Lie symmetries, conservation laws, optimal system and power series solutions of (3+1)-dimensional fractional Zakharov-Kuznetsov equation

Document Type : Research Paper


1 School of Science, Wuhan University of Science and Technology, Wuhan 430081, Hubei, China

2 Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan 430081, Hubei, China



In this paper, the Lie symmetry analysis method is applied to the high dimensional fractional Zakharov-Kuznetsov equation. All Lie symmetries and the corresponding conserved vectors for the equation are obtained. The one-dimensional optimal system is utilized to reduce the aimed equation with Riemann-Liouville fractional derivative to a low-dimensional fractional partial differential equation with Erdelyi-Kober fractional derivative. Then the power series solution of the reduced equation is given. Moreover, some other low dimensional reduced fractional differential equations with Riemann-Liouville fractional derivatives are obtained and can be solved by different methods in the literatures herein.


[1] M.A. Abdou, On the quantum Zakharov-Kuznetsov equation, Int. J. Nonlinear Sci. 26 (2018), 89–96.
[2] R. Al-deiakeh, M. Alquran, M. Ali, A. Yusuf, and S. Momani, On group of Lie symmetry analysis, explicit series solutions and conservation laws for the time-fractional (2+1)-dimensional Zakharov-Kuznetsov (q, p, r) equation, J. Geom. Phys. 176 (2022), 104512.
[3] X.Y. Cheng and L.Z. Wang, Invariant analysis, exact solutions and conservation laws of (2+1)-dimensional time fractional Navier-Stokes equations, Proc. R. Soc. A 477 (2021), 20210220.
[4] K.S. Chou and G.X. Li, A Note on optimal systems for the heat equation, J. Math. Anal. Appl. 261 (2001), 741–751.
[5] V. Daftardar-Gejji and H. Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl. 301 (2005), 508–518.
[6] E.H. El Kinani and A. Ouhadan, Lie symmetry analysis of some time fractional partial differential equations, Int. J. Mod. Phys. Conf. Ser. 38 (2015), 1560075.
[7] Y.Q. Feng and J.C. Yu, Lie symmetry analysis of fractional ordinary differential equation with neutral delay, AIMS Math. 6 (2021), no. 4, 3592–3605.
[8] R.K. Gazizov and A.A. Kasatkin, Construction of exact solutions for fractional order differential equations by the invariant subspace method, Comput. Math. Appl. 66 (2013), 576–584.
[9] R.K. Gazizov, A.A. Kasatkin, and S.Y. Lukashchuk, Continuous transformation groups of fractional differential equations, Vestnik USATU 9 (2007), 125–135.
[10] R.K. Gazizov, A.A. Kasatkin, and S.Y. Lukashchuk, Symmetry properties of fractional diffusion equations, Phys. Scr. T136 (2009), 014016.
[11] S. Herr and S. Kinoshita, The Zakharov-Kuznetsov equation in high dimensions: small initial data of critical regularity, J. Evol. Equ. 21 (2021), 2105–2121.
[12] R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000.
[13] X.R. Hu, Y.Q. Li, and Y. Chen, A direct algorithm of one-dimensional optimal system for the group invariant solutions, J. Math. Phys. 56 (2015), 053504.
[14] N.H. Ibragimov, Lie Group Analysis of Differential Equations, Symmetries, Exact Solutions and Conservation Laws, Volume 1, Boca Raton, FL: CRC Press, 1993.
[15] N.H. Ibragimov, Lie Group Analysis of Differential Equations, Applications in Engineering and Physical Sciences, Volume 2, Boca Raton, FL: CRC Press, 1994.
[16] N.H. Ibragimov, Lie Group Analysis of Differential Equations, New Trends in Theoretical Developments and Computational Methods, Volume 3, Boca Raton, FL: CRC Press, 1995.
[17] N.H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl. 333 (2007), 311–328.
[18] N.H. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A-Math. Theor. 44 (2011), 432002.
[19] A.A. Imani, D.G. Davood, B.R. Houman, H. Latifizadeh, E. Hesameddini, and M. Rafiee, Solutions of the Zakharov-Kuznetsov, Helmholtz and one-dimensional nonhomogeneous parabolic partial differential equations using differential transformation method (DTM), Int. J. Contemp. Math. Sci. 4 (2009), 1779–1790.
[20] A. Jafarian, P. Ghaderi, A.K. Golmankhaneh, and D. Baleanu, Analytical approximate solutions of solutions of the Zakharov-Kuznetsov equations, Rom. Rep. Phys. 66 (2014), 296–306.
[21] A.A. Kilbas, H.M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, New York: Elsevier, 2006.
[22] Y. Liu and X. Wang, The construction of solutions to Zakharov-Kuznetsov equation with fractional power nonlinear terms, Adv. Differ. Equ. 2019 (2019), 134.
[23] J.G. Liu, X.J. Yang, L.L. Geng, and Y.R. Fan, Group analysis of the time fractional (3+1)-dimensional KdV-type equation, Fractals 29 (2021), 2150169.
[24] M.M. Meerschaert, H.P. Scheffler, and C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys. 211 (2006), 249–261.
[25] S. Momani and Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A 365 (2007), 345–350.
[26] S. Momani and Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solitons Fractals 31 (2007), 1248–1255.
[27] S. Monro and E.J. Parkes, The derivation of a modified Zakharov-Kuznetsov equation and the stability of its solutions, J. Plasma Phys. 62 (1999), 305–317.
[28] S. Monro and E.J. Parkes, Stability of solitary-wave solutions to a modified Zakharov-Kuznetsov equation, J. Plasma Phys. 64 (2000), 411–426.
[29] K. Mpungu and A.M. Nass, Symmetry analysis of time fractional convection-reaction-diffusion equation with a delay, Results Nonlinear Anal. 2 (2019), 113–124.
[30] P.J. Olver, Applications of Lie Groups to Differential Equations, Heidelberg, Springer, 1986.
[31] L.V. Ovsiannikov, Group Analysis of Differential Equations, New York, Academic Press, 1982.
[32] I. Podlubny, Fractional Differential Equations, San Diego: Academic Press, 1999.
[33] R.C. Ren and S.L. Zhang, Invariant analysis, conservation laws, and some exact solutions for (2+1)-dimension fractional long-wave dispersive system, Comp. Appl. Math. 39 (2020), 249.
[34] S. Saha Ray, Invariant analysis and conservation laws for the time fractional (2+1)-dimensional ZakharovKuznetsov modified equal width equation using Lie group analysis, Comput. Math. Appl. 76 (2018), 2110–2118.
[35] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Yverdon: Gordon and Breach Science Publishers, 1993.
[36] M.V. Tanwar, Optimal system, symmetry reductions and group-invariant solutions of (2+1)-dimensional ZK-BBM equation, Phys. Scr. 96 (2021), 065215.
[37] A.M. Wazwaz, Exact solutions with solitons and periodic structures for the Zakharov-Kuznetsov (ZK) equation and its modified form, Commun. Nonlinear Sci. Numer. Simul. 10 (2005), 597–606.
[38] H.W. Yang, M. Guo, and H.L. He, Conservation laws of space-time fractional mZK equation for Rossby solitary waves with complete Coriolis force, Int. J. Nonlin. Sci. Num. 20 (2019), 17–32.
[39] M. Yourdkhany and M. Nadjafikhah, Symmetries, similarity invariant solution, conservation laws and exact solutions of the time-fractional Harmonic Oscillator equation, J. Geom. Phys. 153 (2020), 103661.
[40] J.C. Yu, Lie symmetry analysis of time fractional Burgers equation, Korteweg-de Vries equation and generalized reaction-diffusion equation with delays, Int. J. Geom. Methods M. 19 (2022), 2250219.
[41] J.C. Yu and Y.Q. Feng, Lie symmetry analysis and exact solutions of space-time fractional cubic Schrodinger equation, Int. J. Geom. Methods M. 19 (2022), 2250077.
[42] J.C. Yu and Y.Q. Feng, Lie symmetry, exact solutions and conservation laws of some fractional partial differential equations, J. Appl. Anal. Comput. 13 (2023), 1872-1889.
[43] J.C. Yu, Y.Q. Feng, and X.J. Wang, Lie symmetry analysis and exact solutions of time fractional Black-Scholes equation, Int. J. Financ. Eng. 9 (2022), 2250023.
[44] V.E. Zakharov and E.A. Kuznetsov, On three-dimensional solitons, Sov. Phys. 39 (1974), 285–288.
[45] Z.Y. Zhang, Symmetry determination and nonlinearization of a nonlinear time-fractional partial differential equation, Proc. R. Soc. A 476 (2020), 20190564.
[46] Z.Y. Zhang and G.F. Li, Lie symmetry analysis and exact solutions of the time-fractional biological population model, Phys. A 540 (2020), 123134.
[47] S. Zhang and H.Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A 375 (2011), 1069–1073.
[48] H.M. Zhu, Z.Y. Zhang, and J. Zheng, The time-fractional (2+1)-dimensional Hirota-Satsuma-Ito equations: Lie symmetries, power series solutions and conservation laws, Commun. Nonlinear Sci. 115 (2022), 106724.

Articles in Press, Corrected Proof
Available Online from 14 March 2024
  • Receive Date: 10 December 2023
  • Revise Date: 13 January 2024
  • Accept Date: 15 January 2024