Skip to main content
SpringerLink
Log in

Comparative Study of Rogue Wave Solutions for Autonomous and Non-autonomous Saturable Discrete Nonlinear Schrödinger Equation

  • RESEARCH
  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

The existence of rogue waves in the saturable discrete nonlinear Schrodinger equation (SDNLSE) has been established numerically. These extreme waves are created using the Runge–Kutta 4 algorithm. The implementation of this technique is extended further to generate rogue waves for non-autonomous SDNLSE. The time dependence of the nonlinearity coefficient has been observed to play a significant role in manipulating the behavior of rogue waves. Three specific time dependences have been investigated. The periodic variation causes the rogue waves to split among various lattice points, whereas this wave channels along one direction as a result of the kink-shaped variation, and the intensity of the wave also rises due to this change. In contrast to these two Lorentzian variations of the nonlinearity coefficient results in the spread of the rogue waves at the lattice point, lowering the peak strength.

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. Gatz, S., Herrmann, J.: Soliton propagation and soliton collision in double-doped fibers with a non-Kerr-like nonlinear refractive-index change. Opt. Lett. 17(7), 484–486 (1992)

    Article  ADS  Google Scholar 

  2. Kumar, P., Raina, K.K.: Morphological and electro-optical responses of dichroic polymer dispersed liquid crystal films. Curr. Appl. Phys. 7(6), 636–642 (2007). https://doi.org/10.1016/j.cap.2007.01.004

  3. Hickmann, J.M., Cavalcanti, S.B., Borges, N.M., Gouveia, E.A., Gouveia-Neto, A.S.: Modulational instability in semiconductor-doped glass fibers with saturable nonlinearity. Opt. Lett. 18(3), 182 (1993). https://doi.org/10.1364/ol.18.000182

    Article  ADS  Google Scholar 

  4. Kumar, S., Rani, R., Kumar, R.: Shell closure effects studied via cluster decay in heavy nuclei. J. Phys. G: Nucl. Part. Phys. 36(1), 015110 (2009). https://doi.org/10.1088/0954-3899/36/1/015110

  5. Hadžievski, L., Maluckov, A., Stepić, M., Kip, D.: Power controlled soliton stability and steering in lattices with saturable nonlinearity. Phys. Rev. Lett. 93(3), 1–4 (2004). https://doi.org/10.1103/PhysRevLett.93.033901

    Article  Google Scholar 

  6. Khare, A., Rasmussen, K., Samuelsen, M.R., Saxena, A.: Exact solutions of the saturable discrete nonlinear Schrödinger equation. J. Phys. Math. Gen. 38(4), 807–814 (2005). https://doi.org/10.1088/0305-4470/38/4/002

    Article  ADS  MATH  Google Scholar 

  7. Khare, A., Rasmussen, K.Ø., Samuelsen, M.R., Saxena, A.: Staggered and short-period solutions of the saturable discrete nonlinear Schrödinger equation. J. Phys. Math. Theor. 42(8), 085002 (2009)

    Article  ADS  MATH  Google Scholar 

  8. Pankov, A., Rothos, V.: Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity. Proc. R. Soc. Math. Phys. Eng. Sci. 464(2100), 3219–3236 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  9. Yan, Z.: Envelope solution profiles of the discrete nonlinear Schrödinger equation with a saturable nonlinearity. Appl. Math. Lett. 22(4), 448–452 (2009). https://doi.org/10.1016/j.aml.2008.06.015

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhou, Z., Yu, J., Chen, Y.: On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity. Nonlinearity 23(7), 1727–1740 (2010). https://doi.org/10.1088/0951-7715/23/7/011

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Aslan, I. “Exact and explicit solutions to the discrete nonlinear Schrödinger equation with a saturable nonlinearity,” Phys Lett Sect Gen At. Solid State Phys., vol. 375, no. 47, pp. 4214–4217, (2011) https://doi.org/10.1016/j.physleta.2011.10.009.

  12. Akhmediev, N., Ankiewicz, A., and M. Taki, M. “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. Sect. Gen. At. Solid State Phys., vol. 373, no. 6, pp. 675–678, (2009). https://doi.org/10.1016/j.physleta.2008.12.036.

  13. Essama, B. G. O., Atangana, J. Frederick, B. M., Mokhtari, B., Eddeqaqi, N. C., and Kofane, T. C. “Rogue wave train generation in a metamaterial induced by cubic-quintic nonlinearities and second-order dispersion,” Phys. Rev. E - Stat. Nonlinear Soft Matter Phys., vol. 90, no. 3, pp. 1–12, (2014). https://doi.org/10.1103/PhysRevE.90.032911.

  14. Temgoua, D. D. E., and Kofane, T. C., “Nonparaxial rogue waves in optical Kerr media,” Phys. Rev. E - Stat. Nonlinear Soft Matter Phys., vol. 91, no. 6, pp. 26–29, 2015. https://doi.org/10.1103/PhysRevE.91.063201.

  15. Baronio, F., Degasperis, A., Conforti, M., Wabnitz, S.: Solutions of the vector nonlinear Schrödinger equations: Evidence for deterministic rogue waves. Phys. Rev. Lett. 109(4), 2–5 (2012). https://doi.org/10.1103/PhysRevLett.109.044102

    Article  Google Scholar 

  16. Tchinang, J. D., Tchameu, A. B. 2016. Togueu Motcheyo, and C. Tchawoua, “Biological multi-rogue waves in discrete nonlinear Schrödinger equation with saturable nonlinearities. Phys Lett Sect Gen At Solid State Phys., vol. 380, no. 38, pp. 3057–3060. https://doi.org/10.1016/j.physleta.2016.07.011

  17. Akhmediev, N., and Ankiewicz, A. “Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation,” Phys. Rev. E - Stat. Nonlinear Soft Matter Phys., vol. 83, no. 4, pp. 1–10, (2011) https://doi.org/10.1103/PhysRevE.83.046603.

  18. Akhmediev, N., Ankiewicz, A., and Soto-Crespo, J. M. “Rogue waves and rational solutions of the nonlinear Schrödinger equation,” Phys. Rev. E - Stat. Nonlinear Soft Matter Phys., vol. 80, no. 2, (2009) https://doi.org/10.1103/PhysRevE.80.026601.

  19. Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Rogue waves as spatial energy concentrators in arrays of nonlinear waveguides. Opt. Lett. 34(19), 3015 (2009). https://doi.org/10.1364/ol.34.003015

    Article  ADS  Google Scholar 

  20. Gupta, M. Malhotra, S. and Gupta, R. Rogue Waves generation by Using higher order rational solutions of Discrete Nonlinear Schrödinger Equation. Mater. Today Proc., vol. 71, pp. 402-407, (2022) https://doi.org/10.1016/j.matpr.2022.09.545.

  21. Gupta, M., Malhotra, S., Gupta, R.: Numerical generation and investigation of rogue waves for discrete nonlinear Schrödinger equations. J. Nonlinear. Opt. Phys. Mater. 2350026, 1–12 (2022). https://doi.org/10.1142/S0218863523500261

    Article  Google Scholar 

  22. Efe, S., and Yuce, C. “Discrete rogue waves in an array of waveguides,” Phys. Lett. Sect. Gen. At. Solid State Phys., vol. 379, no. 18–19, pp. 1251–1255, (2015). https://doi.org/10.1016/j.physleta.2015.02.031.

  23. Yu, F.: Multi-rogue waves for a higher-order nonlinear Schrödinger equation in optical fibers. Appl. Math. Comput. 220, 176–184 (2013). https://doi.org/10.1016/j.amc.2013.05.031

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, B., and Yang, J. “Partial-rogue waves that come from nowhere but leave with a trace in the Sasa-Satsuma equation,” Phys. Lett. A, pp. 128573, (2022).

  25. Grinevich, P. G., and Santini, P. M. “The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes,” Phys. Lett. Sect. A Gen. At. Solid State Phys., vol. 382, no. 14, pp. 973–979, (2018).

  26. Yin, H.M., Chow, K.W.: Breathers, cascading instabilities and Fermi–Pasta–Ulam–Tsingou recurrence of the derivative nonlinear Schrödinger equation: Effects of ‘self-steepening’ nonlinearity. Phys. D Nonlinear Phenom. 428, 133033 (2021)

    Article  MATH  Google Scholar 

  27. Jagtap, A.D., Kawaguchi, K., Karniadakis, G.E.: Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. J. Comput. Phys. 404, 109136 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Salerno, M., Kh, F.: Abdullaev “Matter wave compactons in deep optical lattices with strong nonlinearity management”, Nanosystems: Physics. Chemistry, Mathematics 6(6), 742–750 (2015)

    MATH  Google Scholar 

  29. Thakur, S., et al.: Experimental characterization of the ultrafast, tunable and broadband optical Kerr nonlinearity in grapheme. Sci. Rep. 9, 10540 (2019)

    Article  ADS  Google Scholar 

Download references

Funding

Funding for this research was provided to Dr. Rama Gupta from CSIR India for the financial support (Scheme number: 03/(1426)/18/EMR-II) for the accomplishment of this work. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Author information

Authors and Affiliations

  1. Department of Physics, D. A. V. University, Jalandhar, 144 012, Punjab, India

    Mishu Gupta & Rama Gupta

  2. Chitkara University Institute of Engineering and Technology, Chitkara University, 140401, Punjab, India

    Shivani Malhotra

Authors
  1. Mishu Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shivani Malhotra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rama Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Mishu Gupta: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Project administration, Software, Validation, Visualization, Writing-original draft.

Shivani Malhotra: Conceptualization, Methodology, Project administrator, Supervision, Validation, Writing–review, editing.

Rama Gupta: Conceptualization, Funding acquisition, Methodology, Project administration, Resources, Supervision, Validation, Writing – review & editing.

Corresponding authors

Correspondence to Shivani Malhotra or Rama Gupta.

Ethics declarations

Competing interests

It is a Comparative study for generation of rogue waves for different input wave functions such as sinusoidal, hyperbolic, and Lorentzian function

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gupta, M., Malhotra, S. & Gupta, R. Comparative Study of Rogue Wave Solutions for Autonomous and Non-autonomous Saturable Discrete Nonlinear Schrödinger Equation. Int J Theor Phys 62, 105 (2023). https://doi.org/10.1007/s10773-023-05365-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10773-023-05365-1

Keywords

Search

Navigation