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Dynamics of exact solitary wave solutions to the conformable time-space fractional model with reliable analytical approaches

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Abstract

This article elucidates the dynamical behavior of exact solitary waves to the conformable three dimensional Wazwaz-Benjamin-Bona-Mahony (3D-WBBM) equation with its spatial and temporal variables. A variety of solitary wave solutions with unknown parameters are extracted in different shapes such as bell-type, shock-type, singular, combine, and complex solitons by utilizing two mathematical tools namely the extended rational sine-cosine/sinh-cosh and extended sinh-Gordon equation expansion method (ShGEEM). Besides, we also secure singular periodic wave solutions with unknown parameters. All the secured solutions are verified by substituting back to the original equation through soft computation Mathematica. The physical characterization of some reported results is figured out graphically in 3D, 2D, and their corresponding contour profiles by using different scales of parameters to clarify and visualize the physical features of the problem. On the basis of achieved results, we may claim that the proposed computational methods are direct, dynamics, well organized, and will be useful for solving the more complicated nonlinear problems in diverse areas together with symbolic computations.

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References

  • Abdel-Gawad, H.I., Osman, M.S.: On the variational approach for analyzing the stability of solutions of evolution equations. Kyungpook Math. J. 53(4), 661–680 (2013)

    MathSciNet  MATH  Google Scholar 

  • Abdel-Gawad, H.I., Osman, M.S.: On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients. J. Adv. Res. 6(5), 593–599 (2015)

    Google Scholar 

  • Akbar, M.A., Akinyemi, L., Yao, S.W., Jhangeer, A., Rezazadeh, H., Khater, M.M.A., Ahmad, H., Inc, M.: Soliton solutions to the Boussinesq equation through sine-Gordon method and Kudryashov method. Results Phys. 25, 104228 (2021)

    Google Scholar 

  • Akbar, M.A., Kayum, M.A., Osman, M.S.: Bright, periodic, compacton and bell-shape soliton solutions of the extended QZK and (3+ 1)-dimensional ZK equations. Commun. Theor. Phys. 73(10), 105003 (2021)

    ADS  MathSciNet  Google Scholar 

  • Akinyemi, L., Hosseini, K., Salahshour, S.: The bright and singular solitons of (2+1)-dimensional nonlinear Schrödinger equation with spatio-temporal dispersions. Optik 242, 167120 (2021)

    ADS  Google Scholar 

  • Akinyemi, L., Rezazadeh, H., Yao, S.W., Akbar, M.A., Khater, M.M.A., Jhangeer, A., Inc, M., Ahmad, H.: Nonlinear dispersion in parabolic law medium and its optical solitons. Results Phys. 26, 104411 (2021)

    Google Scholar 

  • Akinyemi, L., Senol, M., Osman, M.S.: Analytical and approximate solutions of nonlinear Schrödinger equation with higher dimension in the anomalous dispersion regime. J. Ocean Eng. Sci. (2021). https://doi.org/10.1016/j.joes.2021.07.006

    Article  Google Scholar 

  • Akram, U., Seadawy, A.R., Rizvi, S.T.R., Younis, M., Althobaiti, S., Sayed, S.: Traveling wave solutions for the fractional Wazwaz-Benjamin-Bona-Mahony model in arising shallow water waves. Results Phys. 20, 103725 (2021)

    Google Scholar 

  • Akturk, T., Sulaiman, T.A., Baskonus, H.M., Bulut, H.: Complex acoustic gravity wave behaviors to a mathematical model arising in nonlinear mathematical physics. ITM Web Conf. 22, 01032 (2018)

    Google Scholar 

  • Albert, J.: Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation. J. Differ. Equ. 63, 117–134 (1986)

    ADS  MathSciNet  MATH  Google Scholar 

  • Ali, K.K., Osman, M.S., Baskonus, H.M., Elazab, N.S., Ilhan, E.: Analytical and numerical study of the HIV-1 infection of CD4+T-cells conformable fractional mathematical model that causes acquired immunodeficiency syndrome (AIDS) with the effect of antiviral drug therapy. Math. Meth. Appl. Sci. (2020). https://doi.org/10.1002/mma.7022

    Article  Google Scholar 

  • Ali, K.K., Yilmazer, R., Bulut, H., Aktürk, T., Osman, M.S.: Abundant exact solutions to the strain wave equation in micro-structured solids. Mod. Phy. Lett. B 35(26), 2150439 (2021)

    ADS  MathSciNet  Google Scholar 

  • Amick, C.J., Bona, J.L., Schonbeks, M.E.: Decay of solutions of some nonlinear wave equation. J. Differ. Equ. 81, 1–49 (1989)

    ADS  MathSciNet  Google Scholar 

  • Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm Sci. (2016)

  • Atas, S.S., Sulaiman, T.A., Bulut, H.: Optical solitons and other solutions to the (2+ 1)-dimensional cubic nonlinear Schrödinger equation with fractional temporal evolution. ITM Web Conf. 22, 01053 (2018)

    Google Scholar 

  • Baskonus, H.M., Bulut, H., Sulaiman, T.A.: Investigation of various travelling wave solutions to the extended (2+ 1)-dimensional quantum ZK equation. Eur. Phys. J. Plus 132, 11 (2017)

    Google Scholar 

  • Baskonus, H.M., Osman, M.S., 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. Quantum Electron. 53(10), 556 (2021)

    Google Scholar 

  • Benjamin, T. B.: Lectures on nonlinear wave motion. Lectures in Appl. Math. 15, AMS,Providence, RI,. (1974)

  • Bilal, M., Hu, W., Ren, J.: Different wave structures to the Chen-Lee-Liu equation of monomode fibers and its modulation instability analysis. Eur. Phys. J. Plus 136, 385 (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. Quantam Electron 53, 406 (2021)

    Google Scholar 

  • Bilal, M., Younas, U., Ren, J.: Dynamics of exact soliton solutions in the double-chain model of deoxyribonucleic acid. Math. Meth Appl. Sci. (2021). https://doi.org/10.1002/mma.7631

    Article  MathSciNet  MATH  Google Scholar 

  • Bilal, M., Younas, U., Ren, J.: Dynamics of exact soliton solutions to the coupled nonlinear system using reliable analytical mathematical approaches. Commun. Theor. Phys. 73, 085005 (2021)

    ADS  MathSciNet  Google Scholar 

  • Bilal, M., Younas, U., Ren, J.: Propagation of diverse solitary wave structures to the dynamical soliton model in mathematical physics. Opt. Quantam Electron 53, 522 (2021)

    Google Scholar 

  • Bona, J.L., Dougalis, V.A.: An initial and boundry value problem for a model equation propagation of long waves. J. Math. Anal. Appl. 75, 503–522 (1980)

    MathSciNet  Google Scholar 

  • Bulut, H., Aksan, E.N., Kayhan, M., Sulaiman, T.A.: New solitary wave structures to the (3+1) dimensional Kadomtsev-Petviashvili and Schrödinger equation. J. Ocean Eng. Sci. 4(4), 373–378 (2019)

    Google Scholar 

  • Debnath, L.: Recent applications of fractional calculus to science and engineering. Int J. Math. Math. Sci. 54, 3413–3442 (2003)

    MathSciNet  MATH  Google Scholar 

  • Estevez, P.G., Kuru, S., Negro, J., Nieto, L.M.: Travelling wave solutions of the generalized Benjamin-Bona-Mahony equation. Chaos Solitons Fractals 40(4), 2031–2040 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  • Gao, F., Chi, C.: Improvement on conformable fractional derivative and its applications in fractional differential equations. J. Funct. Spaces 2020, 5852414 (2020)

    MathSciNet  MATH  Google Scholar 

  • Gao, W., Yel, G., Baskonus, H.M., Cattani, C.: Complex solitons in the conformable (2+1)-dimensional Ablowitz-Kaup-Newell-Segur Equation. AIMS Math. 5(1), 507–521 (2020)

    MathSciNet  Google Scholar 

  • Ilhan, O.A., Bulut, H., Sulaiman, T.A., Baskonus, H.M.: On the new wave behavior of the Magneto-Electro-Elastic (MEE) circular rod longitudinal wave equation. Int. J. Optim. Control: Theor. Appl. 10(1), 1–8 (2019)

    MathSciNet  Google Scholar 

  • Inc, M., Kilic, B., Baleanu, D.: Optical soliton solutions of the pulse propagation generalized equation in parabolic-law media with space-modulated coefficients. Optik 127, 1056–1058 (2016)

    ADS  Google Scholar 

  • Ismael, H.F., Atas, S.S., Bulut, H., Osman, M.S.: Analytical solutions to the M-derivative resonant Davey-Stewartson equations. Mod. Phy. Lett. B 35(30), 2150455 (2021)

    ADS  MathSciNet  Google Scholar 

  • Kaabar, M.K.A. Kaplan, ,M., Siri, Z.: New exact soliton solutions of the (3 + 1)-dimensional conformable Wazwaz-Benjamin-Bona-Mahony equation via two novel techniques. J. Funct. Spaces 4659905 (2021)

  • Khalil, R., Al-Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 265, 65–701 (2014)

    MathSciNet  MATH  Google Scholar 

  • Kilbas, A..A., Srivastava, H..M., Trujillo, J..J.: Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  • Kilic, B., Inc, M.: Optical solitons for the Schrodinger-Hirota equation with power law nonlinearity by the Backlund transformation. Optik 138, 64–67 (2017)

    ADS  Google Scholar 

  • Leta, W., Liu, T.D., El-Achab, A., Rezazadeh, H., Bekir, A.: Dynamical behavior of traveling wave solutions for a (2+ 1)-Dimensional Bogoyavlenskii coupled system. Qual. Theory Dyn. Syst. 20(1), 1–22 (2021)

    MathSciNet  MATH  Google Scholar 

  • Lin, X.X., Shi, T.J.: Travelling wave solutions for Konopelchenko-Dubrovsky equation using an extended sinh-gordon equation expansion method. Commun. Theor. Phys. 50, 1047 (2008)

    ADS  MathSciNet  MATH  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(7), e07483 (2021)

    Google Scholar 

  • Nisar, K.S., Ciancio, A., Ali, K.K., Osman, M.S., Cattani, C., Baleanu, D., Zafar, A., Raheel, M., Azeem, M.: On beta-time fractional biological population model with abundant solitary wave structures. Alex Eng. J. (2021). https://doi.org/10.1016/j.aej.2021.06.106

    Article  Google Scholar 

  • Osman, M.S.: On multi-soliton solutions for the (2+1)-dimensional breaking soliton equation with variable coefficients in a graded-index waveguide. Comput. Math. with Appl. 75(1), 1–6 (2018)

    MathSciNet  MATH  Google Scholar 

  • Osman, M.S., Abdel-Gawad, H.I., El Mahdy, M.A.: Two-Layer-Atmospheric blocking in a medium with high nonlinearity and lateral dispersion. Results Phys. 8(3), 1054–1060 (2018)

    ADS  Google Scholar 

  • Osman, M.S., Machado, J.A.T.: New nonautonomous combined multi-wave solutions for (2+1)-dimensional variable coefficients KdV equation. Nonlin. Dyn. 93(2), 733–740 (2018)

    MATH  Google Scholar 

  • Rasheed, N.M., Al-Amr, M.O., Az-Zobi, E.A., Tashtoush, M.A., Akinyemi, L.: Stable optical solitons for the Higher-order Non-Kerr NLSE via the modified simple equation method. Math 9(16), 1986 (2021)

    Google Scholar 

  • Rehman, S.U., Ahmad, J.: Modulation instability analysis and optical solitons in birefringent fibers to RKL equation without four wave mixing. Alex. Eng. J. 60, 1339–1354 (2020)

    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. 8, 332 (2020)

    Google Scholar 

  • Saliou, Y., Abbagari, S., Houwe, A., Osman, M.S., Yamigno, D.S., Crépin, K.T., Inc, M.: W-shape bright and several other solutions to the (3+ 1)-dimensional nonlinear evolution equations. Mod. Phy. Lett. B 35(30), 2150468 (2021)

    ADS  MathSciNet  Google Scholar 

  • Seadawy, A.R., Ali, A., Albarakati, W.A.: Analytical wave solutions of the(2+1)-dimensional first integro-differential Kadomtsev Petviashivili hierarchy equation by using modified mathematical methods. Results Phys. 15, 102775 (2019)

    Google Scholar 

  • Seadawy, A.R., Ali, K.K., Nuruddeen, R.I.: A variety of soliton solutions for fractional Wazwaz-Benjamin-Bona-Mahony equation. Results Phys. 12, 2234–2241 (2019)

    ADS  Google Scholar 

  • Siddique, I., Jaradat, M.M.M., Zafar, A., Mehdi, K.B., Osman, M.S.: Exact traveling wave solutions for two prolific conformable M-Fractional differential equations via three diverse approaches. Results Phys. 28, 104557 (2021)

    Google Scholar 

  • Silambarasan, R., Baskonus, H.M., Bulut, H.: Jacobi elliptic function solutions of the double dispersive equation in the Murnaghan’s rod. Eur. Phys. J. Plus. 134, 125 (2019)

    Google Scholar 

  • Srivastava, H.M., Baleanu, D., Machado, J.A.T., Osman, M.S., Rezazadeh, H., Arshed, S., Gunerhan, H.: Traveling wave solutions to nonlinear directional couplers by modified Kudryashov method. Phys. Scr. 95(7), 075217 (2020)

    ADS  Google Scholar 

  • Sulaiman, T.A.: Three-component coupled nonlinear Schrödinger equation: optical soliton and modulation instability analysis. Phys. Scr. 95(6), 065201 (2020)

    ADS  Google Scholar 

  • Sulaiman, T.A., Bulut, H., Baskonus, H.M.: Construction of various soliton solutions via the simplified extended sinh-Gordon equation expansion method. ITM Web Conf. 22, 01062 (2018)

    Google Scholar 

  • Sulaiman, T.A., Bulut, H., Baskonus, H.M.: On the exact solutions to some system of complex nonlinear models. AMNS 6(1), 29–42 (2021)

    MathSciNet  Google Scholar 

  • Sulaiman, T.A., Nuruddeen, R.I., Mikail, B.B.: Dark and singular solitons to the two nonlinear Schrödinger equations. Optik 186, 423–430 (2019)

    ADS  Google Scholar 

  • Sulaiman, T.A., Yusuf, A., Atangana, A.: New lump, lump-kink, breather waves and other interaction solutions to the (3+ 1)-dimensional soliton equation. Commun. Theor. Phys. 72(8), 085004 (2020)

    ADS  MathSciNet  MATH  Google Scholar 

  • Tso, T.: Exixtence of solutions of the modified Benjamin-Bona- Mahony-equaton. Chin. J. Math. 24, 327–336 (1996)

    MATH  Google Scholar 

  • Wang, B.: Strong attractors for the BBM equation. Appl. Math. Lett. 10(2), 23–28 (1997)

    MathSciNet  Google Scholar 

  • Weisstein, E.W.: Concise encyclopedia of mathematics. CRC Press, New York (2002)

    MATH  Google Scholar 

  • Yamamoto, M.: One unique continuation for a linearized Benjamin-Bona- Mahony equation. J. Inverse III-Probl. 11(5), 537–543 (2003)

    MathSciNet  MATH  Google Scholar 

  • Younas, U., Bilal, M., Ren, J.: Propagation of the pure-cubic optical solitons and stability analysis in the absence of chromatic dispersion. Opt. Quantam Electron 53, 490 (2021)

    Google Scholar 

  • Younas, U., Ren, J.: Investigation of exact soliton solutions in magneto-optic waveguides and its stability analysis. Results Phys. 21, 103816 (2021)

    Google Scholar 

  • Younis, M., Younas, U., Rehman, S.U., Bilal, M., Waheed, A.: Optical bright-dark and Gaussian soliton with third order dispersion. Optik 134, 233–238 (2017)

    ADS  Google Scholar 

  • Yusuf, A., Sulaiman, T.A., Inc, M., Bayram, M.: Breather wave, lump-periodic solutions and some other interaction phenomena to the Caudrey-Dodd-Gibbon equation. Eur. Phys. J. Plus 135(7), 1–8 (2020)

    Google Scholar 

  • Zafar, A., Raheel, M., Asif, M., Hosseini, K., Mirzazadeh, M., Akinyemi, L.: Some novel integration techniques to explore the conformable M-fractional Schrödinger-Hirota equation. J. Ocean Eng. Sci. (2021). https://doi.org/10.1016/j.joes.2021.09.007

    Article  Google Scholar 

  • Zhang, J.L., Wang, M.L., Wang, U.M., Fang, Z.D.: The improved F-expansion method and its applications. Phy. Lett. A 350, 103–9 (2006)

    ADS  MATH  Google Scholar 

  • Zhao, X., Wei, X.: Travelling wave solutions for a class of the generalized Benjamin-Bona-Mahony equation. Appl. Math. Comput. 192, 507–519 (2007)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to acknowledge the financial support provided for this research via the National Natural Science Foundation of China (11771407-52071298), Zhong Yuan Science and Technology Innovation Leadership Program (214200510010), and the MOST Innovation Method project (2019IM050400). They also thank the reviewers for their valuable reviews and kind suggestions.

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  1. Henan Academy of Big Data/School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Muhammad Bilal & Jingli Ren

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  1. Muhammad Bilal
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Bilal, M., Ren, J. Dynamics of exact solitary wave solutions to the conformable time-space fractional model with reliable analytical approaches. Opt Quant Electron 54, 40 (2022). https://doi.org/10.1007/s11082-021-03408-7

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