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Dispersive optical soliton solutions to the truncated time M-fractional paraxial wave equation with its stability analysis

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Abstract

This article investigates the truncated time M-fractional paraxial wave equation. This model is frequently used to depict the activation of waves in utterly different physical frameworks, such as quantum mechanics and optics. Two trustworthy methodologies, the improved F-expansion and modified exponent function method, are used to obtain the different soliton solutions to the truncated time M-fractional paraxial wave equation. The obtained solutions offer a valuable understanding of the underlying physical events. Discussion is also had over the equation’s modulation instability, which confirms the given equation is stable. Several graphical charts, including 2D, 3D, density, and contour, are produced using symbolic software. These visual representations have a significant positive impact on qualitative evaluations of diverse natural occurrences. The evaluated findings showed that the approaches employed in this work to obtain inclusive and standard solutions are efficient and speedier in computing, they will be beneficial in addressing more difficult higher order nonlinear perturbed truncated time M-fractional models.

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References

  • Acay, B., Bas, E., Abdeljawad, T.: Non-local fractional calculus from different viewpoints generated by truncated M-derivative. J. Comput. Appl. Math. 366, 112410 (2020)

    Article  MathSciNet  Google Scholar 

  • Ahmad, J., Akram, S., Ali, A: Analysis of new soliton type solutions to generalized extended (2+ 1)-dimensional Kadomtsev-Petviashvili equation via two techniques. Ain Shams Eng. J, 102302 (2023)

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

  • Ahmad, J., Mustafa, Z.: (2023). Dynamics of exact solutions of nonlinear resonant Schrödinger equation utilizing conformable derivatives and stability analysis. Eur. Phys. J. D, 77(6), 123, (2021)

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

  • Akbar Ali, M., Ali, N.H.M.: The improved F-expansion method with Riccati equation and its applications in mathematical physics. Cogent Math. 4(1), 1282577 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Akram, S., Ahmad, J., Rehman, S.: U: Stability analysis and dynamical behavior of solitons in nonlinear optics modelled by Lakshmanan-Porsezian-Daniel equation. Opt Quantum Electron 55, 685 (2023)

    Article  Google Scholar 

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

    Article  MathSciNet  ADS  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 Quantum Electron 55, 810 (2023)

    Article  Google Scholar 

  • Bas, E., Acay, B.: The direct spectral problem via local derivative, including truncated Mittag-Leffler function. Appl. Math. Comput. 367, 124787 (2020)

    MathSciNet  Google Scholar 

  • Bilal, M., Haris, H., Waheed, A., Faheem, M.: The analysis of exact solitons solutions in monomode optical fibers to the generalized nonlinear Schrödinger system by the compatible techniques. Int. J. Math. Comput. Sci. 1(2), 149–170 (2023)

    Google Scholar 

  • Golmankhaneh, A.K., Welch, K., Serpa, C.: & Jørgensen. Fuzzification of Fractal Calculus, P. E. (2023)

  • Hussain, A., Ali, H., Zaman, F., Abbas, N.: New closed form solutions of some nonlinear pseudo-parabolic models via a new extended direct algebraic method. Int. J. Math. Comput. Sci. (2023)

  • Ibrahim, R.W., Jalab, H.A., Karim, F.K., Alabdulkreem, E., Ayub, M.N.: A medical image enhancement based on generalized fractional partial differential equations classes. Quant. Imaging Med. Surg. 12, 172 (2022)

    Article  Google Scholar 

  • İlhan, E., Kıymaz, İO.: (2020) Generalization of truncated \(M\)-fractional derivative and applications to fractional differential equations. Appl. Math. Nonlinear Sci. 5(1), 171–188 (2023)

    Article  Google Scholar 

  • Ismael, H.F., Baskonus, H.M., Bulut, H., Gao, W.: Instability modulation and novel optical soliton solutions to the Gerdjikov-Ivanov equation with \(M\)-fractional. Opt. Quantum Electron. 55, 303 (2021)

    Article  Google Scholar 

  • Jin, T., Yang, X.: Monotonicity theorem for the uncertain fractional differential equation and application to the uncertain financial market. Math. Comput. Simul. 190, 203–221 (2021)

    Article  MathSciNet  Google Scholar 

  • Khater, M.M.: Computational and numerical wave solutions of the Caudrey–Dodd–Gibbon equation. Heliyon, 9(2), (2023r)

  • Khater, M.M.: Soliton propagation under diffusive and nonlinear effects in physical systems;(1+ 1)–dimensional MNW integrable equation. Phys. Lett. A, (2023c)

  • Khater, M.M.: Analyzing pulse behavior in optical fiber: novel solitary wave solutions of the perturbed Chen-Lee-Liu equation. Mod. Phys. Lett. B 37, 2350177 (2023a)

    Article  MathSciNet  ADS  Google Scholar 

  • Khater, M.M.: Computational simulations of propagation of a tsunami wave across the ocean. Chaos, Solitons Fract. 174, 113806 (2023b)

    Article  MathSciNet  Google Scholar 

  • Khater, M.M.: Novel computational simulation of the propagation of pulses in optical fibers regarding the dispersion effect. Int. J. Mod. Phys. B 37(09), 2350083 (2023d)

    Article  ADS  Google Scholar 

  • Khater, M.M.: In surface tension; gravity-capillary, magneto-acoustic, and shallow water waves’ propagation. Eur. Phys. J. Plus 138(4), 320 (2023e)

    Article  Google Scholar 

  • Khater, M.M.: Abundant and accurate computational wave structures of the nonlinear fractional biological population model. Int. J. Mod. Phys. B 37(18), 2350176 (2023f)

    Article  ADS  Google Scholar 

  • Khater, M.M.: Long waves with a small amplitude on the surface of the water behave dynamically in nonlinear lattices on a non-dimensional grid. Int. J. Mod. Phys. B 37(19), 2350188 (2023g)

    Article  ADS  Google Scholar 

  • Khater, M.M.: Numerous Accurate and Stable Solitary Wave Solutions to the Generalized Modified Equal-Width Equation. Int. J. Theor. Phys. 62(7), 151 (2023h)

    Article  MathSciNet  ADS  Google Scholar 

  • Khater, M.M.: Advancements in computational techniques for precise solitary wave solutions in the (1+ 1)-dimensional mikhailov-novikov-wang equation. Int. J. Theor. Phys. 62(7), 152 (2023i)

    Article  MathSciNet  Google Scholar 

  • Khater, M.M.: Characterizing shallow water waves in channels with variable width and depth; computational and numerical simulations. Chaos, Solitons Fract. 173, 113652 (2023j)

    Article  MathSciNet  Google Scholar 

  • Khater, M.M.: Horizontal stratification of fluids and the behavior of long waves. Eur. Phys. J. Plus 138(8), 715 (2023k)

    Article  Google Scholar 

  • Khater, M.M.: Horizontal stratification of fluids and the behavior of long waves. Eur. Phys. J. Plus 138(8), 715 (2023l)

    Article  Google Scholar 

  • Khater, M.M.: Multi-vector with nonlocal and non-singular kernel ultrashort optical solitons pulses waves in birefringent fibers. Chaos, Solitons Fract. 167, 113098 (2023m)

    Article  MathSciNet  Google Scholar 

  • Khater, M.M.: A hybrid analytical and numerical analysis of ultra-short pulse phase shifts. Chaos, Solitons & Fract. 169, 113232 (2023n)

    Article  MathSciNet  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(08), 2350071 (2023o)

    Article  ADS  Google Scholar 

  • Khater, M.M.: In solid physics equations, accurate and novel soliton wave structures for heating a single crystal of sodium fluoride. Int. J. Mod. Phys. B 37(07), 2350068 (2023p)

    Article  ADS  Google Scholar 

  • Khater, M.M.: Nonlinear elastic circular rod with lateral inertia and finite radius: Dynamical attributive of longitudinal oscillation. Int. J. Mod. Phys. B 37(06), 2350052 (2023q)

    Article  ADS  Google Scholar 

  • Khater, M.M.: Physics of crystal lattices and plasma; analytical and numerical simulations of the Gilson-Pickering equation. Results Phys. 44, 106193 (2023s)

    Article  Google Scholar 

  • Kumar, A., Kumar, S.: Dynamic nature of analytical soliton solutions of the (1+ 1)-dimensional Mikhailov-Novikov-Wang equation using the unified approach. Int. J. Math. Comput. Sci., (2023)

  • Milici, C., Drăgănescu, G., Machado, J.T.: Introduction to fractional differential equations 25, Springer (2018)

  • Rasool, T., Hussain, R., Al Sharif, M.A., Mahmoud, W., Osman, M.S.: Various optical soliton solutions for the M-truncated Paraxial wave equation use Sardar-subequation technique. Opt. Quantum Electron 55(5), 396 (2023)

    Article  Google Scholar 

  • Riaz, M.B., Jhangeer, A., Awrejcewicz, J., Baleanu, D., Tahir, S.: Fractional propagation of short light pulses mono mode mode optical fibers: comparison of beta and truncated \(M\)-fractional derivatives. J. Comput. Nonlinear Dyn. 17(3), 031002 (2022)

    Article  Google Scholar 

  • Roshid, M.M., Uddin, M., Mostafa, G.: Dynamic optical soliton solutions for \(M\)-fractional Paraxial Wave equation using unified technique. Results Phys., 106632, (2023)

  • Sivasundaram, S., Kumar, A., Singh, R.K.: On the complex properties to the first equation of the Kadomtsev-Petviashvili hierarchy. Int. J. Comput. Math., (2024)

  • Sousa, J.V.D.C., de Oliveira, E.C.: A new truncated \(M\)-fractional derivative type unifies some fractional derivative types with classical properties. (2017) arXiv preprint arXiv:1704.08187

  • Sylvere, A.S., Justin, M., David, V., Joseph, M., Betchewe, G.: Impact of fractional effects on modulational instability and bright soliton in fractional optical metamaterials. Waves in Random and Complex Media, 1-14, (2021)

  • Tagne, A.T., Ema’a Ema’a, J.M., Ben-Bolie, G.H., Buske, D.: A new truncated \(M\)-fractional derivative for air pollutant dispersion. Indian J. Phys. Appl. 94, 1777–1784 (2020)

    Article  ADS  Google Scholar 

  • Tian, Y., Liu, J.: A modified exp-function method for fractional partial differential equations. Therm. Sci. 25, 1237–1241 (2021)

    Article  Google Scholar 

  • Wu, G.C., Baleanu, D., Zeng, S.D., Luo, W.H.: Mittag-Leffler function for discrete fractional modelling. J. King Saud. Univ. Sci. 28, 99–102 (2016)

    Article  Google Scholar 

  • Yao, S.W., Manzoor, R., Zafar, A., Inc, M., Abbagari, S., Houwe, A.: Exact soliton solutions to the Cahn-Allen equation and Predator-Prey model with truncated \(M\)-fractional derivative. Results Phys. 37, 105455 (2022)

    Article  Google Scholar 

  • Zafar, A., Bekir, A., Raheel, M., Razzaq, W.: Optical soliton solutions to Biswas-Arshed model with truncated \(M\)-fractional derivative. Optik 222, 165355 (2020)

    Article  ADS  Google Scholar 

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Authors and Affiliations

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

    Jamshad Ahmad, Kanza Noor & Sonia Akram

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  1. Jamshad Ahmad
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  2. Kanza Noor
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  3. Sonia Akram
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Contributions

JA: Supervision, methodology, formal analysis, review and editing. KN: Formal analysis, supervision, review and editing, conceptualization, writing the original draft, software implementation. SA: Formal analysis, Investigation, Software, Validation.

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Correspondence to Jamshad Ahmad.

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Ahmad, J., Noor, K. & Akram, S. Dispersive optical soliton solutions to the truncated time M-fractional paraxial wave equation with its stability analysis. Opt Quant Electron 56, 896 (2024). https://doi.org/10.1007/s11082-024-06663-6

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