Skip to main content
SpringerLink
Log in

Exact optical solitons to the perturbed nonlinear Schrödinger equation with dual-power law of nonlinearity

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

Abstract

In this paper, the perturbed nonlinear Schrödinger equation with dual-power law of nonlinearity is studied by adopting four mathematical methods namely the Modified Kudryashov method, the extended Tanh–Coth method, the modified simple equation method and soliton ansatz method. As a results, a set of various types of solitons that contains dark, singular, dark–singular, bright optical solitons and other form of optical soliton solutions are obtained. Firstly, we solve the perturbed nonlinear Schrödinger equation by considering the dual power law parameter using the first three integration methods and secondly we set this parameter equal to one in order, to solve the problem with the fourth method. The used methods in this paper present various applications in fields of nonlinear wave equations. Comparing our new results with well-known in literature are also given. Moreover, the graphical representations of the modulus of some optical soliton solutions and their 2-D profile are also depicted.

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

  • Abbagari, S., et al.: N-rotating loop-soliton solution of the coupled integrable dispersionless equation. J. Appl. Math. Phys. 5, 1370–1379 (2014)

    Article  Google Scholar 

  • Bekir, A.: Application of the (G’/G)-expansion method for nonlinear evolution equations. Phys. Lett. A 372, 3400–3406 (2008a)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Bekir, A.: Applications of the extended tanh method for coupled nonlinear evolution equations. Commun. Nonlinear Sci. Numer. Simul. 13, 1748–1757 (2008b)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Bekir, A.: New solitons and periodic wave solutions for some nonlinear physical models by using the sine-cosine method. Phys. Scr. 77, 045008 (2008c)

    Article  ADS  MATH  Google Scholar 

  • Bekir, A., Boz, A.: Exact solutions for nonlinear evolution equations using exp-function method. Phys. Lett A 372, 1619–1625 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Biswas, A.: Temporal 1-soliton solution of the complex Ginzburg-Landau equation with power law nonlinearity. Progr. Electromagn. Res. 96, 1–7 (2009)

    Article  Google Scholar 

  • Biswas, A., Konar, S.: Introduction to Non-Kerr Law Optical Solitons. CRC Press, Boca Raton (2006)

    Book  MATH  Google Scholar 

  • Boudoue, M.H., et al.: Dispersive solitons in optical metamaterials having parabolic form of nonlinearity. Optik Int. J. Light Electron Opt. 179, 1009–1018 (2018)

    Google Scholar 

  • Doka, S.Y., Mibaile, J., Gambo, B., Kofane, T.C.: Optical chirped soliton in metamaterials. Nonlinear Dyn. 90, 13–18 (2017)

    Article  MathSciNet  Google Scholar 

  • Gambo, B., Bouetou, B., Kamgang, V.K., Kofane, T.C.: Dynamical survey of a generalized-Zakharov equation and its exact travelling wave solutions. Appl. Math. Comput. 217, 203–211 (2010)

    MathSciNet  MATH  Google Scholar 

  • Ghanbari, B., Osman, M.S., Baleanu, D.: Generalized exponential rational function method for extended Zakharov–Kuzetsov equation with conformable derivative. Mod. Phys. Lett. A 34, 1950155 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Houwe, A., et al.: Solitons solutions of nonlinear Schrödinger equation in the left-handed metamaterials by three different techniques. J. Phys. Commun. 3, 011002 (2019)

    Article  Google Scholar 

  • Javid, A., Raza, N., Osman, M.S.: Multi-solitons of thermophoretic motion equation depicting the wrinkle propagation in substrate-supported graphene sheets. Commun. Theor. Phys. 71, 362–366 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  • Jawad, A.J.M., Biswas, A., et al.: Hamiltonian perturbation of optical solitons with parabolic law nonlinearity using three integration methodologies. Optik 160, 248–254 (2018)

    Article  ADS  Google Scholar 

  • Kohl, R., Milovic, D., Zerrad, E., Biswas, A.: Soliton perturbation theory for dispersion-managed optical fibers. J. Nonlinear Opt. Phys. Mater. 18, 227–270 (2009)

    Article  ADS  MATH  Google Scholar 

  • Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 2248–2253 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Liu, J.G., Osman, M.S., et al.: Different complex wave structures described by the Hirota equation with variable coefficients in inhomogeneous optical fibers. Appl. Phys. B 125, 175 (2019)

    Article  ADS  Google Scholar 

  • Nestor, S., et al.: New solitary waves for the Klein–Gordon–Zakharov equations. Mod. Phys. Lett. B 34, 2050246 (2020a)

    Article  Google Scholar 

  • Nestor, S., Betchewe, G., Inc, M., Doka, S.Y.: Exact traveling wave solutions to the higher-order nonlinear Schrödinger equation having Kerr nonlinearity form using two strategic integrations. Eur. Phys. J. Plus 135, 380 (2020bb)

    Article  Google Scholar 

  • Njikue, R., Bogning, J.R., Kofane, T.C.: Exact bright and dark solitary wave solutions of the generalized higher-order nonlinear Schrödinger equation describing the propagation of ultra-short pulse in optical fiber. J. Phys. Commun. 2, 025030 (2018)

    Article  Google Scholar 

  • Osman, M.S.: Nonlinear interaction of solitary waves described by multi-rational wave solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili equation with variable coefficients. Nonlinear Dyn. 87(2), 1209–1216 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Osman, M.S.: One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada–Kotera equation. Nonlinear Dyn. 96, 1491–1496 (2019a)

    Article  Google Scholar 

  • Osman, M.S.: New analytical study of water waves described by coupled fractional variant Boussinesq equation in fluid dynamics. Pramana 93, 26 (2019b)

    Article  ADS  Google Scholar 

  • Osman, M.S., Wazwaz, A.M.: A general bilinear form to generate different wave structures of solitons for a (3+1)-dimensional Boiti-Leon–Manna–Pempinelli equation. Math. Methods Appl. Sci. 42, 6277–6283 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Osman, M.S., Rezazadeh, H., Eslami, M.: Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov–Kuznetsov equation with power law nonlinearity. Nonlinear Eng. 8, 559–567 (2019a)

    Article  ADS  Google Scholar 

  • Osman, M.S., Lu, D., Khater, M.M.A.: A study of optical wave propagation in the nonautonomous Schrödinger-Hirota equation with power-law nonlinearity. Results Phys. 13, 102157 (2019b)

    Article  Google Scholar 

  • Rezazadeh, H.: New solitons solutions of the complex Ginzburg–Landau equation with Kerr law nonlinearity. Optik 167, 218–227 (2018)

    Article  ADS  Google Scholar 

  • Rezazadeh, H., et al.: New optical solitons of nonlinear conformable fractional Schrödinger-Hirota equation. Optik 172, 545–553 (2018a)

    Article  ADS  Google Scholar 

  • Rezazadeh, H., et al.: Mitigating internet bottleneck with fractional temporal evolution of optical solitons having quadratic-cubic nonlinearity. Optik 164, 84–92 (2018b)

    Article  ADS  Google Scholar 

  • Rezazadeh, H., Manafian, J., Khodadad, F.S., Nazari, F.: Traveling wave solutions for density-dependent conformable fractional diffusion-reaction equation by the first integral method and the improved tan(1/2())-expansion method. Opt. Quant. Electron. 50(3), 121 (2018c)

    Article  Google Scholar 

  • Rezazadeh, H., Osman, M.S., et al.: Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations. Nonlinear Eng. 8, 224–230 (2019)

    Article  ADS  Google Scholar 

  • Savaissou, N., et al.: New Jacobi elliptic solutions and other solutions with quadratic-cubic nonlinearity using two mathematical methods. Asian Eur. J. Math. 13, 2050043 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R.: Two-dimensional interaction of a shear flow with a free surface in a stratified fluid and its solitary-wave solutions via mathematical methods. Eur. Phys. J. Plus 132, 518 (2017a)

    Article  Google Scholar 

  • Seadawy, A.R.: Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev–Petviashvili dynamical equation for dispersive shallow-water waves. Eur. Phys. J. Plus 132, 29 (2017b)

    Article  Google Scholar 

  • Seadawy, A.R., Arshad, M., Lu, D.: Stability analysis of new exact traveling-wave solutions of new coupled KdV and new coupled Zakharov–Kuznetsov systems. Eur. Phys. J. Plus 132, 162 (2017)

    Article  Google Scholar 

  • Wang, M., Li, X., Zhang, J.L.: Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation. Phys. Lett. A 363, 96–101 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Wazwaz, A.M.: The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. Appl. Math. Comput. 187, 1131–1142 (2007)

    MathSciNet  MATH  Google Scholar 

  • Yepez-Martinez, H., Rezazadeh, H., et al.: The extended modified method applied to optical solitons solutions in birefringent fibers with weak nonlocal nonlinearity and four wave mixing. Chin. J. Phys. 58, 137–150 (2019)

    Article  Google Scholar 

  • Zayed, E.M.E., Abdelaziz, M.A.M.: The tanh function method using a generalized wave transformation for nonlinear equations. Int. J. Nonlinear Sci. Numer. Simul. 11, 595–601 (2010)

    Google Scholar 

  • Zayed, E.M., Al-Nowehy, A.G.: Many new exact solutions to the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms using three different techniques. Optik 143, 84–103 (2017)

    Article  ADS  Google Scholar 

  • Zhang, J.L., Wang, M., Li, X.: The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrödinger equation. Phys. Lett. A 357, 188–195 (2006)

    Article  ADS  MATH  Google Scholar 

  • Zhou, Q., Yao, D.Z., Chen, F., et al.: Optical solitons with nonlinear dispersion in polynomial law medium. J. Optoelectron. Adv. Mater. 17, 82–86 (2015)

    Google Scholar 

Download references

Acknowledgements

The five authors of this work wish to thank the referees for their comments on this paper.

Funding

This research received no external funding.

Author information

Authors and Affiliations

  1. Department of Physics, Faculty of Science, University of Maroua, P.O. Box 814, Maroua, Cameroon

    Nestor Savaissou, B. Gambo & Serge Y. Doka

  2. Faculty of Engineering Technology, Amol University of Special Modern Technologies, Amol, Iran

    Hadi Rezazadeh

  3. Neighbourhood of Akcaglan, Imarli Street, Number: 28/4, 26030, Eskisehir, Turkey

    Ahmet Bekir

  4. Department of Physics, Faculty of Science, University of Ngaoundere, P.O. Box 454, Ngaoundere, Cameroon

    Serge Y. Doka

Authors
  1. Nestor Savaissou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Gambo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hadi Rezazadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ahmet Bekir
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Serge Y. Doka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmet Bekir.

Ethics declarations

Conflict of interest

No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Savaissou, N., Gambo, B., Rezazadeh, H. et al. Exact optical solitons to the perturbed nonlinear Schrödinger equation with dual-power law of nonlinearity. Opt Quant Electron 52, 318 (2020). https://doi.org/10.1007/s11082-020-02412-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-020-02412-7

Keywords

Search

Navigation