Abstract
This work studies soliton solutions of time-fractional coupled Konopelchenko–Dubrovsky (CKDE) and (3 + 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznestsov (mKdVZKE) equations. These models are used to define the physical phenomena of ocean dynamics, plasma physics, and soliton theory. The unified method is used to solve these fractional models analytically. To deal with the time fractional part, conformable and local M derivatives are used. A fractional wave transformation is used to transform a fractional partial differential equation to an ordinary differential equation. Using the proposed scheme, soliton solutions are obtained in polynomial and rational forms. The behavior of a soliton solution is also analyzed at different fractional parameters. The results show that the proposed scheme is simple and easy to apply to all types of time-fractional nonlinear homogenous evolution equations encountered in various fields of science.
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The data sets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
References
Alam, M.N., Li, X.: Exact traveling wave solutions to higher order nonlinear equations. J. Ocean Eng. Sci. 4(3), 276–288 (2019)
Alfalqi, S.H., Alzaidi, J.F., Lu, D., Khater, M.: On exact and approximate solutions of (2 + 1)-dimensional Konopelchenko–Dubrovsky equation via modified simplest equation and cubic B-spline schemes. Therm. Sci. 23(Suppl. 6), 1889–1899 (2019)
Ali, K.K., Mehanna, M.S.: Traveling wave solutions and numerical solutions of Gilson–Pickering equation. Results Phys. 28, 104596 (2021)
Baleanu, D., Kilic, B., Ugurlu, Y., Inc, M.: The first integral method for the (3 + 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov and Hirota equations. Rom. J. Phys. 60, 111–125 (2015)
Bashar, M.H., Islam, S.R., Kumar, D.: Construction of traveling wave solutions of the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation. Partial Differ. Equ. Appl. Math. 4, 100040 (2021)
Cenesiz, Y., Tasbozan, O., Kurt, A.: Functional variable method for conformable fractional modified KdV–ZK equation and Maccari system. Tbilisi Math. J. 10(1), 117–125 (2017)
Gepreel, K.A., Omran, S., Elagan, S.K.: The traveling wave solutions for some nonlinear PDEs in mathematical physics. Appl. Math. 2(3), 343 (2011)
Islam, M.H., Khan, K., Akbar, M.A., Salam, M.A.: Exact traveling wave solutions of modified KdV–Zakharov–Kuznetsov equation and viscous Burgers equation. Springer Plus 3(1), 105 (2014)
Kajouni, A., Chafiki, A., Hilal, K., Oukessou, M.: A new conformable fractional derivative and applications. Int. J. Differ. Equ. 2021, 6245435 (2021)
Khan, H., Shah, R., Gomez-Aguilar, J.F., Baleanu, D., Kumam, P.: Travelling waves solution for fractional-order biological population model. Math. Model. Nat. Phenom. 16, 32 (2021)
Kumar, S., Hama, A., Biswas, A.: Solutions of Konopelchenko–Dubrovsky equation by traveling wave hypothesis and Lie symmetry approach. Appl. Math. Inf. Sci. 8(4), 1533 (2014)
Rafiq, M.N., Majeed, A., Yao, S.W., Kamran, M., Rafiq, M.H., Inc, M.: Analytical solutions of nonlinear time fractional evaluation equations via unified method with different derivatives and their comparison. Results Phys. 26, 104357 (2021)
Raza, N., Rafiq, M.H., Kaplan, M., Kumar, S., Chu, Y.M.: The unified method for abundant soliton solutions of local time fractional nonlinear evolution equations. Results Phys. 22, 103979 (2021)
Sousa, J.V.D.C., de Oliveira, E.C.: On the local \(M\)-derivative (2017). arXiv preprint arXiv:1704.08186
Taghizadeh, N., Mirzazadeh, M.: Exact travelling wave solutions for Konopelchenko–Dubrovsky equation by the first integral method. Appl. Appl. Math. Int. J. (AAM) 6(1), 153–161 (2011)
Taqi, A. H., Shallal, M. A., Jomaa, B. F., and Ali, K. K. (2019). Travelling wave solution for some partial differential equations. In: AIP Conference Proceedings. 2096, 020015. Chaos Solit Fractals 28(2), 448–453 (2006)
Tasbozan, O., Cenesiz, Y., Kurt, A., Baleanu, D.: New analytical solutions for conformable fractional PDEs arising in mathematical physics by exp-function method. Open Phys. 15(1), 647–651 (2017)
Wang, K.J.: Abundant analytical solutions to the new coupled Konno–Oono equation arising in magnetic field. Results Phys. 31, 104931 (2021)
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 (2022a)
Wang, K.: Exact traveling wave solution for the fractal Riemann wave model arising in ocean science. Fractals 30, 2250143 (2022b)
Wang, K.: Exact travelling wave solution for the local fractional Camassa–Holm–Kadomtsev–Petviashvili equation. Alexand. Eng. J. 63, 371–376 (2023)
Wang, K.J., Si, J.: Investigation into the explicit solutions of the integrable (2+ 1)-dimensional maccari system via the variational approach. Axioms 11(5), 234 (2022)
Wang, K.J., Wang, G.D., Shi, F.: Abundant exact traveling wave solutions to the local fractional (3 + 1)-dimensional Boiti–Leon equation. Fractals 30(3), 2250064–13893 (2022a)
Wang, X., Javed, S.A., Majeed, A., Kamran, M., Abbas, M.: Investigation of exact solutions of nonlinear evolution equations using unified method. Mathematics 10(16), 2996 (2022b)
Yaslan, H.C., Girgin, A.: Exp-function method for the conformable space-time fractional STO, ZKBBM and coupled Boussinesq equations. Arab J. Basic Appl. Sci. 26(1), 163–170 (2019)
Zayed, E.M., Gepreel, K.A.: New applications of an improved (G’/G)-expansion method to construct the exact solutions of nonlinear PDEs. Int. J. Nonlinear Sci. Numer. Simul. 11(4), 273–284 (2010)
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The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) for funding and supporting this work through Research Partnership Program no RP-21-09-07.
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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by [AA], [AM] and [MI]. The first draft of the manuscript was written by [AA], [AM], [MK], [RTA] and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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Aslam, A., Majeed, A., Kamran, M. et al. Dynamical behavior of the fractional coupled Konopelchenko–Dubrovsky and (3 + 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznestsov equations. Opt Quant Electron 55, 543 (2023). https://doi.org/10.1007/s11082-023-04704-0
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DOI: https://doi.org/10.1007/s11082-023-04704-0