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Optical soliton solutions: the evolution with changing fractional-order derivative in Biswas–Arshed and Schrödinger Kerr law equations

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Abstract

The space–time fractional Biswas–Arshed and Schrödinger Kerr law equations featuring beta derivative hold substantial application in nonlinear optics, optical solitons, ultrafast optical signal, nonlinear photonics, quantum optics, biophotonics, photonic crystals photonics, etc. In this study, a wide variety of geometric shape solitons have been established that include hyperbolic, exponential, trigonometric, and rational functions, as well as their assimilation to the considered equations, through the two-variable (\(R^{\prime } /R, 1/R\))-expansion approach. The implication of the fractional parameter \(\mu \) on the wave shape has also been examined by depicting two-dimensional and three-dimensional plots for particular parameter values. The solitons include irregular periodic, pulse like, V-shaped, bell-shaped, positive periodic, asymptotic, general solitons, and some others. It is significant to note that the changes in the wave pattern result from the adjustments to substantive and auxiliary parameters. The outcomes demonstrate the efficiency, acceptability, and dependability of the (\(R^{\prime } /R, 1/R\))-expansion approach for obtaining solutions to the fractional-order evolution equations in the domains of engineering, technology, and sciences. It is evident from the graph that changing the value of μ results in a change in the shape of the soliton. The study explores how these equations change as fractional-order derivatives vary. Soliton solutions, which are stable, localized waveforms, are crucial in optical communication systems. Understanding their behavior under changing fractional-order derivatives is essential for advancing optical signal processing and communication technologies.

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References

  • Ablowitz, M.J., Clarkson, P.A.: Solitons Nonlinear Evolution Equation and Inverse Scattering. Cambridge University Press, New York (1991)

    Book  Google Scholar 

  • Ahamed, J., Rani, S., Turki, N.B., Shah, N.A.: Novel resonant multi-soliton solutions of time fractional coupled nonlinear Schrödinger equation in optical fiber via an analytical method. Results Phys. 52, 106761 (2023)

    Article  Google Scholar 

  • Ahmad, J., Akram, S., Noor, K., Nadeem, M., Bucur, A., Alsayaad, Y.: Soliton solutions of fractional extended nonlinear Schrödinger equation arising in plasma physics and nonlinear optical fiber. Sci. Reports. 13, 10877 (2023a)

    CAS  ADS  Google Scholar 

  • Ahmad, J., Akram, S., Rehman, S.U., Turki, N.B., Shah, N.A.: Description of soliton and lump solutions to M-truncated stochastic Biswas–Arshed model in optical communication. Results Phys. 51, 106719 (2023b)

    Article  Google Scholar 

  • Ahmed, I., Seadawy, A.R., Liu, D.: Kinky breathers, W-shaped and multipeak solitons interactions in (2+1)-dimensional nonlinear Schrödinger equation with Kerr law of nonlinearity. European plus. J. plus. 134, 1–10 (2019)

    CAS  Google Scholar 

  • Akram, S., Ahmed, J., Rehman, S.U., Ali, A.: Dynamics of soliton solutions in optical fibers modelled by perturbed nonlinear Schrödinger equation and stability analysis. Opt. Quant. Electron. 55, 450 (2023)

    Article  Google Scholar 

  • Ali, A., Ahmed, J., Javed, S.: Investigating the dynamics of soliton solutions to the fractional coupled nonlinear Schrödinger model with their bifurcation and stability analysis. Opt. Quant. Electron. 55, 829 (2023a)

    Article  Google Scholar 

  • Ali, A., Ahmad, J., Javed, S.: Exploring the dynamic nature of soliton solutions to the fractional coupled nonlinear Schrödinger model with their sensitivity analysis. Opt. Quant. Electron. 55, 810 (2023b)

    Article  Google Scholar 

  • Alqarni, A.A., Ebaid, A., Alshaery, A.A., Bakodah, H.O., Biswas, A., Khan, S., Ekici, M., Zhou, Q., Moshokoa, S.P., Belic, M.R.: Optical solitons for Lakshmanan–Porsezian–Daniel model by Riccati equation approach. Optik 182, 922–929 (2019)

    Article  ADS  Google Scholar 

  • Asghari, Y., Eslami, M., Rezazadeh, H.: Novel optical solitons for the Ablowitz-Ladik lattice equation with conformable derivatives in the optical fibers. Opt. Quant. Electron. 55(10), 930 (2023a)

    Article  Google Scholar 

  • Asghari, Y., Eslami, M., Rezazadeh, H.: Exact solutions to the conformable time-fractional discretized mKdv lattice system using the fractional transformation method. Opt. Quant. Electron. 55(4), 318 (2023)

    Article  Google Scholar 

  • Asghari, Y., Eslami, M., Rezazadeh, H.: Soliton solutions for the time-fractional nonlinear diferential-diference equation with conformable derivatives in the ferroelectric materials. Opt. Quant. Electron. 55(4), 289 (2023)

    Article  Google Scholar 

  • Bakodah, H.O., Al-Qarni, A.A., Banaja, M.A., Zhou, Q., Moshokoa, S.P., Biswas, A.: Bright and dark thirring optical solitons with improved adomian decomposition method. Optik 130, 1115–1132 (2017)

    Article  ADS  Google Scholar 

  • Baleanu, D., Ghassabzade, F.A., Nieto, J.J., Jajarmi, A.: On a new and generalized fractional model for a real cholera outbreak. Alex. Eng. J. 61(11), 9175–9186 (2022)

    Article  Google Scholar 

  • Bilal, M., Rehman, S.U., Ahmad, J.: Dynamics of diverse optical solitary wave solutions to the Biswas–Arshed equation in nonlinear optics. Int. J. Appl. Comput. Math. 8, 137 (2022)

    Article  MathSciNet  Google Scholar 

  • Biswas, A., Mirzazadeh, M., Eslami, M., Zhou, Q., Bhrawy, A., Belic, M.: Optical solitons in nano-fibers with spatio-temporal dispersion by trial solution method. Optik 127(18), 7250–7257 (2016)

    Article  ADS  Google Scholar 

  • Biswas, A., Ullah, M.Z., Asma, M., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical solutions with quadratic-cubic nonlinearity by semi-inverse variational principle. Optik 139, 16–19 (2017)

    Article  CAS  ADS  Google Scholar 

  • Biswas, A., Yildirim, Y., Yasar, E., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical solitons with differential group delay by trial equation method. Optik 160, 116–123 (2018)

    Article  ADS  Google Scholar 

  • Das, N., Roy, S.S.: Dispersive optical soliton solutions of the (2+1)-dimensional cascaded system governing by coupled nonlinear Schrödinger equation with Kerr law nonlinearity in plasma. Opt. Quant. Electron. 55, 328 (2023)

    Article  CAS  ADS  Google Scholar 

  • Ekici, M., Sonmezoglu, A.: Optical solitons with Biswas–Arshed equation by extended trial function method. Optik 177, 13–20 (2019)

    Article  ADS  Google Scholar 

  • Ekici, M., Sonmezoglu, A., Zhou, Q., Biswas, A., Ullah, M.Z., Asma, M., Moshokoa, S.P., Belic, M.: Optical solitons in dwdm system by extended trial solution method. Optik 141, 157–167 (2017)

    Article  ADS  Google Scholar 

  • Eslami, M., Rezazadeh, H.: The first integral method for Wu-Zhang system with conformable time-fractional derivative. Calcolo 53, 475–485 (2016)

    Article  MathSciNet  Google Scholar 

  • Ghanbari, B.: Abundant soliton solutions for the Hirota–Maccari equation via the generalized exponential rational function method. Mod. Phys. Lett. B. 33(09), 1950106 (2019)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  • Ghanbari, B., Akgul, A.: Abundant new analytical and approximate solutions to the generalized Schamel equation. Phys. Scr. 95(7), 075201 (2020)

    Article  CAS  ADS  Google Scholar 

  • Ghanbari, B., Baleanu, D.: New solutions of gardner’s equation using two analytical methods. Front. Phys. 7, 202 (2019)

    Article  Google Scholar 

  • Ghanbari, B., Baleanu, D.: New optical solutions of the fractional Gerdjikov–Ivanov equation with conformable derivative. Stat. Comput. Phys. 8, 00167 (2020)

    Google Scholar 

  • Ghanbari, B., Gomez-Aguilar, J.F.: Optical soliton solutions for the nonlinear Radhakrishnan–Kundu–Lakshmanan equation. Mod. Phys. Lett. B 33(32), 1950402 (2019a)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  • Ghanbari, B., Gomez-Aguilar, J.F.: New exact optical soliton solutions for nonlinear Schrödinger equation with second-order spatio-temporal dispersion involving M-derivative. Mod. Phys. Lett. B 33(20), 1950235 (2019b)

    Article  CAS  ADS  Google Scholar 

  • Ghanbari, B., Kuo, C.K.: New exact wave solutions of the variable-coefficient (1 + 1)- dimensional Benjamin–Bona-Mahony and (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov equations via the generalized exponential rational function method. Eur. Phys. J. plus. 134, 334 (2019)

    Article  Google Scholar 

  • Ghanbari, B., Baleanu, D., Qurashi, M.A.: New exact solutions of the generalized Benjamin–Bona–Mahony Equation. Symmetry 11(1), 11010020 (2019)

    Google Scholar 

  • Hirota, R.: Exact solutions of Kdv equation for multiple collisions of soliton. Phys. Rev. Lett. 27, 1192–1194 (1971)

    Article  ADS  Google Scholar 

  • Jajarmi, A., Baleanu, D., Sajjadi, S.S., Nieto, J.: Analysis and some applications of a regularized ψ-Hilfer fractional derivative. J. Comput. Appl. Math. 415, 114476 (2022)

    Article  MathSciNet  Google Scholar 

  • Jawad, A.J.M., Petkovic, M.D., Biswas, A.: Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 217, 869–877 (2010)

    MathSciNet  Google Scholar 

  • Khan, A., Alshehri, H.M., Gomez-Aguilar, J.F., Khan, Z.A., Fernandez-Anaya, G.: A predator-prey model involving variable-order fractional differential equations with Mittag–Leffler Kernel. Adv. Differ. Eq. 2021, 183 (2021)

    Article  MathSciNet  Google Scholar 

  • Khater, M., Ghanbari, B.: On the solitary wave solutions and physical characterization of gas diffusion in a homogeneous medium via some efficient techniques. Eur. Phys. J. plus. 136, 447 (2021)

    Article  Google Scholar 

  • Krishnan, E.V., Biswas, A., Zhou, Q., Alfiras, M.: Optical soliton perturbation with Fokas Lenells equation by mapping methods. Optik 178, 104–110 (2019)

    Article  ADS  Google Scholar 

  • Li, Z.: Bifurcation and traveling wave solution to fractional Biswas–Arshed equation with the beta time derivative. Chaos Solitons Fractals 160, 112249 (2022)

    Article  MathSciNet  Google Scholar 

  • Mathanaranjan, T.: An effective technique for the conformable space-time fractional cubic-quartic nonlinear Schrödinger equation with different laws of nonlinearity. Comput. Method Differ. Equ. 10(3), 701–715 (2022)

    MathSciNet  Google Scholar 

  • Mathanaranjan, T.: Optical solitons and stability analysis for the new (3+1)-dimensional nonlinear Schrödinger equation. J. Nonlinear. Opt. Phys. Material. 32(2), 2350016 (2023a)

    Article  ADS  Google Scholar 

  • Mathanaranjan, T.: Optical soliton, linear stability analysis and conservation laws via multipliers to the integrable Kuralay equation. Optik 290, 171266 (2023b)

    Article  ADS  Google Scholar 

  • Mathanaranjan, T., Kumar, D., Rezazadeh, H., Akinyemi, L.: Optical solitons in metamaterials with third and fourth order dispersions. Opt. Quant. Electron. 54(5), 271 (2022)

    Article  Google Scholar 

  • Mathanaranjan, T., Hashemi, M.S., Rezazadeh, H., Akinyemi, L., Bekir, A.: Chirped optical solitons and stability analysis of the nonlinear Schrödinger equation with nonlinear chromatic dispersion. Commun. Theor. Phys. 75(8), 085005 (2023)

    Article  ADS  Google Scholar 

  • Mathanaranjan, T.: New Jacobi elliptic solutions and other solutions in optical metamaterials having higher-order dispersion and its stability analysis. Int. J. Comput. Math. 9(5), 66 (2023)

    MathSciNet  Google Scholar 

  • Neirameh, A., Eslami, M.: New optical soliton of stochastic chiral nonlinear Schrödinger equation. Opt. Quant. Electron. 55(5), 444 (2023)

    Article  Google Scholar 

  • Odabasi, M.: Traveling wave solutions of conformable time fractional Zakharov Kuznetsov and Zoomeron equations. Chin. J. Phys. 64, 194–202 (2020)

    Article  MathSciNet  CAS  Google Scholar 

  • Ozisik, M., Secer, A., Bayram, M.: The bell-shaped perturbed dispersive optical solitons of Biswas–Arshed equation using the new Kudryashov’s approach. Optik 267, 169650 (2022)

    Article  ADS  Google Scholar 

  • Pinar, Z., Rezazadeh, H., Eslami, M.: Generalized logistic equation method for Kerr law and dual power law Schrödinger equations. Opt. Quant. Electron. 52, 504 (2020)

    Article  Google Scholar 

  • Rehman, H.U., Ullah, N., Imran, M.A.: Optical solutions of Biswas–Arshed equation in birefringent fibers using extended direct algebraic method. Optik 226(2), 165378 (2021)

    Article  ADS  Google Scholar 

  • Sajid, N., Akram, G.: Novel solutions of Biswas–Arshed equation by newly ϕ6-model expansion method. Optik 211(6), 64564 (2020)

    ADS  Google Scholar 

  • Samir, I., Badra, N., Ahmed, H.M., Arnous, A.H.: Exploring soliton solutions to sixth order dispersive nonlinear Schrödinger equation with Kerr law nonlinearity using modified extended direct algebraic method. Opt. Quant. Electron. 184, 152 (2023)

    Google Scholar 

  • Sarma, A.K., Saha, M., Biswas, A.: Optical solutions with power law nonlinearity and Hamiltonian perturbations an exact solution. J. Infrared Milli Terahz wave. 31, 1048–1056 (2010)

    Article  CAS  Google Scholar 

  • Wazwaz, A.M.: The tanh and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants. Commu. Nonlinear Sci. Nurner. Simul. 11, 148–160 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  • Wiss, J., Tabor, M., Carnevale, G.: The Painleve property for partial differential equations. J. Math. Phys. 24, 522–526 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  • Yel, G., Bulut, H.: New wave approach to the conformable resonant nonlinear Schrödinger equation with Kerr law nonlinearity. Opt. Quant. Electron. 54, 1–13 (2022)

    Article  Google Scholar 

  • Yildirim, Y.: Optical solitons of Biswas–Arshed equation by trial equation technique. Optik 182, 876–883 (2019)

    Article  ADS  Google Scholar 

  • Zayed, E.M.E., Al-Nowehy, A.G.: Exact solutions for nonlinear foam drainage equation. Ind. J. Phys. 91, 209–218 (2017)

    Article  CAS  Google Scholar 

  • Zayed, E.M.E., Arnous, A.H.: DNA dynamics studied using the homogeneous balance method. Chin. Phys. Lett. 29, 080203–080205 (2012)

    Article  Google Scholar 

  • Zhang, Z.Y., Huang, J.H., Zhong, J.N., Dou, S.S., Liu, J., Peng, D., Gao, T.: The extended G′/G-expansion method and travelling wave solutions for the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity. Pramana-J. Phys. 82, 1011–1029 (2014)

    Article  ADS  Google Scholar 

  • Zhao, Y.H., Mathanaranjan, T., Rezazadeh, H., Akinyemi, L., Inc, M.: New solitary wave solutions and stability analysis for the generalized (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles. Result Phys. 43, 106083 (2022)

    Article  Google Scholar 

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Acknowledgements

The authors would like to extend their gratitude to the unidentified reviewers for their valuable feedback and suggestions aimed at enhancing the article’s quality.

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  1. Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh

    Asaduzzaman & M. Ali Akbar

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  1. Asaduzzaman
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  2. M. Ali Akbar
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A: Conceptualization, Resources, Project administration, Funding acquisition, Supervision, Writing-Original Draft. MAA: Methodology, Data Curation, Formal Analysis, Software, Investigation, Validation, Visualization, Writing-Review Editing.

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Correspondence to M. Ali Akbar.

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Asaduzzaman, Akbar, M.A. Optical soliton solutions: the evolution with changing fractional-order derivative in Biswas–Arshed and Schrödinger Kerr law equations. Opt Quant Electron 56, 465 (2024). https://doi.org/10.1007/s11082-023-05955-7

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