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Optical soliton solutions to the fractional nonlinear Fokas–Lenells and paraxial Schrödinger equations

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Abstract

In nonlinear optics, photonics, plasma, condensed matter physics, and other domains, the space–time fractional nonlinear Fokas–Lenells and paraxial Schrödinger equations associated with beta derivative have significant applications. The fractional wave transformation has been used to turn the space–time fractional nonlinear equations into integer order equations. To obtain optical soliton solutions relating to exponential, trigonometric, and hyperbolic functions and their integration with free parameters, the improved Bernoulli sub-equation function (IBSEF) scheme has been exploited. Different shapes of solitons have been extracted from the attained solutions, including kink, periodic, bell-shaped, anti-kink, dark-bright soliton, single kink type soliton, etc. A kink soliton is an optical shock front that keeps its shape while traveling through optical fibers. The characteristics of the solitons have been studied by describing profiles in 3D, 2D, contour, and density plots. The results imply that the IBSEF technique is simple, efficient, and capable of generating comprehensive soliton solutions of nonlinear models related to telecommunication and optics.

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References

  • Akram, U., Seadawy, A.R., Rizvi, S.T.R., Mustafa, B.: Applications of the Resonanat nonlinear Schrödinger equation with self-steeping phenomena for chirped periodic waves. Opt. Quant. Electron. 54(4), 1–36 (2022)

    Article  Google Scholar 

  • Ali, K.K., Osman, M.S., Abdel-Aty, M.: New optical solitary wave solutions of Fokas–Lenells equation in optical fiber via Sine-Gordon expansion method. Alex. Eng. J. 59(3), 1191–1196 (2020)

    Article  Google Scholar 

  • Al-Amin, M., Islam, M.N., Akbar, M.A.: Adequate wide-ranging closed-form wave solutions to a nonlinear biological model. Partial Diff. Eqn. Appl. Math. 4, 100042 (2021)

    Google Scholar 

  • Ali, K., Rizvi, S.T.R., Nawaz, B., Younis, M.: Optical solitons for paraxial wave equation in Kerr media. Mod. Phys. Lett. B 33(03), 1950020 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  • Arqub, O.A., Osman, M.S., Abdel-Aty, A.H., Mohamed, A.B.A., Momani, S.: A numerical algorithm for the solutions of ABC singular Lane-Emden type models arising in astrophysics using reproducing kernel discretization method. Mathematics 8(6), 923 (2020)

    Article  Google Scholar 

  • , A., Alqahtani, R.T.: Modelling the spread of river blindness disease via the caputo fractional derivative and the beta-derivative. Entropy 18(2), 40 (2016)

    Article  ADS  Google Scholar 

  • Atangana, A., Baleanu, D., Alsaedi, A.: Analysis of time-fractional Hunter-Saxton equation: a model of neumatic liquid crystal. Open Physics 14(1), 145–149 (2016)

    Article  Google Scholar 

  • Barman, H.K., Roy, R., Mahmud, F., Akbar, M.A., Osman, M.S.: Harmonizing wave solutions to the Fokas–Lenells model through the generalized Kudryashov method. Optik 229, 166294 (2021)

    Article  ADS  Google Scholar 

  • Biswas, A., Yıldırım, Y., Yaşar, E., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical soliton solutions to Fokas–Lenells equation using some different methods. Optik 173, 21–31 (2018a)

    Article  ADS  Google Scholar 

  • Biswas, A., Ekici, M., Sonmezoglu, A., Alqahtani, R.T.: Optical solitons with differential group delay for coupled Fokas–Lenells equation by extended trial function scheme. Optik 165, 102–110 (2018b)

    Article  ADS  Google Scholar 

  • Bulut, H., Sulaiman, T.A., Baskonus, H.M., Rezazadeh, H., Eslami, M., Mirzazadeh, M.: Optical solitons and other solutions to the conformable space-time fractional Fokas–Lenells equation. Optik 172, 20–27 (2018)

    Article  ADS  Google Scholar 

  • Darvishi, M.T., Najafi, M., Wazwaz, A.M.: Some optical soliton solutions of space-time conformable fractional Schrödinger-type models. Phys. Scr. 96(6), 065213 (2021)

    Article  ADS  Google Scholar 

  • Das, N., Saha Ray, S.: Novel optical soliton solutions for time-fractional resonant nonlinear Schrödinger equation in optical fiber. Opt. Quant. Electron. 54(2), 1–23 (2022)

    Article  Google Scholar 

  • Demiray, S.T., Bulut, H.: New exact solutions of the new Hamiltonian amplitude-equation and Fokas–Lenells equation. Entropy 17(9), 6025–6043 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Demirbileko, U., Ala, V., Mamedov, K.R.: An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional equation. Tbilisi Math. J. 14(3), 59–70 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Durur, H., Yokuş, A.: Discussions on diffraction and the dispersion for traveling wave solutions of the (2+1)-dimensional paraxial wave equation. Mathematical Sciences, pp. 1–11 (2021)

  • Gao, W., Ismael, H.F., Mohammed, S.A., Baskonus, H.M., Bulut, H.: Complex and real optical soliton properties of the paraxial nonlinear Schrödinger equation in Kerr media with M-fractional. Frontiers in Physics 7, 197 (2019)

    Article  ADS  Google Scholar 

  • Hashemi, M.S.: A novel approach to find exact solutions of fractional evolution equations with non-singular kernel derivative. Chaos Solitons Fractals 152, 111367 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Hashemi, M.S., Haji-Badali, A., Vafadar, P.: Group invariant solutions and conservation laws of the Fornberg-Whitham equation. Zeitschrift Für Naturforschung A 69(8–9), 489–496 (2014)

    Article  ADS  Google Scholar 

  • Hashemi, M.S., Bahrami, F., Najafi, R.: Lie symmetry analysis of steady-state fractional reaction-convection-diffusion equation. Optik 138, 240–324 (2017)

    Article  ADS  Google Scholar 

  • Hashemi, M.S. and Baleanu, D., 2020. Lie symmetry analysis of fractional differential equations. Chapman and Hall/CRC.

  • Islam, M.E., Akbar, M.A.: Stable wave solutions to the Landau-Ginzburg-Higgs equation and the modified equal width wave equation using the IBSEF method. Arab J. Basic Appl. Sci. 27(1), 270–278 (2020)

    Article  Google Scholar 

  • Islam, M., Akbar, M.A.: Study of the parametric effects on soliton propagation in optical fibers through two analytical methods. Opt. Quant. Electron. 53(10), 1–20 (2021)

    Article  Google Scholar 

  • Islam, M.N., Miah, M.M., Rahman, M.A., Akbar, M.A.: Adequate closed form wave solutions to the space-time fractional nonlinear equations in physical sciences. Partial Differential Equations in Applied Mathematics 3, 100024 (2021)

    Article  Google Scholar 

  • Ismael, H.F., Bulut, H., Baskonus, H.M., Gao, W.: Dynamical behaviors to the coupled Schrödinger-Boussinesq system with the beta derivative. AIMS Mathematics 6(7), 7909–7928 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Islam, M., Akter, M., Gómez-Aguilar, J.F., Akbar, M.: Novel and diverse soliton constructions for nonlinear space-time fractional modified Camassa-Holm equation and Schrödinger equation. Opt. Quant. Electron. 54(4), 1–23 (2022)

    Article  Google Scholar 

  • Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar, S., Kumar, A., Samet, B., Gómez-Aguilar, J.F., Osman, M.S.: A chaos study of tumor and effector cells in fractional tumor-immune model for cancer treatment. Chaos, Solitons Fractals 141, 110321 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Miller, K.S., Ross, B.: An Introduction to the Fractional calculus and Fractional Differential Equations. Wiley (1993)

    MATH  Google Scholar 

  • Mirzazadeh, M., Akinyemi, L., Şenol, M., Hosseini, K.: A variety of solitons to the sixth-order dispersive (3+1)-dimensional nonlinear time-fractional Schrödinger equation with cubic-quintic-septic nonlinearities. Optik 241, 166318 (2021)

    Article  ADS  Google Scholar 

  • Osman, M.S., Ghanbari, B.: New optical solitary wave solutions of Fokas–Lenells equation in presence of perturbation terms by a novel approach. Optik 175, 328–333 (2018)

    Article  ADS  Google Scholar 

  • Pashayi, S., Hashemi, M.S., Shahmorad, S.: Analytical lie group approach for solving fractional integro-differential equations. Commun. Nonlinear Sci. Numer. Simul. 51, 66–77 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Rezazadeh, H., Osman, M.S., Eslami, M., Ekici, M., Sonmezoglu, A., Asma, M., Othman, W.A.M., Wong, B.R., Mirzazadeh, M., Zhou, Q., Biswas, A.: Mitigating Internet bottleneck with fractional temporal evolution of optical solitons having quadratic–cubic nonlinearity. Optik 164, 84–92 (2018)

    Article  ADS  Google Scholar 

  • Rezazadeh, H., Ullah, N., Akinyemi, L., Shah, A., Mirhosseini-Alizamin, S.M., Chu, Y.M., Ahmad, H.: Optical soliton solutions of the generalized non-autonomous nonlinear Schrödinger equations by the new Kudryashov’s method. Res. Phys. 24, 104179 (2021)

    Google Scholar 

  • Rizvi, S.R., Afzal, I., Ali, K., Younis, M.: Stationary solutions for nonlinear Schrödinger equations by Lie group analysis. Acta Phys. Pol. A 136(1), 187–189 (2019)

    Article  ADS  Google Scholar 

  • Rizvi, S.T.R., Seadawy, A.R., Younis, M., Javed, I., Iqbal, H.: Lump and optical dromions for paraxial nonlinear Schrödinger equation. Int. J. Mod. Phys. B 35(05), 2150078 (2021)

    Article  ADS  MATH  Google Scholar 

  • Sajid, N., Akram, G.: Optical solitons with full nonlinearity for the conformable space-time fractional Fokas–Lenells equation. Optik 196, 163131 (2019)

    Article  ADS  Google Scholar 

  • Sajid, N., Akram, G.: Dark, singular, bright, rational and periodic solutions of the space–time fractional Fokas–Lenells equation by the ϕ6-model expansion method. Optik 228, 165843 (2021)

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  • Wang, B.H., Wang, Y.Y., Dai, C.Q., Chen, Y.X.: Dynamical characteristic of analytical fractional solitons for the space-time fractional Fokas–Lenells equation. Alex. Eng. J. 59(6), 4699–4707 (2020)

    Article  Google Scholar 

  • Wang, L., Luan, Z., Zhou, Q., Biswas, A., Alzahrani, A.K., Liu, W.: Bright soliton solutions of the (2+ 1)-dimensional generalized coupled nonlinear Schrödinger equation with the four-wave mixing term. Nonlinear Dyn. 104(3), 2613–2620 (2021)

    Article  Google Scholar 

  • Wang, Q., Zhang, L., Malmode, B.A., Mihalache, D., Zeng, L.: Transformation of multipole and vortex solitons in the nonlocal nonlinear fractional Schrödinger equation by means of Levy-index management. Chaos Solitons Fractals 157, 111995 (2022)

    Article  Google Scholar 

  • Wazwaz, A.M.: Solitary Waves Theory. In Partial Differential Equations and Solitary Waves Theory, pp. 479–502. Springer, Berlin (2009)

  • Xia, F.L., Jarad, F., Hashemi, M.S., Riaz, M.B.: A reduction technique to solve the generalized nonlinear dispersive mK (m, n) equation with new local derivative. Res. Phys. 38, 105512 (2022)

    Google Scholar 

  • Yel, G., Bulut, H., İlhan, E.: A new analytical method to the conformable chiral nonlinear Schrödinger equation in the quantum Hall effect. Pramana J. Phys. 96(1), 1–11 (2022)

    Article  Google Scholar 

  • Zafar, A., Raheel, M., Bekir, A., Razzaq, W.: The conformable space-time fractional Fokas–Lenells equation and its optical soliton solutions based on three analytical schemes. Int. J. Mod. Phys. B 35(01), 2150004 (2021a)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Zafar, A., Raheel, M., Bekir, A., Razzaq, W.: The conformable space–time fractional Fokas–Lenells equation and its optical soliton solutions based on three analytical schemes. Int. J. Mod. Phys. B 35(01), 2150004 (2021b)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Zulfiqar, A., Ahmad, J.: Dynamics of new optical solutions of fractional perturbed Schrödinger equation with Kerr law nonlinearity using a mathematical method. Opt. Quant. Electron. 54(3), 1–18 (2022)

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: (22UQU4410172DSR14).

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Authors and Affiliations

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

    Md. Morshedul Haque & M. Ali Akbar

  2. Department of Mathematics, Faculty of Applied Science, Umm Al-Qura University, Makkah, 21955, Saudi Arabia

    M. S. Osman

  3. Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

    M. S. Osman

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  1. Md. Morshedul Haque
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  2. M. Ali Akbar
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  3. M. S. Osman
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Contributions

MMH: Conceptualization, Methodology, Formal Analysis, Resources, Validation, Visualization, Writing-Original Draft. MAA: Software, Investigation, Formal Analysis, Writing-Review Editing, Project administration, Funding acquisition. MSO: Data Curation, Writing-Review Editing, Supervision.

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Correspondence to M. S. Osman.

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Haque, M.M., Akbar, M.A. & Osman, M.S. Optical soliton solutions to the fractional nonlinear Fokas–Lenells and paraxial Schrödinger equations. Opt Quant Electron 54, 764 (2022). https://doi.org/10.1007/s11082-022-04145-1

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