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New abundant analytical solutions of coupled nonlinear Schrödinger (FNSE) equation in fractal order arising in quantum mechanics

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Abstract

In this work, we investigate coupled space–time fractional nonlinear Schrödinger equation (FNSE) arising in physics. The FNSE can be utilized to explain non-relativistic quantum mechanical phenomena. With the aid of the conformable fractional derivative (CFD) and fractional complex transform (FCT), we implement the extensive direct algebraic approach (EDAA), the behaviors of some of the generated solutions are shown as 3D-graphics for various with different fractal orders. The optical soliton solutions that are bright periodic, kink bright and kink-bright periodic are among these precise solutions. The acquired results demonstrate the simplicity, effectiveness, and capacity to produce additional kinds of exact solutions of these proposed methods, which are useful in deciphering the intricate physical interpretation of space–time FNSE.

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No data from other sources have been used in this study.

References

  • Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)

    MathSciNet  Google Scholar 

  • Abdou, M.A., Owyed, S., Saha Ray, S., Inc, M., Chu, Y.M., Ouahid, L.: Fractal Ion acoustic waves of the space-time fractional three dimensional KP equation. Adv. Math. Phys. 2020, 8323148 (2020)

    MathSciNet  Google Scholar 

  • Abdou, M.A., Ouahid, L., Al Shahrani, J.S., Owyed, S.: Novel analytical techniques for HIV-1 infection of CD4+T-cells on fractional order in mathematical biology. Indian J. Phys. 97, 2319–2325 (2023a)

    ADS  Google Scholar 

  • Abdou, M.A., Ouahid, L., Kumar, S.: Plenteous specific analytical solutions for new extended deoxyribonucleic acid (DNA) model arising in mathematical biology. Mod. Phys. Lett. B 37(34), 2350173 (2023b)

    ADS  MathSciNet  Google Scholar 

  • Ahmad, J., Rani, S.: Study of soliton solutions with different wave formations to model of nonlinear Schrödinger equation with mixed derivative and applications. Opt. Quant. Electron. 55, 1195 (2023)

    Google Scholar 

  • Akbar, M.A., Wazwaz, A.M., Mahmud, F., Baleanu, D., Roy, R., Barman, H.K., Mahmoud, W., Sharif, M.A.A., Osman, M.S.: Dynamical behavior of solitons of the perturbed nonlinear Schrodinger equation and microtubules through the generalized Kudryashov scheme. Results Phys. 43, 106079 (2022)

    Google Scholar 

  • Akbar, M.A., Abdullah, F.A., Islam, M.D.T., Al Sharif, M.A., Osman, M.S.: Newsolutions of the soliton type of shallow water waves and superconductivity models. Res. Phys. 44, 106180 (2023)

    Google Scholar 

  • Akram, S., Ahmad, J., Ur-Rehman, S., Sarwar, S., Ali, A.: Dynamics of soliton solutions in optical fibers modeled by perturbed nonlinear Schrodinger equation and stability analysis. Opt. Quantum Electron. 55, 450 (2023)

    Google Scholar 

  • Al Alwan, B., Abu Bakar, M., Faridi, W.A., Turcu, A.C., Akgül, A., Sallah, M.: The propagating exact solitary waves formation of generalized Calogero–Bogoyavlenskii–Schiff equation with robust computational approaches. Fractal Fract. 7, 191 (2023)

    Google Scholar 

  • Ala, V.: New exact solutions of space-time fractional Schrodinger–Hirota equation. Bull. Karaganda Univer. Math. Ser. 107, 3 (2022)

    Google Scholar 

  • Ala, V.: Exact solutions of nonlinear time fractional schrödinger equation with beta-derivative. Fundam. Contemp. Math. Sci. 4(1), 1–8 (2023)

    MathSciNet  Google Scholar 

  • Ala, V., Shaikhova, G.: Analytical solutions of nonlinear beta fractional schrödinger equation via sine-cosine method. Lobachevskii J. Math. 43(11), 3033–3038 (2022)

    MathSciNet  Google Scholar 

  • Alabedalhadi, M.: Exact travelling wave solutions for nonlinear system of spatiotemporal fractionlquantum mechanics equations. Alex. Eng. J. 61, 1033–1044 (2022)

    Google Scholar 

  • Alanazi, M.M., Ouahid, L., Al Shahrani, J.S., Abdou, M.A., Kumar, S.: Novel soliton solutions to the Atangana Baleanu (AB) fractional for ion sound and Langmuir waves (ISALWs) equations. Opt. Quantum Electron. 55, 462 (2023)

    Google Scholar 

  • Ali, A., Ahmad, J., Javed, S.: Solitary wave solutions for the originating waves that propagate of the fractional Wazwaz–Benjamin–Bona–Mahony system. Alex. Eng. J. 69, 121–133 (2023)

    Google Scholar 

  • Alqhtani, M., Saad, K.M., Shah, R., Hamanah, W.M.: Discovering novel soliton solutions for (3+1)-modified fractional Zakharov–Kuznetsov equation in electrical engineering through an analytical approach. Opt. Quant. Electron. 55, 1149 (2023)

    Google Scholar 

  • Aniqa, A., Ahmad, J.: Soliton solution of fractional Sharma–Tasso–Olever equation via an efficient (G′/G)-expansion method. Ain Shams Eng. J. 13, 101528 (2022)

    Google Scholar 

  • Zulfiqar, A., Ahmad, J.: Soliton solutions of fractional modified unstable Schrödinger equation using Exp-function method. Results Phys. 19, 103476 (2020a)

    Google Scholar 

  • Faridi, W.A., Yusuf, A., Akgül, A., Tawfiq, F.M., Tchier, F., Al-deiakeh, R., Sulaiman, T.A., Hassan, A.M., Ma, W.X.: The computation of Lie point symmetry generators, modulational instability classification of conserved quantities, and explicit power series solutions of the coupled system. Results Phys. 54, 107126 (2023a)

    Google Scholar 

  • Faridi, W.A., Bakar, M.A., Myrzakulova, Z., Myrzakulova, R., Akgul, A., El Din, S.M.: The formation of solitary wave solutions and their propagation for Kuralay equation. Results Phys. 52, 106774 (2023ab)

    Google Scholar 

  • Faridi, W.A., Asjad, M.I., Jhangeer, A., Yusuf, A., Sulaiman, T.A.: The weakly non-linear waves propagation for Kelvin-Helmholtz instability in the magnetohydrodynamics flow impelled by fractional theory. Opt. Quant. Electron. 55, 172 (2023bc)

    Google Scholar 

  • Faridi, W.A., Bakar, M.A., Akgül, A., Abd El-Rahman, M., El Din, S.M.: Exact fractional soliton solutions of thin-film ferroelectric material equation by analytical approaches. Alex. Eng. J. 78, 483–497 (2023d)

    Google Scholar 

  • Faridi, W.A., Yusuf, A., Akgul, A., Tawfiq, F.M.O., Tchier, F., Al-deiakeh, R., Sulaiman, T.A., Hassan, A.M., Ma, W.-X.: The computation of Lie point symmetry generators, modulational instability classification of conserved quantities, and explicit power series solutions of the coupled system. Results Phys. 54, 107126 (2023de)

    Google Scholar 

  • Faridi, W.A., Hussain, G., Myrzakulova, Z., Myrzakulov, R., Akinyemi, L.: Formation of optical soliton wave profles of Shynaray-IIA equation via two improved techniques: a comparative study. Opt. Quant. Electron. 56, 132 (2024)

    ADS  Google Scholar 

  • Fendzi-Donfack, E., Tala-Tebue, E., Inc, M., Kenfack-Jiotsa, A., Nguenang, J.P., Nana, L.: Dynamical behaviours and fractional alphabetical-exotic solitons in a coupled nonlinear electrical transmission lattice including wave obliqueness. Opt. Quant. Electron. 55, 35 (2023)

    Google Scholar 

  • Ganaini, E.S.: Solitons and other solutions to a new coupled nonlinear Schrödinger type equation. J. Egypt. Math. Soc. 25, 19–27 (2017)

    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)

    ADS  MathSciNet  Google Scholar 

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

    ADS  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  • Ghanbari, B., Gómez-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)

    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)

    Google Scholar 

  • Ghanbari, B., Baleanu, D., Al Qurashi, M.: New exact solutions of the generalized Benjamin–Bona–Mahony equation. Symmetry 11(1), 19 (2019)

    Google Scholar 

  • Hamid, I., Kumar, S.: Symbolic computation and Novel solitons, traveling waves and soliton-like solutions for the highly nonlinear (2+1)-dimensional Schrödinger equation in the anomalous dispersion regime via newly proposed modified approach. Opt. Quant. Electron. 55(9), 755 (2023)

    Google Scholar 

  • Hoan, L.V.C., Owyed, S., Inc, M., Ouahid, L., Abdou, M.A., Chu, Y.-M.: New explicit optical solitons of fractional nonlinear evolution equation via three different methods. Results Phys. 18, 103209 (2020)

    Google Scholar 

  • Islam, M.D.T., Akbar, M.D.A., Gómez-Aguilar, J.F., Bonyah, E., Fernandez-Anaya, G.: Assorted soliton structures of solutions for fractional nonlinear Schrodinger types evolution equations. J. Ocean Eng. Sci. 7, 528–535 (2022)

    Google Scholar 

  • Islam, M.D.T., Ryehan, S., Abdullah, F.A., Gómez-Aguilar, J.F.: The effect of Brownian motion and noise strength on solutions of stochastic Bogoyavlenskii model alongside conformable fractional derivative. Optik 287, 171140 (2023)

    ADS  Google Scholar 

  • Islama, M.D.T., Akter, M.S.T.A., Ryehana, S., Gómez-Aguilar, J.F., Ali Akbar, M.D.: A variety of solitons on the oceans exposed by the Kadomtsev Petviashvili-modified equal width equation adopting different techniques. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.07.001

    Article  Google Scholar 

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

    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. 136, 447 (2021)

    Google Scholar 

  • Kılıç, S.ŞŞ, Çelik, E., Bulut, H.: Solitary wave solutions to some nonlinear conformable partial differential equations. Opt. Quant. Electron. 55, 693 (2023)

    Google Scholar 

  • Kumar, S., Hamid, I., Abdou, M.A.: Specific wave profiles and closed-form soliton solutions for generalized nonlinear wave equation in (3+1)-dimensions with gas bubbles in hydrodynamics and fluids. J. Ocean Eng. Sci. 8, 91–102 (2023)

    Google Scholar 

  • Ma, Y., Geng, X.: A coupled nonlinear Schrödinger type equation and its explicit solutions. Chaos Soliton Fract. 42, 2949–2953 (2009)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  • Ouahid, L., Owyed, S., Abdou, M.A., Alshehri, N.A., Elagan, S.K.: New optical soliton solutions via generalized Kudryashov’s scheme for Ginzburg–Landau equation in fractal order. Alex. Eng. J. 60(6), 5495–5510 (2021)

    Google Scholar 

  • Ozdemir, N., Esen, H., Secer, A., Bayram, M., Yusuf, A., Sulaiman, T.A.: Optical solitons and other solutions to the Hirota-Maccari system with conformable, M-truncated and beta derivatives. Mod. Phys. Lett. B 36(11), 2150625 (2022)

    ADS  MathSciNet  Google Scholar 

  • Rani, A., Zulfiqar, A., Ahmad, J., Ul Hassan, Q.M.: New soliton wave structures of fractional Gilson–Pickering equation using tanh–coth method and their application. Results Phys. 29, 104724 (2021)

    Google Scholar 

  • Shi, L., Zhou, X.: Exact solutions of a coupled space-time fractional nonlinear Schrödinger type equation in quantum mechanics. Results Phys. 42, 105967 (2022)

    Google Scholar 

  • Tozar, A., Kurt, A., Tasbozan, O.: New wave solutions of an integrable dispersive wave equation with a fractional time derivative arising in ocean engineering models. Kuwait J. Sci. 47(2), 22–23 (2020)

  • Ur-Rehman, S., Bilal, M., Ahmad, J.: New exact solitary wave solutions for the 3D-FWBBM model in arising shallow water waves by two analytical methods. Results Phys. 25, 104230 (2021)

    Google Scholar 

  • Vahidi, J., Zabihi, A., Rezazadeh, H., Ansari, R.: New extended direct algebraic method for the resonant nonlinear Schrödinger equation with Kerr law nonlinearity. Optik 227, 165936 (2021)

    ADS  Google Scholar 

  • Wang, B.H., Lu, P.H., Dai, C.Q., Chen, Y.X.: Vector optical soliton and periodic solutions of a coupled fractional nonlinear Schrödinger equation. Results Phys. 17, 103036 (2020)

    Google Scholar 

  • Xu, Y., Feng, Y., Jiang, J.: Exact solutions of the fractional resonant nonlinear Schrödinger equation. Opt. Quant. Electron. 55, 1208 (2023)

    Google Scholar 

  • Yokuş, A., Durur, H., Duran, S., Islam, M.D.T.: Ample felicitous wave structures for fractional foam drainage equation modeling for fluid-flow mechanism. Comput. Appl. Math. 41, 174 (2022)

    MathSciNet  Google Scholar 

  • Zhou, G.Z., Dai, C.Q.: Nonautonomous soliton solutions of variable-coefficient fractional nonlinear Schrödinger equation. Appl. Math. Lett. 106, 106365 (2020)

    MathSciNet  Google Scholar 

  • Zulfiqar, A., Ahmad, J.: Soliton solutions of fractional modified unstable Schrödinger equation using Exp-function method. Results Phys. 19, 103476 (2020b)

    Google Scholar 

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Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at University of Bisha, Saudi Arabia for funding this research work through the Promising Program under Grant Number (UB-Promising-20-1445).

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Authors and Affiliations

  1. Department of Physics, College of Science, University of Bisha, Bisha, 61922, P.O. Box 551, Saudi Arabia

    Maryam Alshahrani, Loubna Ouahid & M. A. Abdou

  2. Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, 110007, India

    Sachin Kumar

  3. Department of Mathematics, College of Science, University of Bisha, Bisha, 61922, P.O. Box 551, Saudi Arabia

    Jameelah S. Al Shahrani

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  1. Maryam Alshahrani
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  2. Loubna Ouahid
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  3. M. A. Abdou
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  4. Sachin Kumar
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  5. Jameelah S. Al Shahrani
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Alshahrani, M., Ouahid, L., Abdou, M.A. et al. New abundant analytical solutions of coupled nonlinear Schrödinger (FNSE) equation in fractal order arising in quantum mechanics. Opt Quant Electron 56, 735 (2024). https://doi.org/10.1007/s11082-024-06378-8

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