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Investigating the dynamics of soliton solutions to the fractional coupled nonlinear Schrödinger model with their bifurcation and stability analysis

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Abstract

The fractional coupled nonlinear Schrödinger model (FCNLSM) is widely utilized in various fields such as nonlinear optics, optical communication systems, plasmas and mathematical physics. In this study, we aim to achieve three primary objectives. Firstly, we seek to obtain novel soliton solutions for the FCNLSM, which have not been previously reported in the literature. Secondly, we employ the Sardar sub-equation method and the improved generalized tanh-function method to effectively analyze the dynamics of solutions and solve the studied model. These methods provide valuable insights into the behavior of the system. Lastly, we conduct bifurcation and stability analyses to explore the dynamical properties of the model. To ensure the physical validity of our findings, we present 2-dimensional, 3-dimensional and contour plots using carefully selected parameter values. The obtained results demonstrate the feasibility, efficiency and computational speed of the employed techniques in obtaining comprehensive and reliable solutions. The study represents a novel and significant contribution to the field by expanding the understanding of soliton solutions in the FCNLSM, introducing new techniques for their investigation and conducting a comprehensive analysis of bifurcation and stability properties. The findings of this research open new avenues for exploration and application in the areas of nonlinear optics, optical communication systems and other related fields.

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References

  • Abouelregal, A.E., Fahmy, M.A.: Generalized Moore-Gibson-Thompson thermoelastic fractional derivative model without singular kernels for an infinite orthotropic thermoelastic body with temperature-dependent properties. ZAMM J. Appl. Math. Mech. 102(7), e202100533 (2022)

    MathSciNet  Google Scholar 

  • Agussuryani, Q., Sudarmin, S., Sumarni, W., Cahyono, E., Ellianawati, E.: STEM Literacy in growing vocational school student hots in science learning: a meta-analysis. Int. J. Eval. Res. Educ. 11(1), 51–60 (2022)

    Google Scholar 

  • Ahmad, J.: Dispersive multiple lump solutions and soliton’s interaction to the nonlinear dynamical model and its stability analysis. Eur. Phys. J. D 76–1, 1–13 (2022)

    Google Scholar 

  • Ahmed, K.K., Badra, N.M., Ahmed, H.M., Rabie, W.B.: Soliton solutions of generalized Kundu-Eckhaus equation with an extra-dispersion via improved modified extended tanh-function technique. Opt. Quant. Electron. 55(4), 299 (2023)

    Google Scholar 

  • Al Qarni, A.A., Bodaqah, A.M., Mohammed, A.S.H.F., Alshaery, A.A., Bakodah, H.O., Biswas, A.: Dark and singular cubic-quartic optical solitons with Lakshmanan-Porsezian-Daniel equation by the improved Adomian decomposition scheme. Ukr. J. Phys. Opt. 24(1), 46 (2023)

    Google Scholar 

  • Aniqa, A., Ahmad, J.: Soliton solution of fractional Sharma-Tasso-Olever equation via an efficient \(\frac{G^{\prime }}{G}\)-expansion method. Ain Shams Eng. J. 13(1), 101528 (2022)

    Google Scholar 

  • Arnous, A.H., Biswas, A., Yildirim, Y., Moraru, L., Aphane, M., Moshokoa, S., Alshehri, H.: Quiescent optical solitons with Kudryashov’s generalized quintuple-power and nonlocal nonlinearity having nonlinear chromatic dispersion: generalized temporal evolution. Ukr. J. Phys. Opt. 24(2), 105–13 (2023)

    Google Scholar 

  • Asjad, M.I., Inc, M., Iqbal, I.: Exact solutions for new coupled Konno-Oono equation via Sardar subequation method. Opt. Quant. Electron. 54(12), 798 (2022)

    Google Scholar 

  • Athron, P., Fowlie, A., Lu, C.T., Wu, L., Wu, Y., Zhu, B.: The \( W \) boson Mass and Muon \( g-2\): Hadronic Uncertainties or New Physics?. arXiv preprint arXiv: 2204.03996. (2022)

  • Balcı, E., Öztürk, İ, Kartal, S.: Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative. Chaos Solitons Fractals 1(123), 43–51 (2019)

    MathSciNet  MATH  ADS  Google Scholar 

  • Baskonus, H.M., Osman, M.S., Rehman, H.U., Ramzan, M., Tahir, M., Ashraf, S.: On pulse propagation of soliton wave solutions related to the perturbed Chen-Lee-Liu equation in an optical fiber. Opt. Quant. Electron. 53, 1–7 (2021)

    Google Scholar 

  • Bayram, M.: Optical bullets with Biswas-Milovic equation having Kerr and parabolic laws of nonlinearity. Optik 1(270), 170046 (2022)

    Google Scholar 

  • Beregi, S., Barton, D.A., Rezgui, D., Neild, S.: Using scientific machine learning for experimental bifurcation analysis of dynamic systems. Mech. Syst. Signal Process. 184, 109649 (2023)

    Google Scholar 

  • Bilal, M., Ahmad, J.: Dynamics of soliton solutions in saturated ferromagnetic materials by a novel mathematical method. J. Magn. Magn. Mater. 538, 168245 (2021)

    Google Scholar 

  • Bilal, M., Ahmad, J.: A variety of exact optical soliton solutions to the generalized (2+ 1)-dimensional dynamical conformable fractional Schrödinger model. Results Phys. 33, 105198 (2022)

    Google Scholar 

  • Bilal, M., Ahmad, J.: Stability analysis and diverse nonlinear optical pluses of dynamical model in birefringent fibers without four-wave mixing. Opt. Quant. Electron. 54(5), 277 (2022)

    Google Scholar 

  • Bilal, M., Rehman, S.U., Ahmad, J.: The study of new optical soliton solutions to the time-space fractional nonlinear dynamical model with novel mechanisms. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.05.027

    Article  Google Scholar 

  • Biswas, A.: Highly dispersive optical soliton perturbation with complex-Ginzburg Landau model by semi-inverse variation. In: 10th (Online) International Conference on Applied Analysis and Mathematical Modeling (ICAAMM22), Istanbul-Turkey 1-3, 72, (2022)

  • Biswas, A., Milovic, D., Edwards, M.: Mathematical Theory of Dispersion-Managed Optical Solitons. Springer, Berlin (2010)

    MATH  Google Scholar 

  • Bonyah, E., Hammouch, Z., Koksal, M.E.: Mathematical modeling of coronavirus dynamics with conformable derivative in Liouville-Caputo sense. J. Math. (2022). https://doi.org/10.1155/2022/8353343

    Article  MathSciNet  Google Scholar 

  • Chen, W., Sun, H., Li, X.C.: Fractional Derivative Modelling in Mechanics and Engineering. Springer, Berlin (2022)

    MATH  Google Scholar 

  • Darvishi, M.T., Najafi, M.: A modification of extended homoclinic test approach to solve the (3+ 1)-dimensional potential-YTSF equation. Chin. Phys. Lett. 28(4), 040202 (2011)

    ADS  Google Scholar 

  • Darvishi, M.T., Najafi, M., Seadawy, A.R.: Dispersive bright, dark and singular optical soliton solutions in conformable fractional optical fiber Schrödinger models and its applications. Opt. Quant. Electron. 50, 1–6 (2018)

    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)

    ADS  Google Scholar 

  • Darvishi, M.T., Najafi, M., Wazwaz, A.M.: Conformable space-time fractional nonlinear (1+ 1)-dimensional Schrödinger-type models and their traveling wave solutions. Chaos Solitons Fractals 1(150), 111187 (2021)

    MATH  Google Scholar 

  • Dubey, S., Chakraverty, S.: Application of modified extended tanh method in solving fractional order coupled wave equations. Math. Comput. Simul. 198, 509–520 (2022)

    MathSciNet  MATH  Google Scholar 

  • González-Gaxiola, O., Biswas, A., Yildirim, Y., Alshehri, H.M.: of the Estonian academy of sciences (2023)

  • González-Gaxiola, O., Biswas, A., Yildirim, Y., Alshehri, H.M.: Highly dispersive optical solitons in birefringent fibres with non) local form of nonlinear refractive index: Laplace-Adomian decomposition. Ukr. J. Phys. Opt. 23(2), 68 (2022)

    Google Scholar 

  • He, X.J., Lü, X.: M-lump solution, soliton solution and rational solution to a (3+ 1)-dimensional nonlinear model. Math. Comput. Simul. 197, 327–340 (2022)

    MathSciNet  MATH  Google Scholar 

  • Islam, S.R., Arafat, S.Y., Wang, H.: Abundant closed-form wave solutions to the simplified modified Camassa-Holm equation. J. Ocean Eng. Sci. 8(3), 238–245 (2023)

    Google Scholar 

  • Jiang, Y., Wang, F., Salama, S.A., Botmart, T., Khater, M.M.: Computational investigation on a nonlinear dispersion model with the weak non-local nonlinearity in quantum mechanics. Results Phys. 38, 105583 (2022)

    Google Scholar 

  • Joseph, S.P.: Exact traveling wave doubly periodic solutions for generalized double sine-Gordon equation. Int. J. Appl. Comput. Math. 8–1, 1–19 (2022)

    MathSciNet  Google Scholar 

  • Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 1(264), 65–70 (2014)

    MathSciNet  MATH  Google Scholar 

  • Khater, M.M.: Prorogation of waves in shallow water through unidirectional Dullin-Gottwald-Holm model; computational simulations. Int. J. Mod. Phys. B 37, 2350071 (2022)

    ADS  Google Scholar 

  • Kudryashov, N.A.: Optical solitons of the model with generalized anti-cubic nonlinearity. Optik 257, 168746 (2022)

    ADS  Google Scholar 

  • Kutovyy, S., Savchuk, R., Bashmakova, N., Stanovyi, O., Palchykivska, L.: Vibrational spectra of quercetin and their interpretation with quantum-mechanical density-functional method. Ukr. J. Phys. Opt. 22, 4 (2021)

    Google Scholar 

  • Lechelon, M., Meriguet, Y., Gori, M., Ruffenach, S., Nardecchia, I., Floriani, E., Coquillat, D., Teppe, F., Mailfert, S., Marguet, D., Ferrier, P.: Experimental evidence for long-distance electrodynamic intermolecular forces. Sci. Adv. 8(7), eabl5855 (2022)

    ADS  Google Scholar 

  • Li, Q., Shan, W., Wang, P., Cui, H.: Breather, lump and N-soliton wave solutions of the (2+ 1)-dimensional coupled nonlinear partial differential equation with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 106, 106098 (2022)

    MathSciNet  MATH  Google Scholar 

  • Mahmood, M.F., Alshehri, H.M.: Highly dispersive optical soliton perturbation with Kudryashov’s sextic-power law of nonlinear refractive index. Ukr. J. Phys. Opt. 23(1), 24 (2022)

    Google Scholar 

  • Mahmoud, A.Z., Abdel-Rahim, M.A., Mohamed, M.: Role of the annealing temperature for optimizing the optical and electronic parameters of Ge 10 Se 75 Ag 15 films for optoelectronic applications. Opt. Quant. Electron. 53, 1–20 (2021)

    Google Scholar 

  • Min, R., Hu, X., Pereira, L., Soares, M.S., Silva, L.C., Wang, G., Martins, L., Qu, H., Antunes, P., Marques, C., Li, X.: Polymer optical fiber for monitoring human physiological and body function: a comprehensive review on mechanisms, materials, and applications. Opt. Laser Technol. 147, 107626 (2022)

    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 

  • Najafi, M., Najafi, M., Darvishi, M.T.: New exact solutions to the (2+ 1)-dimensional Ablowitz-Kaup-Newell-Segur equation: modification of the extended homoclinic test approach. Chin. Phys. Lett. 29(4), 040202 (2012)

    Google Scholar 

  • Nwabunwanne, S.F., Donaldson, W.R.: Boosting the external quantum efficiency of AlGaN-based metal-semiconductor-metal ultraviolet photodiodes by electrode geometry variation. IEEE J. Quantum Electron. 57(6), 1–8 (2021)

    Google Scholar 

  • Ozdemir, N.: Optical solitons for Radhakrishnan-Kundu-Lakshmanan equation in the presence of perturbation term and having Kerr law. Optik 271, 170127 (2022)

    ADS  Google Scholar 

  • Rahman, N.: A note on the Sine-Gordon expansion method and its applications. arXiv preprint arXiv: 2101.05460. (2021)

  • Rehman, H.U., Inc, M., Asjad, M.I., Habib, A., Munir, Q.: New soliton solutions for the space-time fractional modified third order Korteweg-de Vries equation. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.05.032

    Article  Google Scholar 

  • Rezazadeh, H., Inc, M., Baleanu, D.: New solitary wave solutions for variants of (3+ 1)-dimensional Wazwaz-Benjamin-Bona-Mahony equations. Front. Phys. 4(8), 332 (2020)

    Google Scholar 

  • Shakeel, M., El-Zahar, E.R., Shah, N.A., Chung, J.D.: Generalized Exp-function method to find closed form solutions of nonlinear dispersive modified Benjamin-Bona-Mahony equation defined by seismic sea waves. Mathematics 10–7, 1026 (2022)

    Google Scholar 

  • Siddique, I., Mehdi, K.B., Jarad, F., Elbrolosy, M.E., Elmandouh, A.A.: Novel precise solutions and bifurcation of traveling wave solutions for the nonlinear fractional (3+ 1)-dimensional WBBM equation. Int. J. Mod. Phys. B 37, 2350011 (2023)

    ADS  Google Scholar 

  • Sultan, A.M., Lu, D., Arshad, M., Rehman, H.U., Saleem, M.S.: Soliton solutions of higher order dispersive cubic-quintic nonlinear Schrödinger equation and its applications. Chin. J. Phys. 1(67), 405–13 (2020). https://doi.org/10.1016/j.cjph.2019.10.003

    Article  Google Scholar 

  • Tarla, S., Ali, K.K., Yilmazer, R., Yusuf, A.: Investigation of the dynamical behavior of the Hirota-Maccari system in single-mode fibers. Opt. Quant. Electron. 54–10, 1–12 (2022)

    Google Scholar 

  • Ur Rehman, S., Ahmad, J.: Modulation instability analysis and optical solitons in birefringent fibers to RKL equation without four-wave mixing. Alex. Eng. J. 60–1, 1339–1354 (2021)

    Google Scholar 

  • Wang, K.J.: Variational principle and diverse wave structures of the modified Benjamin-Bona-Mahony equation arising in the optical illusions field. Axioms 11(9), 445 (2002)

    Google Scholar 

  • Wang, K.J.: Traveling wave solutions of the Gardner equation in dusty plasmas. Results Phys. 33, 105207 (2022)

    Google Scholar 

  • Wang, H., Wang, H.S., Ma, C., Chen, L., Jiang, C., Chen, C., Xie, X., Li, A.P., Wang, X.: Graphene nanoribbons for quantum electronics. Nat. Rev. Phys. 3(12), 791–802 (2021)

    Google Scholar 

  • Wang, K.J., Liu, J.H., Wu, J.: Soliton solutions to the Fokas system arising in monomode optical fibers. Optik 251, 168319 (2022)

    ADS  Google Scholar 

  • Yildrim, Y., Biswas, A., Dakova, A., Guggilla, P., Khan, S., Alshehri, H.M., Belic, M.R.: Cubic-quartic optical solitons having quadratic-cubic nonlinearity by sine-Gordon equation approach. Ukr. J. Phys. Opt. 22(4), 255 (2021)

    Google Scholar 

  • Zayed, E., Shohib, R., Alngar, M., Biswas, A., Ekici, M., Khan, S., Alzahrani, A., Belic, M.: Optical solitons and conservation laws associated with Kudryashov’s sextic power-law nonlinearity of refractive index. Ukr. J. Phys. Opt. 22(1), 38 (2021)

    Google Scholar 

  • Zayed, E.M., Shohib, R., Alngar, M.E., Biswas, A., Yıldırım, Y., Dakova, A., Alshehri, H.M., Belic, M.R.: Optical solitons in the Sasa-Satsuma model with multiplicative noise via Itô calculus. Ukr. J. Phys. Opt. 23(1), 9–16 (2022)

    Google Scholar 

  • Zhang, T., Li, M., Chen, J., Wang, Y., Miao, L., Lu, Y., He, Y.: Multi-component ZnO alloys: bandgap engineering, hetero-structures, and optoelectronic devices. Mater. Sci. Eng. R. Rep. 147, 100661 (2022)

    Google Scholar 

  • Zhao, D., Luo, M.: General conformable fractional derivative and its physical interpretation. Calcolo 54, 903–17 (2017)

    MathSciNet  MATH  Google Scholar 

  • Zhu, Y., Tang, T., Zhao, S., Joralmon, D., Poit, Z., Ahire, B., Keshav, S., Raje, A.R., Blair, J., Zhang, Z., Li, X.: Recent advancements and applications in 3D printing of functional optics. Addit. Manuf. 52, 102682 (2022)

    Google Scholar 

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

    Google Scholar 

  • Zulfiqar, A., Ahmad, J., Ul-Hassan, Q.M.: Analysis of some new wave solutions of fractional order generalized Pochhammer-chree equation using exp-function method. Opt. Quant. Electron. 54(11), 1–21 (2022)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Mirpur University of Science and Technology, (MUST), Mirpur, 10250, AJK, Pakistan

    Asghar Ali & Sara Javed

  2. Department of Mathematics, Faculty of Science, University of Gujrat, 50700, Gujrat, Pakistan

    Jamshad Ahmad

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  1. Asghar Ali
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  2. Jamshad Ahmad
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Contributions

AA: Formal analysis, review and editing. JA: Supervision, reviewed, formal analysis and editing. SJ: Conceptualization, formal analysis, writing the original draft, review, software implementation and editing.

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Correspondence to Jamshad Ahmad.

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Ali, A., Ahmad, 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 (2023). https://doi.org/10.1007/s11082-023-05060-9

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