Skip to main content
SpringerLink
Log in

Analytical solutions along with grey–black optical solitons under the influence of self-steepening effect and third order dispersion

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

Abstract

In this article, the propagation of breathers, multiwave, M-shape solitons, grey–black optical solitons, periodic cross rational (PCR) solitons, kink cross rational (KCR) solitons will be discussed under the effect of self-steepening and third-order of linear dispersion in single-mode fibers. All new analytical solutions are found by utilizing the symbolic computation with ansatz functions scheme and logarithmic transformation. Moreover two kinds of interaction of M-shape soliton with kink wave will be examined. We apply nine different approaches, namely, three-waves technique, homoclinic breather technique, M-shape solitons, single exponential form (one-kink), double exponential form (two-kink), PCR, KCR, grey optical soliton and black optical soliton for governing model. Moreover, stability characteristics of all solutions are evaluated.

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
Fig. 14
Fig. 15

Similar content being viewed by others

Availability of data and materials

Not applicable.

References

  • Agrawal, G. P. Nonlinear fiber optics. In Nonlinear Science at the Dawn of the 21st Century, Springer, Berlin, Heidelberg (2000) vol 542, pp. 195-211

  • Agrawal, G.P.: Nonlinear fiber optics: its history and recent progress. JOSA B 28(12), 1–10 (2011)

    ADS  Google Scholar 

  • Ahmed, S., Ashraf, R., Seadawy, A.R., Rizvi, S.T.R., Younis, M., Althobaiti, A., El-Shehawi, A.M.: Lump, multi-wave, kinky breathers, interactional solutions and stability analysis for general \((2+ 1)\)-rth dispersionless Dym equation. Res. Phys. 25, 104160 (2021)

    Google Scholar 

  • Ahmed, I., Seadawy, A.R., Lu, D.: \(M\)-shaped rational solitons and their interaction with kink waves in the Fokas–Lenells equation. Physica Scripta 94, 055205 (2019)

    ADS  Google Scholar 

  • Ahmed, Iftikhar, Seadawy, A.R., Lu, D.: Kinky breathers, W-shaped and multi-peak solitons interaction in (2+1)-dimensional nonlinear Schrodinger’s equation with kerr law of nonlinearity. Eur. Phys. J. Plus 134(120), 1–11 (2019)

    Google Scholar 

  • Ahmed, I., Seadawy, A.R., Lu, D.: Kinky breathers, W-shaped and multi-peak solitons interaction in \((2+ 1)\)-dimensional nonlinear Schrödinger equation with Kerr law of nonlinearity. Eur. Phys. J. Plus 134(3), 1–10 (2019)

    Google Scholar 

  • Ahmed, S., Seadawy, A.R., Rizvi, S.T.: Study of breathers, rogue waves and lump solutions for the nonlinear chains of atoms. Opt. Quant. Electron. 54(5), 1–28 (2022)

    Google Scholar 

  • Akram, U., Seadawy, A.R., Rizvi, S.T.R., Younis, M., Althobaiti, S., Sayed, S.: Traveling wave solutions for the fractional Wazwaz–Benjamin–Bona–Mahony model in arising shallow water waves. Res. Phys. 20, 103725 (2021)

    Google Scholar 

  • Ali, K., Seadawy, A.R., Ahmed, S., Rizvi, S.T.: Discussion on rational solutions for Nematicons in liquid crystals with Kerr Law. Chaos Solitons Fractals 160, 112218 (2022)

    MathSciNet  MATH  Google Scholar 

  • Aliyu, A.I., Tchier, F., Inc, M., Yusuf, A., Baleanu, D.: Dynamics of optical solitons, multipliers and conservation laws to the nonlinear Schrödinger equation in (2+1)-dimensions with non-Kerr law nonlinearity. J. Mod. Opt. 0950, 1362–3044 (2018)

    Google Scholar 

  • Amiranashvili, Shalva, Bandelow, Uwe, Akhmediev, Nail: Ultrashort optical solitons in transparent nonlinear media with arbitrary dispersion. Opt. Quant. Electron. 46, 1233–1238 (2014)

    Google Scholar 

  • Muniyappan, A., Sahasraari, L.N., Anitha, S., Ilakiya, S., Biswas, A., Yıldırım, Y., Triki, H., Alshehri, H.M., Belic, M.R.: Family of optical solitons for perturbed Fokas–Lenells equation. Optik 249, 168224 (2022)

    ADS  Google Scholar 

  • Bilal, M., Seadawy, A.R., Younis, M., Rizvi, S.T.R., El-Rashidy, K., Mahmoud, S.F.: Analytical wave structures in plasma physics modelled by Gilson–Pickering equation by two integration norms. Res. Phys. 23, 103959 (2021)

    Google Scholar 

  • Bilal, M., Seadawy, A.R., Younis, M., Rizvi, S.T.R., Zahed, H.: Dispersive of propagation wave solutions to unidirectional shallow water wave Dullin–Gottwald–Holm system and modulation instability analysis. Math. Methods Appl. Sci. 44(5), 4094–4104 (2021)

    ADS  MathSciNet  MATH  Google Scholar 

  • Chen, J., Luan, Z., Zhou, Q., Alzahrani, A.K., Biswas, A., Liu, W.: Periodic soliton interactions for higher-order nonlinear Schrödinger equation in optical fibers. Nonlinear Dyn. 100(3), 2817–2821 (2020)

    Google Scholar 

  • Ghaffar, A., Ali, A., Ahmed, S., Akram, S., Baleanu, D., Nisar, K.S.: A novel analytical technique to obtain the solitary solutions for nonlinear evolution equation of fractional order. Adv. Diff. Equ. 1, 1–15 (2020)

    MathSciNet  MATH  Google Scholar 

  • Jiang, C., Cui, J., Qian, X., Song, S.: High-order linearly implicit structure-preserving exponential integrators for the nonlinear Schrödinger equation. J. Sci. Comput. 90(1), 1–27 (2022)

    MATH  Google Scholar 

  • Kalithasan, B., Nakkeeran, K., Porsezian, K., Dinda, P.T., Mariyappa, N.: Ultra-short pulse propagation in birefringent fibers-the projection operator method. J. Opt. A Pure Appl. Opt. 10, 085102 (2008)

    ADS  Google Scholar 

  • Kasapeteva, Z., Dakova, A., Krasteva, S., Slavchev, V., Dakova, D., Kovachev, L., Biswas, A.: Bright solitons under the influence of third-order dispersion and self-steepening effect. Opt. Quant. Electron. 54(6), 1–8 (2022)

    Google Scholar 

  • Kudryashov, N.A.: Almost general solution of the reduced higher-order nonlinear Schrödinger equation. Optik 230, 166347 (2021)

    ADS  Google Scholar 

  • Kudryashov, N.A.: Optical solitons of the resonant nonlinear Schrödinger equation with arbitrary index. Optik 235, 166626 (2021)

    ADS  Google Scholar 

  • Liu, Y., Li, B., Wazwaz, A.M.: Novel high-order breathers and rogue waves in the Boussinesq equation via determinants. Int. J. Mod. Phys. B 43(6), 3701–3715 (2020)

    MathSciNet  MATH  Google Scholar 

  • Lu, D., Seadawy, A., Arshad, M.: Bright-Dark optical soliton and dispersive elliptic function solutions of unstable nonlinear Schrodinger equation and its applications. Opt. Quant. Electron. 50(23), 1–10 (2018)

    Google Scholar 

  • Ma, G., Zhao, J., Zhou, Q., Biswas, A., Liu, W.: Soliton interaction control through dispersion and nonlinear effects for the fifth-order nonlinear Schrödinger equation. Nonlinear Dyn. 106(3), 2479–2484 (2021)

    Google Scholar 

  • Manafian, J., Mohammadi Ivatloo, B., Abapour, M.: Breather wave, periodic, and cross-kink solutions to the generalized Bogoyavlensky–Konopelchenko equation. Math. Methods Appl. Sci. 43(4), 1753–1774 (2020)

    ADS  MathSciNet  MATH  Google Scholar 

  • Marin, M., Seadawy, A., Vlase, S., Chirila, Adina: On mixed problem in thermoelasticity of type III for Cosserat media. J. Taibah Univ. Sci. 16(1), 1264–1274 (2022)

    Google Scholar 

  • Mo, Y., Ling, L., Zeng, D.: Data-driven vector soliton solutions of coupled nonlinear Schrödinger equation using a deep learning algorithm. Phys. Lett. A 421, 127739 (2022)

    MATH  Google Scholar 

  • Muniyappan, A., Monisha, P., Kaviya, Priya E., Nivetha, V.: Generation of wing-shaped dark soliton for perturbed Gerdjikov-Ivanov equation in optical fibre. Optik 30, 166328 (2021)

  • Musammil, N.M., Porsezian, K., Nithyanandan, K., Subha, P.A., Dinda, P.T.: Ultrashort dark solitons interactions and nonlinear tunneling in the modified nonlinear Schrödinger equation with variable coefficient. Opt. Fiber Technol. 37, 11–20 (2017)

    ADS  MATH  Google Scholar 

  • Rizvi, S.T.R., Seadawy, A.R., Raza, U.: Some advanced chirped pulses for generalized mixed for nonlinear Schrödinger dynamical equation. Chaos Solitons Fractals 163, 112575 (2022)

    MATH  Google Scholar 

  • Rizvi, S.T.R., Seadawy, A.R., Sohail, M., Ali, K.: Nonlinear model under anomalous dispersion regime: chirped periodic and solitary waves. Chaos Solitons Fractals 163, 112558 (2022)

    MathSciNet  MATH  Google Scholar 

  • Rizvi, S.T.R., Seadawy, A.R., Younis, M., Ali, I., Althobaiti, S., Mahmoud, S.F.: Soliton solutions, Painleve analysis and conservation laws for a nonlinear evolution equation. Res. Phys. 23, 103999 (2021)

    Google Scholar 

  • Rizvi, S.T.R., Seadawy, A.R., Younis, M., Iqbal, S., Althobaiti, S., El-Shehawi, A.M.: Various optical soliton for a weak fractional nonlinear Schrödinger equation with parabolic law. Res. Phys. 23, 103998 (2021)

    Google Scholar 

  • Savaissou, N., Gambo, B., Rezazadeh, H., Bekir, A., Doka, S.Y.: Exact optical solitons to the perturbed nonlinear Schrödinger equation with dual-power law of nonlinearity. Opt. Quant. Electron. 52, 1–16 (2020)

    Google Scholar 

  • Seadawy, A.R., Ahmed, S., Rizvi, S.T., Ali, K.: Lumps, breathers, interactions and rogue wave solutions for a stochastic gene evolution in double chain deoxyribonucleic acid system. Chaos Solitons Fractals 161, 112307 (2022)

    MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Ahmed, S., Rizvi, S.T., Ali, K.: Various forms of lumps and interaction solutions to generalized Vakhnenko Parkes equation arising from high-frequency wave propagation in electromagnetic physics. J. Geometry Phys. 176, 104507 (2022)

    MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Bilal, M., Younis, M., Rizvi, S.T.R., Althobaiti, S., Makhlouf, M.M.: Analytical mathematical approaches for the double-chain model of DNA by a novel computational technique. Chaos Solitons Fractals 144, 110669 (2021)

    MathSciNet  Google Scholar 

  • Seadawy, A.R., Rizvi, S.T., Ahmed, S.: Multiple lump, generalized breathers, Akhmediev breather, manifold periodic and rogue wave solutions for generalized Fitzhugh-Nagumo equation: Applications in nuclear reactor theory. Chaos Solitons Fractals 161, 112326 (2022)

    MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Rizvi, S.T., Ahmed, S.: Weierstrass and Jacobi elliptic, bell and kink type, lumps, Ma and Kuznetsov breathers with rogue wave solutions to the dissipative nonlinear Schrödinger equation. Chaos Solitons Fractals 160, 112258 (2022)

    MATH  Google Scholar 

  • Seadawy, A.R., Rizvi, S.T., Ashraf, M.A., Younis, M., Hanif, M.: Rational solutions and their interactions with kink and periodic waves for a nonlinear dynamical phenomenon. Int. J. Modern Phys. B 35, 2150236 (2021)

    ADS  MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R.: Stability analysis solutions for nonlinear three-dimensional modified Korteweg-de Vries–Zakharov–Kuznetsov equation in a magnetized electron-positron plasma. Phys. A: Stat. Mech. Appl. 455, 44–51 (2016)

    MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R.: The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions. Optik 139, 31–34 (2017)

    ADS  Google Scholar 

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

    Google Scholar 

  • Seadawy, A.R., Rehman, S.U., Younis, M., Rizvi, S.T.R., Althobaiti, S., Makhlouf, M.M.: Modulation instability analysis and longitudinal wave propagation in an elastic cylindrical rod modelled with Pochhammer-Chree equation. Phys. Scr. 96(4), 045202 (2021)

    ADS  Google Scholar 

  • Seadawy, A.R., Rizvi, S.T.R., Ahmad, S., Younis, M., Baleanu, D.: Lump, lump-one stripe, multiwave and breather solutions for the Hunter–Saxton equation. Open Phys. 19(1), 1–10 (2021)

    Google Scholar 

  • Seadawy, A.R., Rizvi, S.T., Ali, I., Younis, M., Ali, K., Makhlouf, M.M., Althobaiti, A.: Conservation laws, optical molecules, modulation instability and Painlevé analysis for the Chen-Lee-Liu model. Opt. Quant. Electron. 53(4), 1–15 (2021)

    Google Scholar 

  • Serkin, V.N., Hasegawa, A.: Soliton management in the nonlinear Schrödinger equation model with varying dispersion, nonlinearity, and gain. J. Exp. Theor. Phys. Lett. 72(2), 89–92 (2000)

    Google Scholar 

  • Shah, K., Seadawy, A.R., Arfan, Muhammad: Evaluation of one dimensional fuzzy fractional partial differential equations. Alex. Eng. J. 59, 3347–3353 (2020)

    Google Scholar 

  • Sun, J., Liu, D., Huang, Dexiu: Difference-frequency generation among ultrashort optical pulses in Quasi-phase-matching waveguides. Opt. Quant. Electron. 36, 577–587 (2004)

    Google Scholar 

  • Taghizadeh, N., Mirzazadeh, M., Farahrooz, F.: Exact solutions of the nonlinear Schrödinger equation by the first integral method. J. Math. Anal. Appl. 374(2), 549–553 (2011)

    MathSciNet  MATH  Google Scholar 

  • Tariq, K.U., Zainab, H., Seadawy, A.R., Younis, M., Rizvi, S.T.R., Abd Allah, A.M.: On some novel optical wave solutions to the paraxial M-fractional nonlinear Schrödinger dynamical equation. Opt. Quant. Electron. 53(5), 1–14 (2021)

    Google Scholar 

  • Serkin, V.N., Hasegawa, A.: Novel Soliton solutions of the nonlinear Schrödinger equation. Model. Phys. Rev. Lett. 85, 4502 (2003)

    ADS  Google Scholar 

  • Wang, K.J., Wang, G.D.: Variational theory and new abundant solutions to the (1+ 2)-dimensional chiral nonlinear Schrödinger equation in optics. Phys. Lett. A 412, 127588 (2021)

    MATH  Google Scholar 

  • Wazwaz, A.M.: Higher dimensional nonlinear Schrödinger equations in anomalous dispersion and normal dispersive regimes: bright and dark optical solitons. Optik 222, 165327 (2020)

    ADS  Google Scholar 

  • Weng, W., Zhang, G., Zhang, M., Zhou, Z., Yan, Z.: Semi-rational vector Rogon–Soliton solutions and asymptotic analysis for any n-component nonlinear Schrödinger equation with mixed boundary conditions. Phys. D Nonlinear Phenomena 432, 133150 (2022)

    MATH  Google Scholar 

  • Younas, U., Younis, M., Seadawy, A.R., Rizvi, S.T.R., Althobaiti, S., Sayed, S.: Diverse exact solutions for modified nonlinear Schrödinger equation with conformable fractional derivative. Res. Phys. 20, 103766 (2021)

    Google Scholar 

  • Younis, M., Seadawy, A.R., Bilal, M., Rizvi, S.T.R., Althobaiti, S., Alkafafy, M.: Nonlinear dynamical wave structures to the Date–Jimbo–Kashiwara–Miwa equation and its modulation instability analysis. Mod. Phys. Lett. B 35, 2150300 (2021)

    ADS  MathSciNet  Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Mathematics Department, Faculty of Science, Taibah University, 41411, Al-Madinah Al-Munawarah, Kingdom of Saudi Arabia

    Aly R. Seadawy

  2. Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Islamabad, Pakistan

    Syed T. R. Rizvi & Sarfaraz Ahmed

Authors
  1. Aly R. Seadawy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Syed T. R. Rizvi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sarfaraz Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aly R. Seadawy.

Ethics declarations

Conflict of Interest

The authors declare no conflict of interest.

Ethical approval

I hereby declare that this manuscript is the result of my independent creation under the reviewers’ comments. Except for the quoted contents, this manuscript does not contain any research achievements that have been published or written by other individuals or groups.

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

Seadawy, A.R., Rizvi, S.T.R. & Ahmed, S. Analytical solutions along with grey–black optical solitons under the influence of self-steepening effect and third order dispersion. Opt Quant Electron 55, 288 (2023). https://doi.org/10.1007/s11082-023-04559-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-023-04559-5

Keywords

Search

Navigation