International Journal of Mathematics Trends and Technology
Volume 71 | Issue 3 | Year 2025 | Article Id. IJMTT-V71I3P101 | DOI : https://doi.org/10.14445/22315373/IJMTT-V71I3P101
| Received | Revised | Accepted | Published |
|---|---|---|---|
| 05 Jan 2025 | 14 Feb 2025 | 02 Mar 2025 | 15 Mar 2025 |
This article discusses the (2+1)-dimensional case of the Zakharov-Kuznetsov (ZK) equation. The (2+1)-dimensional ZK equation is primarily used to describe wave propagation phenomena in multi-dimensional media. In plasma, liquids, or gases, waves may be influenced by multiple elements. Due to nonlinear effects, the propagation speed, shape and interactions of these waves become complex. We have obtained a variety of exact solutions of the (2+1) dimensional ZK equation by using two effective methods: the improved extended tanh function method and the modified Kudryashov method. The forms of the solutions include exponential solutions, logarithmic solutions, hyperbolic solutions and trigonometric solutions. In addition, by selecting appropriate parameter values,we have plotted three-dimensional and two-dimensional images to illustrate the physical behavior of the exact solutions.
(2+1)-dimensional ZK equation. Wave solution. Modified extended tanh-function method. Modified generalized Kudryashov method.
[1] Jarmo Hietarinta, “A Search for Bilinear Equations Passing Hirota’s Three-Soliton Condition. I. KdV-Type Bilinear Equations,”
Journal of Mathematical Physics, vol. 28, pp. 1732-1742, 1987.
[CrossRef] [Google Scholar] [Publisher Link]
[2] Katuro Sawada, and Takeyasu Kotera, “A Method for Finding N-soliton Solutions of the K.d.V. Equation and K.d.V.-like
Equation,” Progress of Theoretical Physics, vol. 51, no. 5, pp. 1355-1367, 1974.
[CrossRef] [Google Scholar] [Publisher Link]
[3] Abdul-Majid Wazwaz, “The Extended Tanh Method for New Solitons Solutions for Many Forms of the Fifth-order KdV
Equations,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 1002-1014, 2007.
[CrossRef] [Google Scholar] [Publisher Link]
[4] Serena Federico, “Carleman Estimates for Third Order Operators of KdV and Non KdV-type and Applications,” Annali Di
Mathematica Pura Ed Applicata, vol. 203, pp. 2801-2823, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[5] Talat Korpinar, Fatih Sevgin, and Zeliha Korpinar, “Optical Wave Propagation Phase for mKdV Spherical Electric Flux Density in
Sphere Space,” Optical and Quantum Electronics, vol. 56, pp. 1-13, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[6] Renata O. Figueira, and Mahendra Panthee, “New Lower Bounds for the Radius of Analyticity for the mKdV Equation and A
System of mKdV-type Equations,” Journal of Evolution Equations, vol. 24, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[7] Ghazala Akram et al., “Exact Travelling Wave Solutions for Generalized (3+1) Dimensional KP and Modified KP Equations,”
Optical and Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[8] Santanu Raut et al., “Characteristic of Integrability of Nonautonomous KP-Modified KP Equation and Its Qualitative Studies:
Soliton, Shock, Periodic Waves, Breather, Positons and Soliton Interactions,” Nonlinear Dynamics, vol. 112, pp. 9323-9354, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[9] Mostafa M.A. Khater, “Advanced Computational Techniques for Solving the Modified KdV-KP Equation and Modeling Nonlinear
Waves,” Optical and Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[10] Thomas Brooke Benjamin, J.L. Bona, and J.J. Mahony, “Model Equations for Long Waves in Nonlinear Dispersive Systems,”
Philosophical Transactions of the Royal Society a Mathematical, Physical and Engineering Sciences, vol. 272 no. 1220, pp. 47-78,
1972.
[CrossRef] [Google Scholar] [Publisher Link]
[11] Jundong Wang, Lijun Zhang, and Jibin Li, “New Solitary Wave Solutions of a Generalized BBM Equation with Distributed
Delays,” Nonlinear Dynamics, vol. 111, pp. 4631-4643, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[12] Minzhi Wei, and Liping He, “Existence of Periodic Wave of a BBM Equation with Delayed Convection and Weak Diffusion,”
Nonlinear Dynamics, vol. 111, pp. 17413-17425, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[13] Hui Mao, Yu Qian, and Yuanyuan Miao, “Solving the Modified Camassa-Holm Equation Via the Inverse Scattering Transform,”
Theoretical and Mathematical Physics, vol. 216, pp. 1189-1208, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[14] Mohamed R. Ali, Mahmoud A. Khattab, and S.M. Mabrouk, “Travelling Wave Solution for the Landau-Ginburg-Higgs Model Via
the Inverse Scattering Transformation Method,” Nonlinear Dynamics, vol. 111, pp. 7687-7697, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[15] Marwan Alquran, and Rahaf Alhami, “Analysis of Lumps, Single-stripe, Breather-wave, and Two-wave Solutions to the
Generalized Perturbed-KdV Equation by Means of Hirota’s Bilinear Method,” Nonlinear Dynamics, vol. 109, pp. 1985-1992, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[16] Muhammad Shakeel, Xinge Liu, and Abdullah Al-Yaari, “Interaction of Lump, Periodic, Bright and Kink Soliton Solutions of the
(1+1)-Dimensional Boussinesq Equation using Hirota-bilinear Approach,” Journal of Nonlinear Mathematical Physics, vol. 31, pp.
1-18, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[17] Rahaf Alhami, and Marwan Alquran, “Extracted Different Types of Optical Lumps and Breathers to the New Generalized
Stochastic Potential-KdV Equation Via using the Cole-Hopf Transformation and Hirota Bilinear Method,” Optical and Quantum
Electronics, vol. 54, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[18] Asif Yokus, and Muhammad Abubakar Isah, “Stability Analysis and Solutions of (2+1)-Kadomtsev-Petviashvili Equation by
Homoclinic Technique Based on Hirota Bilinear Form,” Nonlinear Dynamics, vol. 109, pp. 3029-3040, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[19] Sevil Culha Unal, “Exact Solutions of the Landau-Ginzburg-Higgs Equation Utilizing the Jacobi Elliptic Functions,” Optical and
Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[20] Aamir Farooq, Muhammad Ishfaq Khan, and Wen Xiu Ma, “Exact Solutions for the Improved mKdv Equation with Conformable
Derivative by Using the Jacobi Elliptic Function Expansion Method,” Optical and Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[21] A.A. Elsadany, and Mohammed K. Elboree, “Construction of Shock, Periodic and Solitary Wave Solutions for Fractional-time
Gardner Equation by Jacobi Elliptic Function Method,” Optical and Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[22] Aamir Farooq, Wen Xiu Ma, and Muhammad Ishfaq Khan, “Exploring Exact Solitary Wave Solutions of Kuralay-II Equation based
on the Truncated M-fractional Derivative using the Jacobi Elliptic Function Expansion Method,” Optical and Quantum Electronics,
vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[23] A.K.M. Kazi Sazzad Hossain, Halida Akter, and M. Ali Akbar, “Soliton Solutions of DSW and Burgers Equations by Generalized (G/G')-Expansion Method,” Optical and Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[24] Fiza Batool et al., “Exploring Soliton Solutions of Stochastic Phi-4 Equation through Extended Sinh-Gordon Expansion Method,”
Optical and Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[25] Abdulla-Al-Mamun et al., “Dynamical Behavior of Water Wave Phenomena for the 3D Fractional WBBM Equations using Rational
Sine-Gordon Expansion Method,” Scientific Reports, pp. 1-19, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[26] Muslum Ozisik, Aydin Secer, and Mustafa Bayram, “On Solitary Wave Solutions for the Extended Nonlinear Schrodinger Equation
via the Modified F-expansion Method,” Optical and Quantum Electronics, vol. 55, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[27] Yosef Jazaa et al., “On the Exploration of Solitary Wave Structures to the Nonlinear Landau-Ginsberg-Higgs Equation under
Improved F-expansion Method,” Optical and Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[28] Saleh M. Hassan, and Abdulmalik A. Altwaty, “Solitons and Other Solutions to the Extended Gerdjikov-Ivanov Equation in
DWDM System by the
exp(
−
( ϕ(ϛ ))) [Google Scholar] [Publisher Link]
)))-Expansion Method,” Ricerche Di Matematica, vol. 73, pp. 2397-2410, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[29] Juan Yang, and Qingjiang Feng, “Using the Improved
exp(
−
(ϕ( ϛ )))
Expansion Method to find the Soliton Solutions of the
Nonlinear Evolution Equation,” The European Physical Journal Plus, vol. 136, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[30] A.A. Elsadany, Fahad Sameer Alshammari, and Mohammed K. Elboree, “Dynamical System Approach and
exp(
−
( ϕ ( ϛ )))
Expansion Method for Optical Solitons in the Complex Nonlinear Fokas-Lenells Model of Optical Fiber,” Optical and Quantum
Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[31] Waleed Hamali et al., “Optical Solitons of M-Fractional Nonlinear Schrodinger’s Complex Hyperbolic Model by Generalized
Kudryashov Method,” Optical and Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[32] Saima Arshed et al., “Solutions of (3+1)-Dimensional Extended Quantum Nonlinear Zakharov-Kuznetsov Equation using the
Generalized Kudryashov Method and The Modified Khater Method,” Optical and Quantum Electronics, vol. 55, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[33] Xiaoxiao Zheng, Lingling Zhao, and Yuanqing Xu, “New Type Solutions to the (2+1)-Dimensional Extended
Bogoyavlenskii-Kadomtsev-Petviashvili Equation Calculated via Generalized Kudryashov Technique,” Nonlinear Dynamics, vol.
112, pp. 1339-1348, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[34] Hasan Cakicioglu et al., “Optical Soliton Solutions of Schrödinger-Hirota Equation with Parabolic Law Nonlinearity via
Generalized Kudryashov Algorithm,” Optical and Quantum Electronics, vol. 55, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[35] Shafqat-Ur-Rehman, Muhammad Bilal, and Jamshad Ahmad, “New Exact Solitary Wave Solutions for the 3D-FWBBM Model in
Arising Shallow Water Waves by Two Analytical Methods,” Results in Physics, vol. 25, pp. 1-15, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[36] Sidheswar Behera, “Multiple Soliton Solutions of Some Conformable Fractional Nonlinear Models using Sine-Cosine Method,”
Optical and Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[37] Volkan Ala, and Gaukhar Shaikhova, “Analytical Solutions of Nonlinear Beta Fractional Schrödinger Equation via Sine-Cosine
Method,” Lobachevskii Journal of Mathematics, vol. 43, pp. 3033-3038, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[38] Hong-Zhun Liu, “New Analytical Solutions to a Medium with Completing Weakly Nonlocal Nonlinearity and Parabolic Law
Nonlinearity by Modified First Integral Method,” Optical and Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[39] Hasan Alzubaidi, “Exact Solutions for Travelling Waves using Tanh Method for Two Dimensional Stochastic Allen-Cahn Equation
with Multiplicative Noise,” Journal of Umm Al-Qura University for Applied Science, vol. 11, pp. 153-158, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[40] Eman H. M. Abdullah et al., “Effect of Higher Order on Constructing the Soliton Waves to Generalized Nonlinear Schrodinger
Equation using Improved Modified Extended Tanh Function Method,” Journal of Optics, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[41] Mostafa Eslami et al., “Solving the Relativistic Toda Lattice Equation via the Generalized Exponential Rational Function Method,”
Optical and Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[42] Rashid Ali et al., “The Analytical Study of Soliton Dynamics in Fractional Coupled Higgs System using the Generalized Khater
Method,” Optical and Quantum Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[43] C. Mabenga, B. Muatjetjeja, and T.G. Motsumi, “Bright, Dark, Periodic Soliton Solutions and Other Analytical Solutions of a
Time-dependent Coefficient (2+1)-Dimensional Zakharov-Kuznetsov Equation,” Optical and Quantum Electronics, vol. 55, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[44] Emad H.M. Zahran et al., “New Diverse Exact Optical Solutions of the Three Dimensional Zakharov-Kuznetsov Equation,” Optical
and Quantum Electronics, vol. 55, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[45] Mahmoud Soliman et al., “Dispersive Perturbations of Solitons for Conformable Fractional Complex Ginzburg-Landau Equation
with Polynomial Law of Nonlinearity using Improved Modified Extended Tanh-function Method,” Optical and Quantum
Electronics, vol. 56, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[46] Emad H.M. Zahran, and Mostafa M.A. Khater, “Modified Extended Tanh-function Method and Its Applications to the
Bogoyavlenskii Equation,” Applied Mathematical Modelling, vol. 40, no. 3, pp. 1769-1775, 2016.
[CrossRef] [Google Scholar] [Publisher Link]
[47] Lohani Md. Badrul Alam, Xingfang Jiang, and Abdulla-Al-Mamun, “Exact and Explicit Traveling Wave Solution to the
Time-fractional Phi-four and (2+1) Dimensional CBS Equations using the Modified Extended Tanh-function Method in
Mathematical Physics,” Partial Differential Equations in Applied Mathematics, vol. 4, pp. 1-11, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[48] Ulviye Demirbilek et al., “Generalized Extended (2+1)-Dimensional Kadomtsev-Petviashvili Equation in Fluid Dynamics:
Analytical Solutions, Sensitivity and Stability Analysis,” Nonlinear Dynamics, vol. 112, pp. 13393-13408, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
Wuming Li, Haoying Zuo, "Two Effective Methods for Solution of the (2+1)-Dimensional Zakharov-Kuznetsov Equation," International Journal of Mathematics Trends and Technology (IJMTT), vol. 71, no. 3, pp. 1-15, 2025. Crossref, https://doi.org/10.14445/22315373/IJMTT-V71I3P101
PDF 
Abstract 
Keywords 
References 
Citation