Skip to main content
SpringerLink
Log in

Bifurcation and optical solutions of the higher order nonlinear Schrödinger equation

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

Abstract

In this paper, we employ the bifurcation to predict and construct the exact solutions of the higher order nonlinear Schrödinger equation (NLSE). We proceed to discussing the bifurcation of phase portraits and we obtain the general solutions of the higher order equation using only analytical approach. We productively achieve exact solutions involving parameters such as hyperbolic solution, Jacobi elliptic function (JEF) and dark soliton which are novel solutions. In addition, we also plot the 3D surface of some solutions obtained and provide some interpretations. It is acknowledged that the method employed here offers a more esteemed mathematical instrument for acquiring analytical answers to several nonlinear equations.

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

Similar content being viewed by others

Dat availability

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

References

  • Abdou, M.A.: The extended F-expansion method and its application for a class of nonlinear evolution equations, Chaos Solitons Fractals 31, 95 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Abdoulkary, S., Mohamadou, A., Dafounansou, O., Doka, S.Y.: Exact solutions of the nonlinear differential-difference equations associated with the nonlinear electrical transmission line through a variable-coefficient discrete (G'/G)-expansion method, Chin. Phys. B 23, 120506 (2014)

    Article  Google Scholar 

  • Almusawa, H., et al.: Analytical and numerical technique for nonlinear fractional PDEs in Engineering, Applied Mathematics and Mathematical Physics, AIP Adv. 11, 065320 (2021)

    Article  ADS  Google Scholar 

  • Apeantia, W.O., Seadawy, A.R., Lu, D.: Complex optical solutions and modulation instability of hyperbolic Schrödinger dynamical equation, Res. Phys. 12, 2091–7 (2019)

    Google Scholar 

  • Arbabi, S., Najafi, M.: Exact solitary wave solutions of the complex nonlinear Schrödinger equations, Optik 127, 4682 (2016)

    Article  ADS  Google Scholar 

  • Arshad, M., Seadawy, A.R., Lu, D.: Study of soliton solutions of higher-order nonlinear Schrödinger dynamical model with derivative non-Kerr nonlinear terms and modulation instability analysis, Res. Phys. 13, 102305 (2019)

    Google Scholar 

  • Arshad, M., Seadawy, A.R., Lu, D., Ali, A.: Dispersive Solitary Wave Solutions of Strain Wave Dynamical Model and Its Stability, Commun. Theor. Phys. 71, 1155–1162 (2019)

    Article  ADS  Google Scholar 

  • Arshad, M., Seadawy, A.R., Lu, D., Saleem, M.S.: Elliptic function solutions, modulation instability and optical solitons analysis of the paraxial wave dynamical model with Kerr media, Opt. Quant. Electron. 53, 7 (2021)

    Article  Google Scholar 

  • Baskonus, H.M., Sulaiman, T.A., Bulut, H.: Dark, bright and other optical solitons to the decoupled nonlinear Schrödinger equation arising in dual-core optical fibers, Opt. Quant. Electron. 50, 165 (2018)

    Article  Google Scholar 

  • Choudhuri, A., Porsezian, K.: Dark-in-the-Bright solitary wave solution of higher-order nonlinear Schrödinger equation with non-Kerr terms, Opt. Commun. 285, 364–367 (2012)

    Article  ADS  Google Scholar 

  • Fan, E.G.: Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277, 212 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  • Feng, L.-L., Tian, S.-F., Yan, H., Wang, L., Zhang, T.-T.: On periodic wave solutions and asymptotic behaviors to a generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation, Eur. Phys. J. Plus 131, 241 (2016)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Inc, M., Evans, D.J.: On traveling wave solutions of some nonlinear evolution equations, Int. J. Comput. Math. 81, 191 (2004)

    Article  MathSciNet  Google Scholar 

  • Jawad, A.J., Marko, D.P., Biswas, A.: Modified simple equation method for nonlinear evolution equations, Appl. Math. Comput. 217, 869 (2010)

    MathSciNet  Google Scholar 

  • Khater, M.A.: On the new solitary wave solution of the generalized Hirota-Satsuma coupled KdV System, Glob. J. Sci. Front. Res. 15, 11–17 (2015)

    Google Scholar 

  • Kudryashov, N.A.: Methods of nonlinear mathematical Physics, Intellect, p. 364 (in Russian) (2010)

  • Liu, S.K., Fu, Z.T., Liu, S.D., Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A 289, 69 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  • Malfliet, W.: Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60, 650 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  • Malwe, B.H., Betchewe, G., Serge Doka, Y., Kofane, T.C.: Travelling wave solutions and soliton solutions for the nonlinear transmission line using the generalized Riccati equation mapping method, Nonlinear Dyn. 84, 171 (2016)

    Article  Google Scholar 

  • Manafian, J.: Optical soliton solutions for Schrödinger type nonlinear evolution equations by the tan (φ(ξ/2))-expansion Method, Optik 127, 4222 (2016)

    Article  ADS  Google Scholar 

  • Njikue, R., Bogning, J.R., Kofane, T.C.: Higher-order nonlinear Schrödinger equation family in optical fiber and solitary wave solutions, Am. J. Opt. 6, 31–41 (2018)

    Google Scholar 

  • Oad, A., Arshad, M., Shoaib, M., Lu, D., Li, X.: Novel soliton solutions of two-mode Sawada-Kotera equation and its applications, IEEE Access 9, 127368–127381 (2021)

    Article  Google Scholar 

  • Porsezian, K., Nakkeeran, K.: Optical solitons in presence of Kerr dispersion and self-frequency shift, Phys. Rev. Lett. 76, 3955 (1996)

    Article  ADS  Google Scholar 

  • Seadawy, A.R., Arshad, M., Lu, D.: The weakly nonlinear wave propagation theory for the Kelvin-Helmholtz instability in magnetohydrodynamics flows, Chaos Solitons Fractals 139, 110141 (2020)

    Article  MathSciNet  Google Scholar 

  • Tala-Tebue, E., Djoufack, Z.I., Fendzi-Donfack, E., Kenfack-Jiotsa, A., Kofane, T.C.: Exact solutions of the unstable nonlinear Schrödinger equation with the new Jacobi elliptic function rational expansion method and the exponential rational function method, Optik 127, 11124 (2016)

    Article  ADS  Google Scholar 

  • Tala-Tebue, E., Tsobgni-Fozap, D.C., Kenfack-Jiotsa, A., Kofane, T.C.: Envelope periodic solutions for a discrete network with the Jacobi elliptic functions and the alternative (G'/G)-expansion method including the generalized Riccati equation, Eur. Phys. J. Plus 129, 136 (2014)

    Article  Google Scholar 

  • Wang, M.L.: Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A 199, 169 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  • Wang, M.L., Li, X.Z.: Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation, Chaos Solitons Fractals 24, 1257 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  • Wazwaz, A.M.: The tanh method for travelling wave solutions to the Zhiber-Shabat equation and other related equations, Commun. Nonlinear Sci. Numer. Simul. 13, 584 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  • Yan, Z.Y.: Abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional Integrable Davey-Stewartson-type equation via a new method, Chaos Solitons Fractals 18, 299 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  • Zayed, E.M.E., Alurrfi, K.A.E.: Solitons and other solutions for two nonlinear Schrödinger equations using the new mapping method, Optik 144, 132–148 (2017)

    Article  ADS  Google Scholar 

  • Zayed, E.M.E., Shahoot, A.M., Alurrfi, K.A.E.: (G'/G,1/G)-expansion method and its applications for constructing many new exact solutions of the higher-order nonlinear Schrödinger equation and the quantum Zakharov-Kuznetsov equation, Opt. Quant. Electron. 50, 96 (2018)

    Article  Google Scholar 

  • Zayed, E.M.E., Zedan, H.A., Gepreel, K.A.: On the solitary wave solutions for nonlinear Hirota-Sasuma coupled KDV equations, Chaos Solitons Fractals 22, 285 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  • Zhao, H., Han, J.-G., Wang, W.-T., An, H.-Y.: Applications of extended hyperbolic function method for Quintic Discrete Nonlinear Schrödinger Equation, Commun. Theor. Phys. 47, 474 (2007)

    Article  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. Unite de Recherche d’Automatique et Informatique Apliquee (UR-AIA), Department of Telecommunication and Network Engineering, Fotso Victor University Institute of Technology, P.O. Box 134, Bandjoun, Cameroon

    Eric Tala-Tebue

  2. Department of Physics, Faculty of Sciences, University of Bamenda Cameroon/University of Dschang, Bandjoun, Cameroon

    Cedric Tetchoka-Manemo

  3. Department of Mathematics, Firat University, 23119, Elazig, Turkey

    Mustafa Inc

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

    Mustafa Inc

  5. IUT-FV Bandjoun, Department of General and Scientific Studies, University of Dschang, Bandjoun, Cameroon

    Geh Wilson Ejuh

  6. Department of Electrical and Electronic Engineering, University of Bamenda, NAHPI, Bandjoun, Cameroon

    Geh Wilson Ejuh

  7. Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, 11432, Saudi Arabia

    Rubayyi T. Alqahtani

Authors
  1. Eric Tala-Tebue
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cedric Tetchoka-Manemo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mustafa Inc
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Geh Wilson Ejuh
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Rubayyi T. Alqahtani
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Conceptualization; Writing - original draft preparation; Methodology: [ET-T]; Formal analysis and investigation: [CT-M]; Writing - review and editing: [MI]; Resources: [GWE]; Resources; Supervision: [RTA]

Corresponding author

Correspondence to Mustafa Inc.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Consent for publication

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

Ethics Approval

Not applicable.

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

Tala-Tebue, E., Tetchoka-Manemo, C., Inc, M. et al. Bifurcation and optical solutions of the higher order nonlinear Schrödinger equation. Opt Quant Electron 55, 445 (2023). https://doi.org/10.1007/s11082-023-04691-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-023-04691-2

Keywords

Search

Navigation