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An enormous diversity of fractional-soliton solutions with sensitive prodigy to the \(Tzitz\acute{e}ica\)–Dodd–Bullough equation

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Abstract

The central objective of this study is to explore the dynamic response of fractional-soliton solutions within a nonlinear \(Tzitz\acute{e}ica\)–Dodd–Bullough (TDB) equation enhancing the long-distance optical communication, developing advanced materials with unique electromagnetic properties, and contributing to a deeper understanding of complex phenomena. This fractional version integrates fractional derivatives to facilitate the modeling of anomalous diffusion and various other non-local phenomena. We approach the governing model using the extended direct algebraic method, leading to the derivation of fractional-soliton solutions. These solutions are not only exhibited but also have their physical implications elucidated, with two fractional derivative definitions serving as the interpretive tools: the \(\beta\)-derivative and a novel local derivative. The aforementioned integration approach enables the derivation of numerous modern optical soliton solutions, encompassing dark, semi-bright, as well as solutions involving trigonometric, mixed hyperbolic, rational functions, and dark singular solitons. This method effectively highlights the fractional impact of the derived physical phenomena on the fTBD equation. Additionally, the fractional dynamical system undergoes a thorough sensitivity analysis, with the results being graphically represented. To facilitate this, the model undergoes transformation into a planar dynamical system via the Galilean transformation, allowing for an evaluation of the sensitivity performance.

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References

  • Abdel Kader, A.H., El Bialy, F., Nour, H.M., Abdel Latif, M.S.: Some new soliton solutions of a semi-discrete fractional complex coupled dispersionless system. Sci. Rep. 13(1), 6502 (2023)

    ADS  Google Scholar 

  • Akram, M., Muhammad, G., Ahmad, D.: Analytical solution of the Atangana–Baleanu–Caputo fractional differential equations using Pythagorean fuzzy sets. Granular Comput. 2023:1–21

  • Akram, G., Sadaf, M., Zainab, I.: Effect of a new local derivative on space-time fractional nonlinear Schrödinger equation and its stability analysis. Opt. Quant. Electron. 55(9), 834 (2023)

  • Ali, TA., Xiao, Z., Jiang, H., Li, B.: A class of digital integrators based on trigonometric quadrature rules. IEEE Trans. Ind. Electron. 2023

  • Almatrafi, M.B.: Solitary wave solutions to a fractional model using the improved modified extended Tanh-function method. Fract. Fract. 7(3), 252 (2023)

    MathSciNet  Google Scholar 

  • Almeida, R., Bastos, N.R., Monteiro, M.T.: Modeling some real phenomena by fractional differential equations. Math. Methods Appl. Sci. 39(16), 4846–55 (2016)

    ADS  MathSciNet  Google Scholar 

  • Alquran, M.: New interesting optical solutions to the quadratic-cubic Schrödinger equation by using the Kudryashov-expansion method and the updated rational sine-cosine functions. Opt. Quant. Electron. 54(10), 666 (2022)

    Google Scholar 

  • Alquran, M.: Classification of single-wave and bi-wave motion through fourth-order equations generated from the Ito model. Phys. Scripta 8(5), 10210–27 (2023)

    Google Scholar 

  • Alquran, M., Alhami, R.: Analysis of lumps, single-stripe, breather-wave, and two-wave solutions to the generalized perturbed-KdV equation by means of Hirota’s bilinear method. Nonlinear Dyn. 109(3), 1985–92 (2022)

    Google Scholar 

  • Alquran, M., Ali, M., Jadallah, H.: New topological and non-topological unidirectional-wave solutions for the modified-mixed KdV equation and bidirectional-waves solutions for the Benjamin Ono equation using recent techniques. J. Ocean Eng. Sci. 7(2), 163–9 (2022)

    Google Scholar 

  • Alquran, M., Najadat, O., Ali, M., Qureshi, S.: New kink-periodic and convex-concave-periodic solutions to the modified regularized long wave equation by means of modified rational trigonometric-hyperbolic functions. Nonlinear Eng. 12(1), 20220307 (2023)

    ADS  Google Scholar 

  • Alquran, M., Ali, M., Gharaibeh, F., Qureshi, S.: Novel investigations of dual-wave solutions to the Kadomtsev–Petviashvili model involving second-order temporal and spatial-temporal dispersion terms. Partial Differ. Equ. Appl. Math. 8, 100543 (2023)

    Google Scholar 

  • Arefin, M.A., Sadiya, U., Inc, M., Uddin, M.H.: Adequate soliton solutions to the space-time fractional telegraph equation and modified third-order KdV equation through a reliable technique. Opt. Quant. Electron. 54(5), 309 (2022)

    Google Scholar 

  • Arefin, M.A., Khatun, M.A., Islam, M.S., Akbar, M.A., Uddin, M.H.: Explicit soliton solutions to the fractional order nonlinear models through the Atangana beta derivative. Int. J. Theor. Phys. 62(6), 134 (2023)

    MathSciNet  Google Scholar 

  • Arqub, O.A., Al-Smadi, M., Almusawa, H., Baleanu, D., Hayat, T., Alhodaly, M., Osman, M.S.: A novel analytical algorithm for generalized fifth-order time-fractional nonlinear evolution equations with conformable time derivative arising in shallow water waves. Alex. Eng. J. 61(7), 5753–69 (2022)

    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), 145–9 (2016)

    Google Scholar 

  • Bairwa, R.K., Tyagi, S.: Solution of non-linear time-fractional generalized Hirota–Satsuma coupled Korteweg-de Vries equation by using new analytical approach. Int. J. Math. Appl. 11(1), 1–4 (2023)

    Google Scholar 

  • Chen, H.X., Chen, W., Liu, X., Liu, X.H.: Establishing the first hidden-charm pentaquark with strangeness. Eur. Phys. J. C 81(5), 409 (2021)

    ADS  MathSciNet  Google Scholar 

  • Chou, D., Ur Rehman, H., Amer, A., Amer, A.: New solitary wave solutions of generalized fractional type evolution equations using Sardar sub-equation method. Opt. Quant. Electron. 55(13), 1–6 (2023)

    Google Scholar 

  • Dong, J., Hu, J., Zhao, Y., Peng, Y.: Opinion formation analysis for Expressed and Private Opinions (EPOs) models: reasoning private opinions from behaviors in group decision-making systems. Expert Syst. Appl. 236, 121292 (2024)

    Google Scholar 

  • Dussel, I.C., Bonder, J.F.: A Bourgain–Brezis–Mironescu formula for anisotropic fractional Sobolev spaces and applications to anisotropic fractional differential equations. J. Math. Anal. Appl. 519(2), 126805 (2023)

    MathSciNet  Google Scholar 

  • Esen, H., Ozdemir, N., Secer, A., Bayram, M., Sulaiman, T.A., Ahmad, H., Yusuf, A., Albalwi, M.D.: On the soliton solutions to the density-dependent space time fractional reaction-diffusion equation with conformable and M-truncated derivatives. Opt. Quant. Electron. 55(10), 923 (2023)

    Google Scholar 

  • Fan, E., Hona, Y.C.: Generalized tanh method extended to special types of nonlinear equations. Zeit. fur Naturforschung A. 57(8), 692–700 (2002)

    ADS  Google Scholar 

  • Ghayad, M.S., Badra, N.M., Ahmed, H.M., Rabie, W.B.: Derivation of optical solitons and other solutions for nonlinear Schrödinger equation using modified extended direct algebraic method. Alex. Eng. J. 64, 801–11 (2023)

    Google Scholar 

  • Gu, Q., Li, S., Liao, Z.: Solving nonlinear equation systems based on evolutionary multitasking with neighborhood-based speciation differential evolution. Expert Syst. Appl. 2023, 122025 (2023)

    Google Scholar 

  • Guzman, P., Langton, G., Lugo, L., Medina, J., Valdes, J.: A new definition of a fractional derivative of local type. J. Math. Anal. 9, 88–98 (2018)

    MathSciNet  Google Scholar 

  • Han, T., Li, Z., Zhang, K.: Exact solutions of the stochastic fractional long-short wave interaction system with multiplicative noise in generalized elastic medium. Results Phys. 44, 106174 (2023)

    Google Scholar 

  • Ilhan, O.A., Baskonus, H.M., Islam, M.N., Akbar, M.A., Soybas, D.: Stable soliton solutions to the time fractional evolution equations in mathematical physics via the new generalized G’/G-expansion method. Int. J. Nonlinear Sci. Numer. Simul. 24(1), 185–200 (2023)

    MathSciNet  Google Scholar 

  • Jaradat, I., Alquran, M.: Construction of solitary two-wave solutions for a new two-mode version of the Zakharov–Kuznetsov equation. Mathematics 8(7), 1127 (2020)

    Google Scholar 

  • Jin, H.Y., Wang, Z.A.: Boundedness, blowup and critical mass phenomenon in competing chemotaxis. J. Differ. Equ. 260(1), 162–96 (2016)

    ADS  MathSciNet  Google Scholar 

  • Khatun, M.A., Arefin, M.A., Akbar, M.A., Uddin, M.H.: Numerous explicit soliton solutions to the fractional simplified Camassa–Holm equation through two reliable techniques. Ain Shams Eng. J. 2023, 102214 (2023)

    Google Scholar 

  • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, New York (2006)

    Google Scholar 

  • Li, H., Peng, R., Wang, Z.A.: On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: analysis, simulations, and comparison with other mechanisms. SIAM J. Appl. Math. 78(4), 2129–53 (2018)

    MathSciNet  Google Scholar 

  • Machado, J.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–53 (2011)

    ADS  MathSciNet  Google Scholar 

  • Majid, S.Z., Faridi, W.A., Asjad, M.I., Abd El-Rahman, M., Eldin, S.M.: Explicit soliton structure formation for the Riemann wave equation and a sensitive demonstration. Fract. Fract. 7(2), 102 (2023)

    Google Scholar 

  • Mirhosseini-Alizamini, S.M., Rezazadeh, H., Srinivasa, K., Bekir, A.: New closed form solutions of the new coupled Konno–Oono equation using the new extended direct algebraic method. Pramana 94, 1–2 (2020)

    Google Scholar 

  • Oldham, K., Spanier, J.: The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier, New York (1974)

    Google Scholar 

  • Peng, Y., Zhao, Y., Hu, J.: On the role of community structure in evolution of opinion formation: a new bounded confidence opinion dynamics. Inf. Sci. 621, 672–90 (2023)

    Google Scholar 

  • Rahman, R.U., Raza, N., Jhangeer, A., Inc, M.: Analysis of analytical solutions of fractional Date–Jimbo–Kashiwara–Miwa equation. Phys. Lett. A 470, 128773 (2023)

    MathSciNet  Google Scholar 

  • Rahman, R.U., Qousini, M.M., Alshehri, A., Eldin, S.M., El-Rashidy, K., Osman, M.S.: Evaluation of the performance of fractional evolution equations based on fractional operators and sensitivity assessment. Results Phys. 49, 106537 (2023)

    Google Scholar 

  • Rahman, R.U., Raza, N., Jhangeer, A., Inc, M.: Analysis of analytical solutions of fractional Date–Jimbo–Kashiwara–Miwa equation. Phys. Lett. A 470, 128773 (2023)

    MathSciNet  Google Scholar 

  • Rani, A., Shakeel, M., Kbiri Alaoui, M., Zidan, A.M., Shah, N.A., Junsawang, P.: Application of the Exp-\(\phi \xi\)-expansion method to find the soliton solutions in biomembranes and nerves. Mathematics 10(18), 3372 (2022)

    Google Scholar 

  • Rasool, T., Hussain, R., Rezazadeh, H., Gholami, D.: The plethora of exact and explicit soliton solutions of the hyperbolic local (4+ 1)-dimensional BLMP model via GERF method. Results Phys. 46, 106298 (2023)

    Google Scholar 

  • Sadaf, M., Akram, G., Inc, M., Dawood, M., Rezazadeh, H., Akgul, A.: Exact special solutions of space-time fractional Cahn–Allen equation by beta and M-truncated derivatives. Int. J. Modern Phys. B 2023, 2450118 (2023)

    Google Scholar 

  • Sadiya, U., Inc, M., Arefin, M.A., Uddin, M.H.: Consistent travelling waves solutions to the non-linear time fractional Klein-Gordon and Sine-Gordon equations through extended tanh-function approach. J. Taibah Univ. Sci. 16(1), 594–607 (2022)

    Google Scholar 

  • Safdar, M., Ijaz Khan, M., Khan, R.A., Taj, S., Abbas, F., Elattar, S., Galal, A.M.: Analytic solutions for the MHD flow and heat transfer in a thin liquid film over an unsteady stretching surface with Lie symmetry and homotopy analysis method. Waves Random Complex Media 33(2), 442–60 (2023)

    ADS  Google Scholar 

  • Sarwar, A., Gang, T., Arshad, M., Ahmed, I., Ahmad, M.O.: Abundant solitary wave solutions for space-time fractional unstable nonlinear Schröinger equations and their applications. Ain Shams Eng. J. 14(2), 101839 (2023)

    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(02), 2350011 (2023)

    ADS  Google Scholar 

  • Singh, J., Kumar, D., Qurashi, M.A., Baleanu, D.: A new fractional model for giving up smoking dynamics. Adv. Differ. Equ. 2017(1), 1–6 (2017)

    MathSciNet  Google Scholar 

  • Tang, L.: Bifurcation analysis and optical soliton solutions for the fractional complex Ginzburg-Landau equation in communication systems. Optik 276, 170639 (2023)

    ADS  Google Scholar 

  • Uddin, M.H., Zaman, U.H., Arefin, M.A., Akbar, M.A.: Nonlinear dispersive wave propagation pattern in optical fiber system. Chaos Solitons Fract. 164, 112596 (2022)

    MathSciNet  Google Scholar 

  • Wang, K.L.: New solitary wave solutions of the fractional modified kdv-kadomtsev-petviashvili equation. Fractals 2023, 2350025 (2023)

    Google Scholar 

  • Wang, K.: Exact travelling wave solution for the local fractional Camassa–Holm–Kadomtsev–Petviashvili equation. Alex. Eng. J. 63, 371–6 (2023)

    Google Scholar 

  • Xu, H.: A generalized analytical approach for highly accurate solutions of fractional differential equations. Chaos Solitons Fract. 166, 112917 (2023)

    Google Scholar 

  • Yasmin, H., Abu Hammad, M.M., Shah, R., Alotaibi, B.M., Ismaeel, S.M., El-Tantawy, S.A.: On the solutions of the fractional-order Sawada–Kotera–Ito equation and modeling nonlinear structures in fluid mediums. Symmetry 15(3), 605 (2023)

    ADS  Google Scholar 

  • Yepez-Martineez, H., Gomez-Aguilar, J.F., Baleanu, D.: Beta-derivative and sub-equation method applied to the optical solitons in medium with parabolic law nonlinearity and higher order dispersion. Optik 155, 357–65 (2018)

    ADS  Google Scholar 

  • Yepez-Martinez H, Rezazadeh H, Inc M, Ali Akinlar M (2021) New solutions to the fractional perturbed Chen–Lee–Liu equation with a new local fractional derivative. Waves Random Complex Media 1–36

  • Yusuf, A., Inc, M., Baleanu, D.: Optical solitons with M-truncated and beta derivatives in nonlinear optics. Front. Phys. 7, 126 (2019)

    Google Scholar 

  • Zahran, E.H., Bekir, A.: New diverse soliton solutions for the coupled Konno–Oono equations. Opt. Quant. Electron. 55(2), 112 (2023)

    Google Scholar 

  • Zaman, U.H., Arefin, M.A., Akbar, M.A., Uddin, M.H.: Analyzing numerous travelling wave behavior to the fractional-order nonlinear Phi-4 and Allen-Cahn equations throughout a novel technique. Results Phys. 37, 105486 (2022)

    Google Scholar 

  • Zaman, U.H., Arefin, M.A., Akbar, M.A., Uddin, M.H.: Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique. PLoS ONE 18(5), e0285178 (2023)

    Google Scholar 

  • Zhang, S.: A generalized auxiliary equation method and its application to (2+ 1)-dimensional Korteweg-de Vries equations. Comput. Math. Appl. 54(7–8), 1028–38 (2007)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Near East University, Operational Research Center in Healthcare, TRNC Mersin 10, Nicosia, 99138, Turkey

    Hijaz Ahmad

  2. Faculty of Science and Information Technology, Department of Mathematics, Al-Zaytoonah University of Jordan, Amman, 11733, Jordan

    Maysoon Qousini

  3. Institute for Advanced Study, Shenzhen University, Shenzhen, 518060, Guangdong, People’s Republic of China

    Riaz Ur Rahman

  4. Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

    Hijaz Ahmad

  5. Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, Mishref, Kuwait

    Hijaz Ahmad

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  1. Hijaz Ahmad
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Conceptualization: MQ Data curation: MQ Formal analysis: RUR Validation: RUR Writing—original draft: RUR Writing—review editing: HA.

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Correspondence to Hijaz Ahmad.

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Ahmad, H., Qousini, M. & Rahman, R.U. An enormous diversity of fractional-soliton solutions with sensitive prodigy to the \(Tzitz\acute{e}ica\)–Dodd–Bullough equation. Opt Quant Electron 56, 726 (2024). https://doi.org/10.1007/s11082-023-06222-5

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