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A variety of optical soliton solutions in closed-form of the nonlinear cubic quintic Schrödinger equations with beta derivative

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Abstract

In the field of nonlinear optics, both fractional and classical-order nonlinear Schrödinger (NS) equations are investigated. However, the fractional-order NS equation has gained widespread acceptance due to its higher compatibility. The space-time fractional nonlinear Schrödinger equation enfolding beta derivative has a wide range of applications in nonlinear optics, quantum computing, Bose-Einstein condensates, wave propagation in complex media, quantum mechanics, and engineering, where understanding wave propagation and nonlinear interactions are diametrical. In this article, the improved Bernoulli sub-equation function (IBSEF) procedure has been used to establish optical soliton solutions in the form of trigonometric, exponential, and hyperbolic functions comprising substantive parameters. These soliton solutions have different shapes, including kink, periodic soliton, singular kink, breathing soliton, and other types. The physical features of the solitons are revealed through three-, two-, contour, and density graphs. The research findings confirm that the IBSEF scheme is effective, straightforward, and applicable for ascertaining soliton solutions in various nonlinear fractional-order models in the fields of physics and communication engineering.

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The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  • Agarwal, R., Belmekki, M., Benchohra, M.: A survey on semi-linear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Differ. Equ. 2009, 1–47 (2009)

    MATH  Google Scholar 

  • Akram, G., Sadaf, M., Zainab, I.: Observations of fractional effects of β-derivative and M-truncated derivative for space time fractional phi-4 equation via two analytical techniques. Chaos Solitons Fractals 154, 111645 (2022)

    MathSciNet  MATH  Google Scholar 

  • Alam, L.M.B., Xingfang, J., Mamun, A.A., Ananna, S.N.: Investigation of lump, soliton, periodic, kink, and rogue waves to the time-fractional phi-four and (2 + 1) dimensional CBS equations in mathematical physics. Partial Differ. Equ. Appl. Math. 4, 100122 (2021)

    Google Scholar 

  • Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  • Alquran, M., Jaradat, I., Yusuf, A., Sulaiman, T.A.: Heart-cusp and bell-shaped-cusp optical solitons for an extended two-mode version of the complex Hirota model: application in optics. Opt. Quant. Electron. 53(1), 1–13 (2021)

    Google Scholar 

  • Ananna, S.N., Mamun, A.A., Tianqing, A.: Periodic wave analysis to the time fractional phi four and (2 + 1) dimensional CBS equations. Int. J. Phys. Res. 9(2), 98–104 (2021)

    Google Scholar 

  • Arnous, A.H., Biswas, A., Ekici, M., Alzahrani, A.K., Belic, M.R.: Optical solitons and conservation laws of Kudryashov’s equation with improved modified extended tanh-function. Optik 225, 165406 (2021)

    ADS  Google Scholar 

  • Arshad, M., Seadawy, A.R., Lu, D.: Exact bright-dark solitary wave solutions of the higher-order cubic-quintic nonlinear Schrödinger equation and its stability. Optik 138, 40–49 (2017)

    ADS  Google Scholar 

  • Atangana, 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)

    ADS  Google Scholar 

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

    Google Scholar 

  • Bezgabadi, A.S., Bolorizadeh, M.A.: Analytic combined bright-dark, bright and dark solitons solutions of generalized nonlinear Schrödinger equation using extended Sinh-Gordon equation expansion method. Results Phys. 30, 104852 (2021)

    Google Scholar 

  • Bilal, M., Ren, J., Younas, U.: Stability analysis and optical soliton solutions to the nonlinear Schrödinger model with efficient computational techniques. Opt. Quant. Electron. 53(7), 1–19 (2021)

    Google Scholar 

  • Bo, W.B., Wang, R.R., Fang, Y., Wang, Y.Y., Dai, C.Q.: Prediction and dynamical evolution of multipole soliton families in fractional Schrodinger equation with the PT-symmetric potential and saturable nonlinearity. Nonlinear Dyn. 111, 1577–1588 (2023)

    Google Scholar 

  • Buryak, A.V., Di Trapani, P., Skryabin, D.V., Trillo, S.: Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications. Phys. Rep. 370(2), 63–235 (2002)

    ADS  MathSciNet  MATH  Google Scholar 

  • Cai, L., Lu, Y., Zhu, H.: Performance enhancement of on-chip optical switch and memory using Ge2Sb2Te5 slot-assisted microring resonator. Opt. Lasers Eng. 162, 107436 (2023)

    Google Scholar 

  • Cao, K., Wang, B., Ding, H., Lv, L., Tian, J., Hu, H., Gong, F.: Achieving reliable and secure communications in wireless-powered NOMA systems. IEEE Trans. Veh. Technol. 70(2), 1978–1983 (2021)

    Google Scholar 

  • Chen, Y.X.: Combined optical soliton solutions of a (1 + 1)-dimensional time fractional resonant cubic-quintic nonlinear Schrödinger equation in weakly nonlocal nonlinear media. Optik 203, 163898 (2020)

    ADS  Google Scholar 

  • Chen, Y.X., Xiao, X.: Combined soliton solutions of a (1 + 1)-dimensional weakly nonlocal conformable fractional nonlinear Schrödinger equation in the cubic–quintic nonlinear material. Opt. Quant. Electron. 53(1), 1–13 (2021)

    Google Scholar 

  • Chung, K.L., Tian, H., Wang, S., Feng, B., Lai, G.: Miniaturization of microwave planar circuits using composite microstrip/coplanar-waveguide transmission lines. Alex. Eng. J. 61(11), 8933–8942 (2022)

    Google Scholar 

  • Demiray, S.T.: New soliton solutions of optical pulse envelope E (Z, τ) with beta time derivative. Optik 223, 165453 (2020)

    ADS  Google Scholar 

  • Gepreel, K.A., Zayed, E.M.E.: Multiple wave solutions for nonlinear burgers equations using the multiple exp-function method. Int. J. Mod. Phys. C 32(11), 2150149 (2021)

    ADS  MathSciNet  Google Scholar 

  • Guo, C., Hu, J., Wu, Y., Čelikovský, S.: Non-singular fixed-time tracking control of uncertain nonlinear pure-feedback systems with practical state constraints. IEEE Trans. Circuits Syst. I Regul. Pap. (2023). https://doi.org/10.1109/TCSI.2023.3291700

    Article  Google Scholar 

  • Hashemi, M.S., Akgül, A.: Solitary wave solutions of time-space nonlinear fractional Schrödinger’s equation: two analytical approaches. J. Comput. Appl. Math. 339, 147–160 (2018)

    MathSciNet  MATH  Google Scholar 

  • Inc, M., Yusuf, A., Aliyu, A.I., Baleanu, D.: Dark and singular optical solitons for the conformable space–time nonlinear Schrödinger equation with Kerr and power law nonlinearity. Optik 162, 65–75 (2018)

    ADS  Google Scholar 

  • Islam, M.N., Ilhan, O.A., Akbar, M.A., Benli, F.B., Soybaş, D.: Wave propagation behavior in nonlinear media and resonant nonlinear interactions. Commun. Nonlinear Sci. Numer. Simul. 108, 106242 (2022)

    MathSciNet  MATH  Google Scholar 

  • Islam, W., Younis, M., Rizvi, S.T.R.: Optical solitons with time fractional nonlinear Schrödinger equation and competing weakly nonlocal nonlinearity. Optik 130, 562–567 (2017)

    ADS  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 Math. 6(7), 7909–7928 (2021)

    MathSciNet  MATH  Google Scholar 

  • Jumarie, G.: Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 51(9–10), 1367–1376 (2006)

    MathSciNet  MATH  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)

    MathSciNet  MATH  Google Scholar 

  • Kong, L., Liu, G.: Synchrotron-based infrared micro-spectroscopy under high pressure: an introduction. Matter Radiat. Extremes 6(6), 68202 (2021)

    Google Scholar 

  • Li, B., Tan, Y., Wu, A., Duan, G.: A distributionally robust optimization based method for stochastic model predictive control. IEEE Trans Autom Control 67(11), 5762–5776 (2021)

    MathSciNet  MATH  Google Scholar 

  • Malik, S., Kumar, S., Kumari, P., Nisar, K.S.: Some analytic and series solutions of integrable generalized broer-kaup system. Alex. Eng. J. 61(9), 7067–7074 (2022)

    Google Scholar 

  • Mamun, A.A., Tianqing, A., Shahen, N.H.M., Ananna, S.N., Foyjonnesa, Hossain, M.F., Muazu, T.: Exact and explicit travelling-wave solutions to the family of new 3D fractional WBBM equations in mathematical physics. Results Phys. 19, 103517 (2020)

    Google Scholar 

  • Mamun, A.A., Ananna, S.N., Tianqing, A., Shahen, N.H.M., Asaduzzaman, M., Foyjonnesa: Dynamical behaviourof travelling wave solutions to the conformable time-fractional modified Liouville and mRLW equations in water wave mechanics. Heliyon 7, e07704 (2021a)

    Google Scholar 

  • Mamun, A.A., Ananna, S.N., Tianqing, A., Shahen, Nur, H.M., Foyjonnesa: Periodic and solitary wave solutions to a family of new 3D fractional WBBM equations using the two-variable method. Partial Differ. Equ. Appl. Math. 3, 100033 (2021b)

    Google Scholar 

  • Mamun, A.A., Shahen, N.H.M., Ananna, S.N., Asaduzzaman, M., Foyjonnesa: Solitary and periodic wave solutions to the family of new 3D fractional WBBM equations in mathematical physics. Heliyon 7, e07483 (2021c)

    Google Scholar 

  • Mamun, A.A., Ananna, S.N., Gharami, P.P., Tianqing, A., Asaduzzaman, M.: The improved modified extended tanh-function method to develop the exact travelling wave solutions of a family of 3D fractional WBBM equations. Results Phys. 41, 105969 (2022a)

    Google Scholar 

  • Mamun, A.A., Ananna, S.N., Tianqing, A., Asaduzzaman, M., Hasan, A.: Optical soliton analysis to a family of 3D WBBM equations with conformable derivative via a dynamical approach. Partial Differ. Equ. Appl. Math. 5, 100238 (2022b)

    Google Scholar 

  • Mamun, A.A., Ananna, S.N., Tianqing, A., Asaduzzaman, M., Miah, M.M.: Solitary wave structures of a family of 3D fractional WBBM equation via the tanh-coth approach. Partial Differ. Equ. Appl. Math. 5, 100237 (2022c)

    Google Scholar 

  • Mamun, A.A., Ananna, S.N., Tianqing, A., Asaduzzaman, M., Rana, M.S.: Sine-Gordon expansion method to construct the solitary wave solutions of a family of 3D fractional WBBM equations. Results Phys. 40, 105845 (2022d)

    Google Scholar 

  • Mamun, A.A., Ananna, S.N., Gharami, P.P., Tianqing, A., Liu, W., Asaduzzaman, M.: An innovative approach for developing the precise traveling wave solutions to a family of 3D fractional WBBM equations. Partial Differ. Equ. Appl. Math. 7, 100522 (2023)

    Google Scholar 

  • Mathanaranjan, T.: Soliton solutions of deformed nonlinear Schrödinger equations using ansatz method. Int. J. Appl. Comput. Math. 7, 159 (2021)

    MathSciNet  MATH  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, 085005 (2023)

    ADS  MATH  Google Scholar 

  • Meng, Q., Ma, Q., Shi, Y.: Adaptive fixed-time stabilization for a class of uncertain nonlinear systems. IEEE Trans. Autom. Control (2023). https://doi.org/10.1109/TAC.2023.3244151

    Article  Google Scholar 

  • Nisar, K.S., Ilhan, O.A., Abdulazeez, S.T., Manafian, J., Mohammed, S.A., Osman, M.S.: Novel multiple soliton solutions for some nonlinear PDEs via multiple exp-function method. Results Phys. 21, 103769 (2021)

    Google Scholar 

  • Raza, N., Rafiq, M.H., Kaplan, M., Kumar, S., Chu, Y.M.: The unified method for abundant soliton solutions of local time fractional nonlinear evolution equations. Results Phys. 22, 103979 (2021)

    Google Scholar 

  • Shahen, N.H.M., Foyjonnesa, Bashar, M.H., Ali, M.S., Mamun, A.A.: Dynamical analysis of long-wave phenomena for the nonlinear conformable space–time fractional (2 + 1)-dimensional AKNS equation in water wave mechanics. Heliyon 6, e05276 (2020)

    Google Scholar 

  • Sharma, V.K., Goyal, A.: Ultrashort double-kink and algebraic solitons of generalized nonlinear Schrödinger equation in the presence of non-kerr terms. J. Nonlinear Opt. Phys. Mater. 23(3), 1450034 (2014)

    ADS  Google Scholar 

  • Sharma, V.K., Goyal, A., Raju, T.S., Kumar, C.N.: Periodic and solitary wave solutions for ultrashort pulses in negative-index materials. J. Mod. Opt. 60(10), 836–840 (2013)

    ADS  Google Scholar 

  • Tripathy, A., Sahoo, S., Rezazadeh, H., Izgi, Z.P., Osman, M.S.: Dynamics of damped and undamped wave natures in ferromagnetic materials. Optik 281, 170817 (2023)

    ADS  Google Scholar 

  • Wang, K.J., Liu, J.H.: Diverse optical solitons to the nonlinear Schrödinger equation via two novel techniques. Eur. Phys. J. Plus 138, 74 (2023)

    Google Scholar 

  • Wu, G.Z., Yu, L.J., Wang, Y.Y.: Fractional optical solitons of the space-time fractional nonlinear Schrödinger equation. Optik 207, 164405 (2020)

    ADS  Google Scholar 

  • Yalçınkaya, Ä., Ahmad, H., Tasbozan, O., Kurt, A.: Soliton solutions for time fractional ocean engineering models with Beta derivative. J. Ocean Eng. Sci. 7(5), 444–448 (2021)

    Google Scholar 

  • Yıldırım, Y.: Optical solitons in birefringent fibers with Biswas-Arshed equation by sine-gordon equation method. Optik 227, 165960 (2021)

    ADS  Google Scholar 

  • Yousif, E.A., Abdel-Salam, E.B., El-Aasser, M.A.: On the solution of the space–time fractional cubic nonlinear Schrödinger equation. Results Phys. 8, 702–708 (2018)

    ADS  Google Scholar 

  • Zhang, Y., He, Y., Wang, H., Sun, L., Su, Y.: Ultra-broadband mode size converter using on-chip meta material-based Luneburg lens. ACS Photonics 8(1), 202–208 (2021)

    Google Scholar 

  • Zhao, C., Cheung, C.F., Xu, P.: High-efficiency sub-microscale uncertainty measurement method using pattern recognition. ISA Trans. 101, 503–514 (2020)

    Google Scholar 

  • Zhao, Q., Liu, J., Yang, H., Liu, H., Zeng, G., Huang, B.: High birefringence D-shaped germanium-doped photonic crystal fiber sensor. Micromachines 13(6), 826 (2022). https://doi.org/10.3390/mi13060826

    Article  Google Scholar 

  • Zhao, Q., Liu, J., Yang, H., Liu, H., Zeng, G., Huang, B., Jia, J.: Double U-groove temperature and refractive index photonic crystal fiber sensor based on surface plasmon resonance. Appl. Opt. 61(24), 7225–7230 (2022)

    ADS  Google Scholar 

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Acknowledgements

The authors wish to extend their gratitude to the anonymous referees for providing valuable feedback and suggestions aimed to improve the quality of the article.

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Authors and Affiliations

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

    Md. Morshedul Haque & M. Ali Akbar

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

    Hadi Rezazadeh

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

    Ahmet Bekir

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  1. Md. Morshedul Haque
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  2. M. Ali Akbar
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  3. Hadi Rezazadeh
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  4. Ahmet Bekir
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Contributions

MMH: Conceptualization, Methodology, Resources, Writing—original draft, Data curation, Visualization. MAA: Writing—review editing, Software, Project administration, Funding acquisition. HR: Writing—review editing, Software, Investigation, Formal analysis. AB: Supervision, Validation, Writing—review editing, Methodology.

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Correspondence to Ahmet Bekir.

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Haque, M.M., Akbar, M.A., Rezazadeh, H. et al. A variety of optical soliton solutions in closed-form of the nonlinear cubic quintic Schrödinger equations with beta derivative. Opt Quant Electron 55, 1144 (2023). https://doi.org/10.1007/s11082-023-05470-9

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