---

Journal of Applied Nonlinear Dynamics 14(1) (2025) 153--162 | DOI:10.5890/JAND.2025.03.010

Rose Maluleka, Khadijo Rashid Adem

Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Pretoria, South Africa

Download Full Text PDF

 

Abstract

In this paper we obtain novell solutions of the modified Kortweg-de Vries- Zakharov-Kuznetsov equation by employing the Lie group analysis and the $(G'/G)$-expansion method. The solutions to be obtained are solitary wave, perodic and rational solutions. Utilizing the Lie group analysis, the the modified Kortweg-de Vries- Zakharov-Kuznetsov equation is integrated. The Lie symmetry technique is distinct from the conventional integrability approaches, which also include Hirota's bilinear method and the multiple exp-function method, among others. The solutions capture the limiting behavior of problems that are far from their intial or boundary conditions. The conservation laws for the underlying equation are also derived by using the multiplier method. The precise solutions provided in this work are anticipated to act as a starting point for numerical simulations of the underlying equation. Furthermore we intend to construct further physical solutions of interest by employing the conservation laws reported here in this paper and these results will be reported elsewhere.

References

1. [1]	El-Tantawy, S.A. and Moslem, W.M. (2014), Nonlinear structures of the Korteweg-de Vries and modified Korteweg-de Vries equations in non-Maxwellian electron-positron-ion plasma: Solitons collision and rogue waves, Physics of Plasmas, 21(5), 052112.

2. [2]	Seadawy, A.R. (2014), Stability analysis for Zakharov–Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma, Computers $\&$ Mathematics with Applications, 67(1), 172-180.

3. [3]	Khater, A.H., Callebaut, D.K., and Seadawy, A.R. (2003), Nonlinear dispersive Kelvin–Helmholtz instabilities in magnetohydrodynamic flows, Physica Scripta, 67, 340-349.

4. [4]	Hirota, R. (2004), The Direct Method in Soliton Theory, Cambridge University Press, Cambridge.

5. [5]	Gu, C.H. (1990), Soliton Theory and Its Application, Zhejiang Science and Technology Press, Zhejiang.

6. [6]	Ma, W.X., Huang, T., and Zhang, Y. (2010), A multiple exp-function method for nonlinear differential equations and its applications, Physica Scripta, 82, 065003

7. [7]	Wang, M., Li, X., and Zhang, J. (2008), The $(G'/G)$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372, 417-423.

8. [8]	Wazwaz, M. (2005), The tanh and sine-cosine method for compact and noncompact solutions of nonlinear Klein Gordon equation, Applied Mathematics and Computation, 167(2), 1179-1195.

9. [9]	Wang, M. and Li, X. (2005), Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations, Physics Letters A, 343(1-3), 48-54.

10. [10]	He, J.H. and Wu, X.H. (2006), Exp-function method for nonlinear wave equations, Chaos, Solitons $\&$ Fractals, 30(3), 700-708.

11. [11]	Mabenga, C., Muatjetjeja, B., and Motsumi, T.G. (2022), Similarity reductions and conservation laws of an extended Bogoyavlenskii–Kadomtsev–Petviashvili equation, International Journal of Applied and Computational Mathematics, 8(1), 43.

12. [12]	Mabenga, C., Muatjetjeja, B., and Motsumi, T.G. (2023), Multiple soliton solutions and other solutions of a variable-coefficient Korteweg–de Vries equation, International Journal of Modern Physics B, 37(09), p.2350090.

13. [13]	Ahmad, H., Seadawy, A.R., and Khan, T.A. (2020), Numerical solution of Korteweg–de Vries-Burgers equation by the modified variational iteration algorithm-II arising in shallow water waves, Physica Scripta, 95(4), p.045210.

14. [14]	Seadawy, A.R., Iqbal, M., and Lu, D. (2020), Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg–de Vries equation arising in ion-acoustic wave in an unmagnetized collisional dusty plasma, Physica A: Statistical Mechanics and its Applications, 544, p.123560.

15. [15]	Seadawy, A.R., Rizvi, S.T.R., Ahmad, S., Younis, M., and Baleanu, D. (2021), Lump, lump-one stripe, multiwave and breather solutions for the Hunter–Saxton equation, Open Physics, 19(1), 1-10.

16. [16]	Tala-Tebue, E., Seadawy, A.R., Kamdoum-Tamo, P.H., and Lu, D. (2018), Dispersive optical soliton solutions of the higher-order nonlinear Schrodinger dynamical equation via two different methods and its applications, The European Physical Journal Plus, 133, 1-10.

17. [17]	Seadawy, A.R. and Cheemaa, N. (2020), Some new families of spiky solitary waves of one-dimensional higher-order K-dV equation with power law nonlinearity in plasma physics, Indian Journal of Physics, 94(1), 117-126.

18. [18]	Wazwaz, A.M. (2008), The extended tanh method for the Zakharov–Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms, Communications in Nonlinear Science and Numerical Simulation, 13(6), 1039-1047.

19. [19]	Islam, M.H., Khan, K., Akbar, M.A., and Salam, M.A. (2014), Exact traveling wave solutions of modified KdV–Zakharov–Kuznetsov equation and viscous Burgers equation, Springer Plus, 3, 1-9.

20. [20]	Sahoo, S., Garai, G., and Saha Ray, S. (2017), Lie symmetry analysis for similarity reduction and exact solutions of modified KdV–Zakharov–Kuznetsov equation, Nonlinear Dynamics, 87, 1995-2000.

21. [21]	Anco, S.C. and Bluman, G. (2002), Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications, European Journal of Applied Mathematics, 13(5), 545-566.

22. [22]	Anthonyrajah, M. and Mason, D.P. (2010), Conservation laws and invariant solutions in the Fanno model for turbulent compressible flow, Mathematical and Computational Applications, 15(4), 529-542.

23. [23]	Ibragimov, N.H. (1995), CRC Handbook of Lie Group Analysis of Differential Equations, 3, CRC press.