Wave Profile Investigation of the Higher Dimensional Nonlinear Evolution Equation through Nonlinear Auxiliary Equation

Hasibun Naher^1, and Farah Aini Abdullah²

¹Department of Mathematics and Natural Sciences, BRAC University, 66 Mohakhali, Dhaka 1212, Bangladesh

²School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

Cite this paper:
Hasibun Naher and Farah Aini Abdullah. Wave Profile Investigation of the Higher Dimensional Nonlinear Evolution Equation through Nonlinear Auxiliary Equation. American Journal of Applied Mathematics and Statistics. 2023; 11(1):1-10. doi: 10.12691/ajams-11-1-1

Abstract

In this article, more general and many new travelling wave solutions have been constructed through new extension of the (G′/G)-expansion method which is known as new generalized (G′/G)-expansion method. The key idea of this technique is to take full advantage of a higher ordinary nonlinear differential equation that has five different general solutions. The presentation of the travelling wave solutions is quite new and additional parameters are also used in the solution form. To illustrate the novelty and efficiency of this method, the (3+1)-dimensional Kadomstev-Petviashvili equation is desired to be investigated. The obtained solutions reveal the wider applicability to handle higher-dimensional nonlinear problems which arising in mathematical physics.

Keywords:
New generalized (G′/G)-expansion method nonlinear auxiliary equation (3+1)-dimensional Kadomstev-Petviashvili equation travelling wave solutions ordinary differential equations and analytical solutions

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	G. L. Lamb Jr, Bäcklund transformations for certain nonlinear evolution equations, J. Math. Phys. 15 (1974): 2157.
[2]	R. Hirota, Exact solution of the Korteweg-de-Vries equation for multiple collisions of solutions, Phys. Rev. Lett. 27 (1971): 1192-1194.
[3]	M. E. Ali, F. Bilkis, G. C. Paul, D. Kumar, H. Naher, Lump, lump-stripe, and breather wave solutions to the (2+ 1)-dimensional Sawada-Kotera equation in fluid mechanics. Heliyon, 7(9) (2021): e07966.
[4]	M. J. Ablowitz, P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering transform, Cambridge Univ. Press, Cambridge, 1991.
[5]	S. Liu, Z. Fu, S. Liu, Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A 289 (2001): 69-74.
[6]	W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (1992): 650-654.
[7]	J. Manafian, M. Lakestani, A. Bekir. Solving the Simplified MCH Equation and the Combined KdV-mKdV Equations via tan Φ(ξ)/2-Expansion Method. International Journal of Nonlinear Science, 22 (1) (2016): 25-36.
[8]	M. A. Abdou, The extended F-expansion method and its application for a class of nonlinear evolution equations, Chaos Solit. Fract. 31 (2007): 95-104.
[9]	J. H. He and X. H. Wu, Exp-function method for nonlinear wave equations, Chaos Solit. Fract. 30 (2006): 700-708.
[10]	H. Naher, F. A. Abdullah, M. A. Akbar, New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method, J. Appl. Math., Article ID 575387, 14 pp.
[11]	Al-Mdallal, Qasem M. "A new family of exact solutions to the unsteady Navier–Stokes equations using canonical transformation with complex coefficients." Applied mathematics and computation 196.1 (2008): 303-308.
[12]	Al-Mdallal, Qasem M., and Muhammad I. Syam. "Sine–Cosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation." Chaos, Solitons & Fractals 33.5 (2007): 1610-1617.
[13]	Ariel, P. Donald, Mohammed I. Syam, and Qasem M. Al-Mdallal. "The extended homotopy perturbation method for the boundary layer flow due to a stretching sheet with partial slip." International Journal of Computer Mathematics 90.9 (2013): 1990- 2002.
[14]	Al Khawaja, U. and Al-Mdallal, Q.M., 2018. Convergent power Series of and solutions to nonlinear differential equations. International Journal of Differential Equations, 2018.
[15]	H. M. Srivastava, D. Baleanu, J. A. T. Machado, M. S. Osman, H. Rezazadeh, S. Arshed, H. Günerhan, Traveling wave solutions to nonlinear directional couplers by modified Kudryashov method. Physica Scripta, 95(7) (2020): 075217.
[16]	Jafari, H., Soltani, R., Khalique, C. M., & Baleanu, D. (2013). Exact solutions of two nonlinear partial differential equations by using the first integral method. Boundary Value Problems, 2013(1), 1-9.
[17]	M. Wang, X. Li, J. Zhang, The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008): 417-423.
[18]	A. Bekir. Application of the (G′/G)-expansion method for nonlinear evolution equations. Physics Letters A, 372(19) (2008): 3400-3406.
[19]	I. Aslan. Exact and explicit solutions to some nonlinear evolution equations by utilizing the (G′/G)-expansion method. Applied Mathematics and Computation, 215(2) (2009): 857-863.
[20]	E. M. E. Zayed. Traveling wave solutions for higher dimensional nonlinear evolution equations using the (G′/G)-expansion method. J. Appl. Math. & Informatics, 28(1-2) (2010): 383-395.
[21]	H. Naher, F.A. Abdullah, M.A. Akbar. The (G′/G)-expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation. Math. Probl. Engr. 2011, 11.
[22]	E. Yasar, I. B. Giresunlu. Exact Traveling Wave Solutions and Conservation Laws of (2+1) Dimensional Konopelchenko-Dubrovsky System, International Journal of Nonlinear Science, 22 (2) (2016): 118-128.
[23]	J. Zhang, F. Jiang, X. Zhao. An improved (G′/G)-expansion method for solving nonlinear evolution equations, International Journal of Computer Mathematics, 87(8) (2010): 1716-1725.
[24]	Y. S. Hamad, M. Sayed, S. K. Elagan, E. R. El-Zahar, The improved (G′/G)-expansion method for solving (3+1)-dimensional potential-YTSF equation, J. Mod. Meth. Numerical Math., 2 (2011): 32-38.
[25]	H. Naher, Analytical Approach to Obtain Some New Traveling Wave Solutions of Coupled Systems of Nonlinear Equations. Advances in Mathematics and Computer Science, 2 (2019): 141-152.
[26]	H. Naher, F. A. Abdullah. The improved (G'/G)-expansion method to the (2+1)-dimensional breaking soliton equation. Journal of Computational Analysis & Applications, 16(2) ( 2014): 220-235.
[27]	H. Naher, F. A. Abdullah. The Improved (G′/G)-expansion method to the (3+1)-dimensional Kadomstev-Petviashvili equation. American Journal of Applied Mathematics and Statistics, 1 (4) (2013): 64-70.
[28]	H. Naher, F. A Abdullah, A. Rashid. Some New Solutions of the (3+1)-dimensional Jimbo-Miwa equation via the Improved (G'/G)-expansion method. Journal of Computational Analysis & Applications, 17(2) (2014): 287-296.
[29]	E. M. E. Zayed. New traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized (G′/G)-expansion method J. Phys. A: Math. Theor. 42 (2009): 195202-14.
[30]	E. M. E. Zayed. The (G′/G)-expansion method combined with the Riccati equation for finding exact solutions of nonlinear PDEs, J. Appl. Math. & Informatics 29 (1-2) (2011): 351-367.
[31]	M. A. Akbar, N. H. M. Ali, E. M. E. Zayed. A generalized and improved (G′/G)-expansion method for nonlinear evolution equations, Math. Prob. Eng., Article ID 459879, 22 pp.
[32]	H. Naher, F. A. Abdullah, M. A. Akbar, Generalized and Improved (G′/G)-Expansion Method for (3+ 1)-Dimensional Modified KdV-Zakharov-Kuznetsev Equation, PloS one, 8(5) (2013): e64618.
[33]	H. Naher, F. A. Abdullah, Further extension of the generalized and improved (G′/G)-expansion method for nonlinear evolution equation. Journal of the Association of Arab Universities for Basic and Applied Sciences, 19 (2016): 52-58.
[34]	A. T. Khan, H. Naher, Solitons and periodic solutions of the Fisher equation with nonlinear ordinary differential equation as auxiliary equation. American Journal of Applied Mathematics and Statistics, 6(6) (2018): 244-252.
[35]	H. Naher, F. A. Abdullah, New Approach of (G′/G)- expansion method and new approach of generalized (G′/G)-expansion method for nonlinear evolution equation AIP Advances, 3, 032116 (2013).
[36]	H. Naher, F. A. Abdullah, New Generalized (G′/G)-expansion Method to the Zhiber-Shabat Equation and Liouville Equations. In Journal of Physics: Conference Series, 890(1) (2017): 012018.
[37]	A. Bekir, F. Uygun. Exact travelling wave solutions of nonlinear evolution equations by using the (G′/G)-expansion method, Arab J. Math. Sci., 18 (2012): 73-85.