Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Akhmediev, N., Ankiewicz, A.: Solitons. Chapman and Hall, London (1997)

    MATH  Google Scholar 

  2. Kivshar, Y., Agrawal, G.: Optical Solitons, from Fibers to Photonic Crystals. Academic, New York (2003)

    Google Scholar 

  3. Hasegawa, A., Matsumoto, M.: Optical Solitons in Fibers. Springer, New York (2003)

    Book  Google Scholar 

  4. Malomed, B.: Soliton Management in Periodic Systems. Springer, New York (2006)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  6. Belić, M., et al.: Analytical light bullet Solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation. Phys. Rev. Lett. 101, 0123904 (2008)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  11. Towers, I., et al.: Stability of spinning ring solitons of the cubicquintic nonlinear Schrdinger equation. Phys. Lett. A 288, 292 (2001)

    Article  Google Scholar 

  12. Schürmann, H.W.: Traveling-wave solutions of the cubic-quintic nonlinear Schrdinger equation. Phys. Rev. E 54, 4313 (1996)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  18. Zhou, Q., et al.: Optical solitons in media with time-modulated nonlinearities and spatiotemporal dispersion. Nonlinear Dyn. 80, 983 (2015)

    Article  MathSciNet  Google Scholar 

  19. Biswas, A.: Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients. Nonlinear Dyn. 58, 345 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Biswas, A., Khalique, C.M.: Stationary solutions for nonlinear dispersive Schrdingers equation. Nonlinear Dyn. 63, 623 (2011)

    Article  Google Scholar 

  21. Eslami, M., Mirzazadeh, M.: Optical solitons with Biswas-Milović equation for power law and dual-power law nonlinearities. Nonlinear Dyn. 83, 731 (2016)

    Article  MATH  Google Scholar 

  22. Micallef, R., et al.: Optical solitons with power-law asymptotics. Phys. Rev. E 54, 2936 (1996)

    Article  Google Scholar 

  23. Biswas, A.: 1-soliton solution of (1+2)-dimensional nonlinear Schrödinger equation in dual-power law media. Phys. Lett. A 372, 5941 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Biswas, A.: Soliton-soliton interaction with dual-power law nonlinearity. Appl. Math. Comput. 198, 605 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Bouzida, A., et al.: Chirped optical solitons in nano optical fibers with dual-power law nonlinearity. Optik 142, 77 (2017)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  29. Wazwaz, A.: Reliable analysis for nonlinear Schrödinger equations with a cubic nonlinearity and a power law nonlinearity. Math. Comput. Model. 43, 178 (2006)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  31. Malomed, B.A., et al.: Spatio-temporal optical solitons. J. Opt. B 7, R53 (2005)

    Article  Google Scholar 

  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. Xu, S.L., et al.: Exact solutions of the (2+1)-dimensional quintic nonlinear Schrdinger equation with variable coefficients. Nonlinear Dyn. 80, 583 (2015)

    Article  Google Scholar 

  34. Adhikari, S.: Nonlinear Schrödinger equation for a superfluid Fermi gas in the BCS-BEC crossover. Phys. Rev. A 77, 045602 (2008)

    Article  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Institute of Physics, University of Belgrade, P.O. Box 68, Belgrade, 11080, Serbia

    Nikola Z. Petrović

Authors
  1. Nikola Z. Petrović
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikola Z. Petrović.

Ethics declarations

Conflict of interest

The author declares that he has no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Petrović, N.Z. Spatiotemporal traveling and solitary wave solutions to the generalized nonlinear Schrödinger equation with single- and dual-power law nonlinearity. Nonlinear Dyn 93, 2389–2397 (2018). https://doi.org/10.1007/s11071-018-4331-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-018-4331-x

Keywords

Search

Navigation