 81, 277 (2015)CrossRefzbMATHGoogle Scholar
  31. 31.
    Malomed, B.A., et al.: Spatio-temporal optical solitons. J. Opt. B 7, R53 (2005)CrossRefGoogle Scholar
  32. 32.
    Koonprasert, S., Punpocha, M.: More exact solutions of Hirota–Ramani partial differential equations by applying F-Expansion method and symbolic computation. Glob. J. Pure Appl. Math. 12(3), 1903 (2006)Google Scholar
  33. 33.
    Xu, S.L., et al.: Exact solutions of the (2+1)-dimensional quintic nonlinear Schrdinger equation with variable coefficients. Nonlinear Dyn. 80, 583 (2015)CrossRefGoogle Scholar
  34. 34.
    Adhikari, S.: Nonlinear Schrödinger equation for a superfluid Fermi gas in the BCS-BEC crossover. Phys. Rev. A 77, 045602 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of PhysicsUniversity of BelgradeBelgradeSerbia

Personalised recommendations