Skip to main content
SpringerLink
Log in

Modulation instability and modulated waves patterns of the generalized non-autonomous nonlinear Schrödinger equation

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

In this work, we studied the behavior of the solitonic waves and localized waves in nonlinear optical fibers. We use the generalized non-autonomous NLSE with space- and time-dependent coefficients. To construct exact soliton solutions, we use the transformation hypothesis and the JEF method. For constant values of the space-and time-dependent coefficients, we have pointed out W-shaped bright soliton, dark soliton, and Jacobi elliptic function solutions. When we consider the dispersion relation as a periodic function, we obtain new characteristics of the solitonic waves. We have used numerical simulation to seek the modulation’s unstable and stable modes. It results in modulated wave patterns emerging for strong enough values of the cubic and quintic nonlinearities, and for specific times of simulation, rogue waves and breather solitons emerge in the structure. We have also demonstrated that when the quintic nonlinearity is absent, the breathing behavior of the solitonic waves generates localized energy in the system.

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
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during this study.

Abbreviations

NLSE:

Nonlinear Schödinger equation

MI:

Modulation instability

MW:

Modulated waves

OFs:

Optical fibers

JEF:

Jacobi elliptic function

CN:

Cubic nonlinearity

QN:

Quintic nonlinearity

GVD:

Group velocity dispersion

FWM:

Four-wave mixing

twt:

Traveling wave transformation

References

  • Abbagari, S., Houwe, A., Doka, S.Y., Bouetou, T.B., Inc, M., Crepin, K.T.: W-shaped profile and multiple optical soliton structure of the coupled nonlinear Schrödinger equation with the four-wave mixing term and modulation instability spectrum. Phys. Lett. A 418, 127710 (2021)

    MATH  Google Scholar 

  • Adem, A.R., Khalique, C.M.: Symmetry reductions exact solutions and conservation laws of a new coupled KdV system. Commun. Nonlinear Sci. Numer. Simul. 17, 3465–3475 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  • Agrawal, G.P.: Nonlinear fiber optics, 6th ed. Academic Press (2019)

  • Akinyemi, L., Mirzazadeh, M., Hosseini, K.: Solitons and other solutions of perturbed nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index. Nonlinear Analysis: Modelling and Control 27, 1–17 (2022)

    MathSciNet  MATH  Google Scholar 

  • Akinyemi, L., Inc, M., Khater, M., Rezazadeh, H.: Dynamical behaviour of Chiral nonlinear Schrödinger equation. Opt. Quantum Electron. 54(3), 1–15 (2022)

    Google Scholar 

  • Al Khawaja, U., Bahlouli, H.: Integrability conditions and solitonic solutions of the nonlinear Schrödinger equation with generalized dual-power nonlinearities PT-symmetric potentials, and space- and time-dependent coefficients. Commun. Nonlinear Sci. Numer. Simul. 69, 248–260 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  • Alterman, D., Rauch, J.: Diffractive short pulse asymptotics for nonlinear wave equation. Phys. Lett. A 264, 390–395 (2000)

    ADS  MathSciNet  MATH  Google Scholar 

  • Aminikhah, H., Sheikhani, A.R., Rezazadeh, H.: Sub-equation method for the fractional regularized long-wave equations with conformable fractional derivatives. Sci. Iran. 23(3), 1048–1054 (2016)

    Google Scholar 

  • Aslan, E.C., Inc, M.: Optical soliton solutions of the NLSE with quadratic-cubic-Hamiltonian perturbations and modulation instability analysis. Optik 196, 162661 (2019)

    ADS  Google Scholar 

  • Az-Zo’bi, E.A., AlZoubi, W.A., Akinyemi, L., Şenol, M., Alsaraireh, I.W., Mamat, M.: Abundant closed-form solitons for time-fractional integro-differential equation in fluid dynamics. Opt. Quantum Electron. 53(3), 1–16 (2021)

    Google Scholar 

  • Biswas, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical solitons having weak non-local nonlinearity by two integration schemes. Optik 164, 380–384 (2018)

    ADS  Google Scholar 

  • Bouetou, T.B., Abbagari, S., Kuetche, V.K., Mouna, F., Kofane, T.C.: Jet bundle formalism towards the extension of the stretched equation. J. Math. Anal. Appl. 177, 269–273 (2011)

    MathSciNet  MATH  Google Scholar 

  • Bulut, H., Sulaiman, T.A., Baskonus, H.M.: On the solitary wave solutions to the longitudinal wave equation in MEE circular rod. Opt. Quantum Electron. 50(2), 87 (2018)

    Google Scholar 

  • Cattani, C., Sulaiman, T.A., Baskonus, H.M., Bulut, H.: On the soliton solutions to the Nizhnik–Novikov-Veselov and the Drinfeld–Sokolov systems. Opt. Quantum Electron. 50(3), 138 (2018)

    Google Scholar 

  • Chung, Y., Jones, C.K.R.T., Schäfer, T., Wayne, C.E.: Ultra-short pulses in linear and nonlinear media. Nonlinearity 18, 1351–1374 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  • Eslami, M.: Trial solution technique to chiral nonlinear Schrödingers equation in \((1+2)\)-dimensions. Nonlinear Dyn. 85(2), 813–816 (2016)

    Google Scholar 

  • Feng, B.F.: An integrable coupled short pulse equation. J. Phys. A. 45, 1–14 (2012)

    MathSciNet  MATH  Google Scholar 

  • Ghanbari, B., Inc, M.: A new generalized exponential rational function method to and exact special solutions for the resonance Schrödinger equation. Eur. Phys. J. Plus. 133, 142 (2018)

    Google Scholar 

  • Grinevich, P.G., 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. A. (2018). https://doi.org/10.1016/j.physleta.2018.02.014

    Article  MathSciNet  MATH  Google Scholar 

  • Guner, O., Bekir, A., Korkmaz, A.: Tanh-type and sech-type solitons for some space-time fractional PDE models. Eur. Phys. J. Plus. 132(92), 1–12 (2017)

    Google Scholar 

  • Guner, O., Korkmaz, A., Bekir, A.: Dark solutions of space-time fractional Sharma-Tasso-Olver and potential Kadomtsev–Petviashvili equations. Commun. Theor. Phys. 67(2), 182–188 (2017)

    ADS  MATH  Google Scholar 

  • Houwe, A., Abbagari, S., Akinyemi, L., Inc, M., Doka, S.Y.: Wave propagation in discrete cold bosonic atoms zig zag optical lattice. Eur. Phys. J. Plus 137(9), 1–11 (2022)

    Google Scholar 

  • Inc, M., Baleanu, D.: Optical Solitons for the Kundu–Eckhaus equation with time dependent coefficient. Optik 159, 324–332 (2018)

    ADS  Google Scholar 

  • Jagtap, A.D., Kawaguchi, K., Karniadakis, G.E.: Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. J. Comput. Phys. 404, 09136 (2020)

  • Kai, C.C., Hai, F.L.: Effect of third-order dispersion on soliton-effect pulse compression. Opt. Lett. 1, 49–51 (1994)

    Google Scholar 

  • Kavitha, L., Parasuraman, E., Gopi, D., Prabhu, A., Vicencio, R.A.: Nonlinear nano-scale localized breather modes in a discrete weak ferromagnetic spin lattice. J. Magn. Magn. Mater. 401, 394 (2016)

    ADS  Google Scholar 

  • Khodadad, F.S., Nazari, F., Eslami, M., Rezazadeh, H.: Soliton solutions of the conformable fractional Zakharov–Kuznetsov equation with dual-power law nonlinearity. Opt. Quantum Electron. 49(11), 1–12 (2017)

    Google Scholar 

  • Konno, K., Jeffrey, A.: Some remarkable properties of two loop soliton solutions. J. Phys. Soc. Jpn. 52(1), 1–3 (1983)

    ADS  MathSciNet  Google Scholar 

  • Korkmaz, A.: Exact Solutions to \((3+1)\) Conformable Time Fractional Jimbo Miwa, Zakharov Kuznetsov and Modified Zakharov Kuznetsov Equations. Commun. Theor. Phys. 67, 479–482 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  • Kozlov, S.A., Sazonov, S.V.: Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media. J. Exp. Theor. Phys. 84, 221–228 (1997)

    ADS  Google Scholar 

  • Liou, L.W., Cao, X.D., McKinstrie, C.J., Agrawal, G.P.: Spatiotemporal instabilities in dispersive nonlinear media. Phys. Rev. A 46(7), 4202–4208 (1992)

    ADS  Google Scholar 

  • Nguepjouo, T.F., Kuetche, V.K., Kofane, T.C.: Soliton interactions between multivalued localized waveguide channels within ferrites. Phys. Rev. E 89, 1–14 (2014)

    Google Scholar 

  • Ntiamoah, D., Ofori-Atta, W., Akinyemi, L.: The higher-order modified Korteweg-de Vries equation: its soliton, breather and approximate solutions. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.06.042

    Article  Google Scholar 

  • Sakovich, S.: Integrability of the vector short pulse equation. J. Phys. Soc. Jpn. 77, 1–4 (2008)

    Google Scholar 

  • Sarma, A.K.: Modulational instability of coupled nonlinear field equations for pulse propagation in a negative index material embedded into a Kerr medium. J. Opt. Soc. Am. B 28(4), 944–948 (2011)

    ADS  Google Scholar 

  • Sokolnikoff, I.S.: Mathematical Theory of Elasticity. Mc. Graw-Hill, New York (1956)

    MATH  Google Scholar 

  • Tang, B., Li, G.-L., Fu, M.: Modulational instability and localized modes in Heisenberg ferromagnetic chains with single-ion easy-axis anisotropy. J. Magn. Magn. Mater. 426, 429–434 (2017)

    ADS  Google Scholar 

  • Tchokouansi, H.T., Kuetche, V.K., Kofane, T.C.: Exact soliton solution to a new coupled integrable short-light pulse system. Chaos, Solitons Fractals 68, 10–19 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  • Tchokouansi, H.T., Kuetche, V.K., Kofane, T.C.: On the propagation of solitons in ferrites the inverse scattering approach. Chaos, Solitons Fractals 86, 64–74 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  • Vakhnenko, V.O., Parkes, E.J.: The two loop soliton solution of the Vakhnenko equation. Nonlinearity 11, 1457–1464 (1998)

    ADS  MathSciNet  MATH  Google Scholar 

  • Wahlquist, H.D., Estabrook, F.B.: Prolongation structure of nonlinear evolution equations. J. Math. Phys. 16, 1–7 (1975)

    ADS  MathSciNet  MATH  Google Scholar 

  • Wang, D.S., Zhang, D.J., Yang, J.: Intergrable properties of the general coupled nonlinear Schrödinger equation. J. Math. Phys. 51, 1–17 (2010)

    Google Scholar 

  • Wen, S.C., Fan, D.Y.J.: Spatiotemporal instabilities in nonlinear Kerr media in the presence of arbitrary higher-order dispersions. Opt. Soc. Am. B. 19(7), 1653–1659 (2002)

    ADS  Google Scholar 

  • 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 (2021). https://doi.org/10.1016/j.physd.2021.133033

    Article  MATH  Google Scholar 

  • Youssoufa, S., Kuetche, V.K., Kofane, T.C.: Head-on collisions of localized pressure excitations in derivative cubic relaxing media dynamical structure survey. Phys. Scr. 90, 1–13 (2015)

    Google Scholar 

  • Yusuf, A., Inc, M., Baleanu, D.: Optical solitons with M-truncated and beta derivatives in nonlinear optics. Front. Phys. 7, 126 (2019)

    Google Scholar 

  • Zakharov, V.E., Shabat, A.B.: Interaction between solitons in a stable medium. J. Exper. Theor. Phys. 37, 823–828 (1973)

    ADS  Google Scholar 

  • Zhong, X., Cheng, K., Chiang, K.S.: Modulation instability with arbitrarily high perturbation frequencies in metamaterials with nonlinear dispersion and saturable nonlinearity. J. Opt. Soc. Am. B 31, 1484–1493 (2014)

    ADS  Google Scholar 

Download references

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) for funding and supporting this work through Research Partnership Program no RP-21–09-07.

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Science Faculty, Department of Mathematics, Fırat University, 23119, Elazig, Turkey

    Mustafa Inc

  2. Department of Medical Research, China Medical University, 40402, Taichung, Taiwan

    Mustafa Inc

  3. Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh, 11432, Saudi Arabia

    Rubayyi T. Alqahtani

Authors
  1. Mustafa Inc
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rubayyi T. Alqahtani
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

“M.I. and R.T.A. wrote the main manuscript text and M.I. prepared figures. All authors reviewed the manuscript.”

Corresponding author

Correspondence to Mustafa Inc.

Ethics declarations

Confict of interest

The authors declare no confict of interest.

Ethics approval

Not applicable.

Consent for publication

All the authors have agreed and given their consent for the publication of this research paper.

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

Inc, M., Alqahtani, R.T. Modulation instability and modulated waves patterns of the generalized non-autonomous nonlinear Schrödinger equation. Opt Quant Electron 55, 362 (2023). https://doi.org/10.1007/s11082-023-04645-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-023-04645-8

Keywords

Search

Navigation