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New exact solutions of (3+1)-dimensional modified KdV-Zakharov-Kuznetsov equation by Sardar-subequation method

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Abstract

New solitary wave solutions are developed for various shapes of three-dimensional nonlinear partial differential equations using the Sardar-Subequation method. Some periodic solitary wave solutions, including bright, dark, dark-bright, and singular periodic soliton solutions, are obtained. These solutions have broad applications in a particular category of applied sciences and physics. Graphical representations of the obtained results have been created using Maple 18 software to illustrate the accuracy and simplicity of the proposed method. Two-dimensional (2-D), three-dimensional (3-D), and 2-D contour plots have been used to depict the physical behavior of the observed patterns by incorporating relevant parameters. To demonstrate the robustness, sustainability, capability, and effectiveness of the present method, we have applied it to solve the nonlinear (3+1)-dimensional modified KdV-Zakharov-Kuznetsov equation.

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References

  • Ahmad, J., Akram, S., Rehman, S.U., Turki, N.B., Shah, N.A.: Description of soliton and lump solutions to M-truncated stochastic Biswas-Arshed model in optical communication. Res. Phys. 51, 106719 (2023)

    Google Scholar 

  • Alam, M.N., Hafez, M.G., Akbar, M.A., Roshid, H.O.: Exact traveling wave solutions to the (3+1)-dimensional mKdV-ZK and the (2+1)-dimensional Burgers equations via \(exp(-\phi (\eta ))\)-expansion method. Alex. Eng. J. 54, 635–644 (2015)

    Google Scholar 

  • Ali, K.K., Abd El Salam, M.A., Mohamed, E.M.H., Samet, B., Kumar, S., Osman, M.S.: Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series. Advan. Differ. Equ. 2020(1), 494 (2020)

  • Alquran, M., Jaradat, I.: Identifying combination of Dark-Bright Binary-Soliton and Binary-Periodic waves for a new two-mode model derived from the (2+ 1)-dimensional Nizhnik-Novikov-Veselov equation. Mathematics 11(4), 861 (2023)

    Google Scholar 

  • Alquran, M., Jaradat, I., Yusuf, A., Sulaiman, T.A.: Heart-cusp and bell-shaped-cusp optical solitons for an extended two-mode version of the complex Hirota model: application in optics. Opt. Quantum Electron. 53, 26 (2021)

    Google Scholar 

  • Az-Zo’bi, E., Al-Maaitah, A.F., Mohammad A.T., Osman, M.S.: New generalised cubicquinticseptic NLSE and its optical solitons. Pramana. 96, 4:184 (2022)

  • Bulut, H., Sulaiman, T.A., Baskonus, H.M.: On the new soliton and optical wave structures to some nonlinear evolution equations. Eur. Phys. J. Plus 132, 459 (2017)

    Google Scholar 

  • Eslami, M.: Trial solution technique to chiral nonlinear Schr\(\ddot{\text{o}}\)dingers equation in(1+2)-dimensions. Nonlinear Dyn. 85(2), 813–816 (2016)

  • Fakhari, A., Donmairry, G., Ebrahimpour, M.: Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution. Phys. Lett. A. 368, 64–68 (2007)

    MathSciNet  CAS  ADS  Google Scholar 

  • Ghanbari, B.: Abundant soliton solutions for the Hirota-Maccari equation via the generalized exponential rational function method. Modern Phys. Lett. B 33(09), 1950106 (2019)

    MathSciNet  CAS  ADS  Google Scholar 

  • Ghanbari, B., Baleanu, D.: New optical solutions of the fractional Gerdjikov-Ivanov equation with conformable derivative. Front. Phys. 8, 167 (2020)

    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(7), 334 (2019)

    Google Scholar 

  • Ghanbari, B., Baleanu, D., Al Qurashi, M.: New exact solutions of the generalized Benjamin-Bona-Mahony equation. Symmetry 11(1), 20 (2018)

    ADS  Google Scholar 

  • Günerhan, H., Rezazadeh, H., Adel, W., et al.: Analytical approximate solution of fractional order smoking epidemic model. Adv. Mech. Eng. 14(9), 1–11 (2022)

  • Günerhan, H.: Optical soliton solutions of nonlinear Davey-Stewartson equation using an efficient method. Rev. Mex. Fís. 67(6) (2021)

  • Günerhan, H., Khodadad, F.S., Rezazadeh, H., Khater, M.M.A.: Exact optical solutions of the (2+ 1) dimensions Kundu-Mukherjee-Naskar model via the new extended direct algebraic method. Modern Phys. Lett. B 34(22), 2050225 (2020)

    MathSciNet  ADS  Google Scholar 

  • Hassan, M.M.: Exact Solitary wave solutions for a generalized KdV-Burgers equation. Chaos Solitons Fractals 19, 1201–1206 (2004)

    MathSciNet  ADS  Google Scholar 

  • Hirota, R.: Direct method of finding exact solutions of nonlinear evolution equations. In: Bullough, R., Caudrey, P. (eds.) Backlund Transformations. Springer, Berlin, Germany (1980)

    Google Scholar 

  • Hosseini, K., Ansari, R., Samadani, F., Zabihi, A., Shafarodi, A., Mirzazadeh, M.: High-order dispersive cubic-quintic Schr\(\ddot{\text{o}}\)dinger equation and its soliton solutions. Acta Physica Polonica A 136, 1 (2019)

    Google Scholar 

  • Hosseini, K., Ansari, R., Zabihi, A., Shafaroody, A., Mirzazadeh, M.: Optical solitons and modulation instability of the resonant nonlinear Schr\(\ddot{\text{o}}\)dinger equations in (3+1)-dimensions. Opt. Int. J. Light Electron. Opt. 209, 164584 (2020)

    Google Scholar 

  • Hu, J.Q.: An algebraic method exactly solving two high-dimensional nonlinear evolution equations. Chaos Solitons Fractals 23, 3918 (2005)

    MathSciNet  Google Scholar 

  • Islam, M.S., Khan, K., Arnous, A.H.: New Trends Math. Sci. 3, 3 (2015)

  • Iqbal, M.A., Miah, M.M., Rasid, M.M., Alshehri, H.M., Osman, M.S.: An investigation of two integro-differential KP hierarchy equations to find out closed form solitons in mathematical physics, Arab J Basic Appl Sci 30(1), 535–545 (2023)

  • 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 

  • Jaradat, I., Alquran, M.: A variety of physical structures to the generalized equal-width equation derived from Wazwaz-Benjamin-Bona-Mahony model. J. Ocean Eng. Sci. 7(3), 244–247 (2022)

    Google Scholar 

  • Jaradat, I., Alquran, M., Ali, M.: A numerical study on weak-dissipative two-mode perturbed Burgers’ and Ostrovsky models: right-left moving waves. Eur. Phys. J. Plus 133, 1–6 (2018)

    Google Scholar 

  • Khater, M.M.A., Seadawy, R.A., Lu, D.: Res. Phys. 7, 2325 (2017)

  • 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. Plus 136(4), 1–28 (2021)

    Google Scholar 

  • Khodadad, F.S., Nazari, F., Eslami, M., Rezazadeh, H.: Soliton solutions of the conformable fractional Zakharov, Kuznetsov equation with dual-power law nonlinearity. Opt. Quantum Electron. 49(11), 384 (2017)

    Google Scholar 

  • Kumar, S., Niwas, M., Osman, M.S., Abdou, M.A.: Abundant different types of exact soliton solution to the (4+1)- dimensional Fokas and (2+1)-dimensional breaking soliton equations. Commun Theor Phys 73(10), 105007 (2021)

  • Kohl, R.W., Biswas, A., Zhou, Q., Ekici, M., Alzahrani, A.K., Belic, M.R.: Optical soliton perturbation with polynominal and triple-power laws of refractive index by semi-inverse variational principle. Cahos Solitons Fractals 135, 109765 (2020)

    Google Scholar 

  • Li, P., Lu, Y., Xu, C., Ren, J.: Insight into Hopf bifurcation and control methods in fractional order BAM neural networks incorporating symmetric structure and delay. Cogn. Comput. 1-43 (2023)

  • Lu, B.Q., Pan, Z.L., Qu, B.Z., Jiang, X.F.: Solitary wave solutions for some system of coupled nonlinear equation. Phys. Lett. A. 180, 61–64 (1993)

    MathSciNet  ADS  Google Scholar 

  • Misirli, E., Gurefe, Y.: Exp-function method to solve the generalized Burgers-Fisher equation. Nonlinear Sci. Lett. A 13, 323–328 (2010)

    Google Scholar 

  • Miura, M.R.: Backlund Transformation. Springer-Verlag, Berlin (1978)

    Google Scholar 

  • Mua, D., Changjin, X., Liu, Z., Pang, Y.: Further insight into bifurcation and hybrid control tactics of a chlorine Dioxide-Iodine-Malonic acid chemical reaction model incorporating delays. MATCH Commun. Math. Comput. Chem. 89(3), 529–566 (2023)

    Google Scholar 

  • Omrani, K.: The convergence of fully discrete Galerkin approximation for the Benjamin-Bona-Mahony (BBM) equation. Appl. Math. Comput. 180, 614–621 (2006)

    MathSciNet  Google Scholar 

  • Osman, M.S., Inc, M., Liu, J.G., Hosseini, K., Yusuf, A.: Different wave structures and stability analysis for the generalized (2+1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili equation. Phys. Scr. 95, 035229 (2020)

    CAS  Google Scholar 

  • Ou, W., Xu, C., Cui, Q., Liu, Z., Pang, Y., Farman, M., Ahmad, S., Zeb, A.: Mathematical study on bifurcation dynamics and control mechanism of tri-neuron bidirectional associative memory neural networks including delay. Math. Methods Appl. Sci. (2023)

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

  • Rehman, S.U., Bilal, M., Ahmad, J.: The study of solitary wave solutions to the time conformable Schr\(\ddot{\text{o}}\)dinger system by a powerful computational technique. Opt. Quantum Electron. 54(4), 228 (2022)

    Google Scholar 

  • Rehman, H.Ur., Rehan, A., Abdul-Majid, W., Hashim, M.A., Osman, M.S.: Analysis of Brownian motion in stochastic Schr\(\ddot{\text{o}}\)dinger wave equation using Sardar sub-equation method. Optik 289, 171305 (2023)

  • 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 

  • Rogers, C., Shadwick, W.F.: Backlund Transformations. Academic Press, New York (1982)

    Google Scholar 

  • Saha Ray, S., Sahoo, S.: A novel analytical method with fractional complex transform for new exact solutions of time-fractional fifth-order Sawada-Kotera equation. Rep. Math. Phys. 75(1), 63–72 (2015)

    MathSciNet  Google Scholar 

  • Saifullah, S., Fatima, N., Abdelmohsen, S.A., Alanazi, M.M., Ahmad, S., Baleanu, D.: Analysis of a conformable generalized geophysical KdV equation with Coriolis effect. Alex. Eng. J. 73, 651–663 (2023)

    Google Scholar 

  • Sirendaoreji.: A new auxiliary equation and exact travelling wave solutions of nonlinear equations. Phys. Lett. A 356, 12430 (2006)

  • Tariq, K.U., Younis, M., Rezazadeh, H., Rizvi, S.T.R., Osman, M.S.: Optical solitons with quadratic-cubic nonlinearity and fractional temporal evolution. Mod. Phys. Lett. B 32(26), 1850317 (2018)

  • Tascan, F., Bekir, A., Koparan, M.: Travelling wave solutions of nonlinear evolution equations by using the first integral method. Commun. Nonlinear Sci. Numer. Simul. 14, 1810–1815 (2009)

    ADS  Google Scholar 

  • Wang, K., Wang, G.-D., Shi, F.: Diverse optical solitons to the Radhakrishnan-Kundu-Lakshmanan equation for the light pulses. J. Nonlinear Opt. Phys. Mater. (2023)

  • Wang, K.-J.: Dynamics of breather, multi-wave, interaction and other wave solutions to the new (3+ 1)-dimensional integrable fourth-order equation for shallow water waves. Int. J. Numer. Methods Heat Fluid Flow (2023a)

  • Wang, K.-J.: Resonant multiple wave, periodic wave and interaction solutions of the new extended (3+ 1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Nonlinear Dyn. 111(17), 16427–16439 (2023b)

    Google Scholar 

  • Wang, K.-J.: Diverse wave structures to the modified Benjamin-Bona-Mahony equation in the optical illusions field. Modern Phys. Lett.B 37(11), 2350012 (2023c)

    MathSciNet  CAS  ADS  Google Scholar 

  • Wang, K., Liu, J.: On abundant wave structures of the unsteady korteweg-de vries equation arising in shallow water. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.04.024

    Article  Google Scholar 

  • Wang, K., Si, J.: Diverse optical solitons to the complex Ginzburg-Landau equation with Kerr law nonlinearity in the nonlinear optical fiber. Eur. Phys. J. Plus 138, 187 (2023)

    ADS  Google Scholar 

  • Wang, K.-J., Peng, X., Shi, F.: Nonlinear dynamic behaviors of the fractional (3+ 1)-dimensional modified Zakharov-Kuznetsov equation. Fractals 31(07), 2350088 (2023)

    ADS  Google Scholar 

  • Wazwaz, A.M.: A sine-cosine method for handling nonlinear wave equations. Math. Comput. Model. 40, 499–508 (2004)

    MathSciNet  Google Scholar 

  • Xu, C., Mu, D., Liu, Z., Pang, Y., Aouiti, C., Tunc, O., Ahmad, S., Zeb, A.: Bifurcation dynamics and control mechanism of a fractional-order delayed Brusselator chemical reaction model. Match 89(1) (2023a)

  • Xu, C., Cui, X., Li, P., Yan, J., Yao, L.: Exploration on dynamics in a discrete predator-prey competitive model involving feedback controls. J. Biol. Dyn. 17(1), 2220349 (2023b)

    MathSciNet  PubMed  Google Scholar 

  • Xu, C., Cui, Q., Liu, Z., Pan, Y., Cui, X., Ou, W., Rahman, M., Farman, M., Ahmad, S., Zeb, A.: Extended hybrid controller design of bifurcation in a delayed chemostat model. MATCH Commun. Math. Comput. Chem. 90(3), 609–648 (2023c)

    Google Scholar 

  • Xu, C., Dan, M., Pan, Y., Aouiti, C., Yao, L.: Exploring bifurcation in a fractional-order predator-prey system with mixed delays. J. Appl. Anal. Comput. 13, 1119–1136 (2023d)

    MathSciNet  Google Scholar 

  • Yang, X.F., Deng, Z.C., Wei, Y.: A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application. Adv. Diff. Equ. 1, 11733 (2015)

    MathSciNet  Google Scholar 

  • Yomba, E.: The extended Fans sub-equation method and its applications o (2+1)- dimensional dispersive long wave and Whitham Broer Kaup equations. Chin. J. Phys. 43(4), 789805 (2005)

    MathSciNet  Google Scholar 

  • Zhang, S., Zong, Q.-A., Liu, D., Gao, Q.: A generalized exp-function method for fractional riccati differential equations. Commun. Fract. Calc. 1(1), 4851 (2010)

    Google Scholar 

  • Zheng, B.: \((\frac{G^{^{\prime }}}{G})\)-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Commun. Theor. Phys. 58, 623 (2012)

    MathSciNet  ADS  Google Scholar 

  • Zhu, W.-H., Pashrashid, A., Adel, W., Gunerhan, H., Nisar, K., Saleel, C.A., Inc, M., Rezazadeh, H.: Dynamical behaviour of the foam drainage equation. Res. Phys. 30, 104844 (2021)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Okara, Okara, Pakistan

    Sadia Yasin

  2. Department of Mathematics, University of Malakand, Chakdara, Dir Lower, Khyber Pakhtunkhwa, Pakistan

    Asif Khan & Shabir Ahmad

  3. Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

    M. S. Osman

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  1. Sadia Yasin
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  4. M. S. Osman
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Yasin, S., Khan, A., Ahmad, S. et al. New exact solutions of (3+1)-dimensional modified KdV-Zakharov-Kuznetsov equation by Sardar-subequation method. Opt Quant Electron 56, 90 (2024). https://doi.org/10.1007/s11082-023-05558-2

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