In this paper, we present a new two equations. The first equation is the (4 + 1)-dimensional Generalized Nonlinear Boussinesq Equation (G-NBE), and the second is the (4+1)-dimensional Generalized Camassa–Holm Kadomtsev–Petviashvili Equation (G-CH-KPE). We use a new exp(φ(ξ))-expansion method for solve our new equations. We determine a variety of exact solutions for each equation and expressed in terms of hyperbolic functions, trigonometric functions, exponential functions and rational functions.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 11, Issue 2) |
| DOI | 10.11648/j.ijtam.20251102.11 |
| Page(s) | 26-33 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2025. Published by Science Publishing Group |
exp(φ(ξ))-expansion method, Generalized Nonlinear Boussinesq Equation, Generalized Camassa–Holm Kadomtsev–Petviashvili Equation, Soliton Solution, Traveling Wave Solutions
| [1] | Wang, Mingliang, Yubin Zhou, and Zhibin Li. “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics.” Physics Letters A 216. 1-5(1996): 67-75. |
| [2] | Nakamura, Akira. “Surface impurity localized diode vibration of the Toda lattice: Perturbation theory based on Hirota’s bilinear transformation method.” Progress of Theoretical Physics 61. 2(1979): 427-442. |
| [3] | Mbusi, S. O., A. R. Adem, and B. Muatjetjeja. “Lie symmetry analysis, multiple exp-function method and conservation laws for the (2+1)-dimensional Boussinesq equation.” Optical and Quantum Electronics 56. 4(2024): 670. |
| [4] | Mawa, H. Z., et al. “Soliton Solutions to the BA Model and (3+1)-Dimensional KP Equation Using Advanced exp (−Φ(ξ))-Expansion Scheme in Mathematical Physics.” Mathematical Problems in Engineering 2023. 1(2023): 5564509. |
| [5] | Akram, Ghazala, Saima Arshed, and Maasoomah Sadaf. “Soliton solutions of generalized time-fractional Boussinesq-like equation via three techniques.” Chaos, Solitons & Fractals 173(2023): 113653. |
| [6] | Ji, Zhe, et al. “Rational solutions of an extended (2+1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili equation in liquid drop.” AIMS Mathematics 8. 2(2023): 3163-3184. |
| [7] | Batool, Fiza, et al. “Exploring soliton solutions of stochastic Phi-4 equation through extended Sinh-Gordon expansion method.” Optical and Quantum Electronics 56. 5(2024): 785. |
| [8] | Ma, Yu-Lan, Abdul-Majid Wazwaz, and Bang-Qing Li. “A new (3+1)-dimensional Kadomtsev–Petviashvili equation and its integrability, multiple-solitons, breathers and lump waves.” Mathematics and Computers in Simulation 187(2021): 505-519. |
| [9] | Alam, Md Nur, Md Ali Akbar, and Syed Tauseef Mohyud-Din. “A novel (G'/G)-expansion method and its application to the Boussinesq equation.” Chinese Physics B 23. 2(2013): 020203. |
| [10] | Al-Amry, M. S., and E. F. Abdullah. “Exact solutions for a new models of nonlinear partial differential equations Using(G'/G2)-Expansion Method.” University of Aden Journal of Natural and Applied Sciences 23. 1(2019): 189-199. |
| [11] | Maher, Ahmed, H. M. El-Hawary, and Mohammed S. Al-Amry. “New exact solutions for new model nonlinear partial differential equation.” Journal of Applied Mathematics 2013. 1(2013): 767380. |
| [12] | Gonzalez-Gaxiola, O., et al. “Highly dispersive optical solitons with non-local law of refractive index by Laplace-Adomian decomposition.” Optical and Quantum Electronics 53(2021): 1-12. |
| [13] | Azizi, Naser, and Reza Pourgholi. “Applications of Sine–Cosine wavelets method for solving the generalized Hirota–Satsuma coupled KdV equation.” Mathematical Sciences 17. 4(2023): 503-516. |
| [14] | Al-Amry, M. S., and E. F. Al-Abdali. ”New Exact Solutions for Generalized of Combined with Negative Calogero-Bogoyavlenskii Schiff and Generalized Yu–Toda–Sassa–Fukuyama Equations.” University of Aden Journal of Natural and Applied Sciences 28. 1(2024): 25-30. |
| [15] | Jisha, C. R., and Ritesh Kumar Dubey. “Wave interactions and structures of (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation.” Nonlinear Dynamics 110. 4(2022): 3685-3697. |
| [16] | Demiray, Şeyma Tülüce, and Sevgi Kastal. “The Modified Exp (−ϑ(σ))-Expansion Function Method for Exact Solutions of the Simplified MCH Equation and the Getmanou Equation.” Adıyaman University Journal of Science 11. 2(2021): 276-289. |
| [17] | Boussinesq, Joseph. “Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond.” Journal de mathématiques pures et appliquées 17(1872): 55-108. |
| [18] | Lu, Xing, et al. “Lump dynamics of a generalized two-dimensional Boussinesq equation in shallow water.” Nonlinear Dynamics 91(2018): 1249-1259. |
| [19] | Rashid, Saima, et al. “A semi-analytical approach for fractional order Boussinesq equation in a gradient unconfined aquifers.” Mathematical Methods in the Applied Sciences 45. 2(2022): 1033-1062. |
| [20] | Akbar, M. Ali, Norhashidah Hj Mohd Ali, and Tasnim Tanjim. “Adequate soliton solutions to the perturbed Boussinesq equation and the KdV-Caudrey- Dodd-Gibbon equation.” Journal of King Saud University-Science 32. 6(2020): 2777-2785. |
| [21] | Ma, Yu-Lan. “N-solitons, breathers and rogue waves for a generalized Boussinesq equation.” International Journal of Computer Mathematics 97. 8(2020): 1648- 1661. |
| [22] | Gai, Xiao-Ling, et al. “Soliton interactions for the generalized (3+1)-dimensional Boussinesq equation.” International Journal of Modern Physics B 26. 7(2012): 1250062. |
| [23] | Hussain, Amjad, et al. “Lie analysis, conserved vectors, nonlinear self-adjoint classification and exact solutions of generalized (N+1)-dimensional nonlinear Boussinesq equation.” AIMS Math 7. 7(2022): 13139â. |
| [24] | De Monvel, Anne Boutet, et al. “Long-time asymptotics for the Camassa–Holm equation.” SIAM journal on mathematical analysis 41. 4(2009): 1559-1588. |
| [25] | Camassa, Roberto, and Darryl D. Holm. “An integrable shallow water equation with peaked solitons.” Physical review letters 71. 11(1993): 1661. |
| [26] | Xie, Shaolong, Lin Wang, and Yuzhong Zhang. “Explicit and implicit solutions of a generalized Camassa–Holm Kadomtsev–Petviashvili equation.” Communications in Nonlinear Science and Numerical Simulation 17.3 (2012): 1130-1141. |
| [27] | Osman, M. S., et al. “Different wave structures and stability analysis for the generalized (2+1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation.” Physica Scripta 95. 3(2020): 035229. |
| [28] | Liu, Zhigang, Kelei Zhang, and Mengyuan Li. “Exact Traveling Wave Solutions and Bifurcation of a Generalized (3+1)-Dimensional Time-Fractional Camassa-Holm-Kadomtsev-Petviashvili Equation.” Journal of Function Spaces 2020. 1(2020): 4532824. |
| [29] | Hussain, Amjad, et al. “Analysis of (1+n)-Dimensional Generalized Camassa–Holm Kadomtsev–Petviashvili Equation Through Lie Symmetries, Nonlinear Self- Adjoint Classification and Travelling Wave Solutions.” Fractals 31. 10(2023): 2340078. |
APA Style
AL-Amry, M. S. A., AL-Abdali, E. F. A. (2025). Exact Soliton Solutions for New (4+1)-Dimensional Nonlinear Partial Differential Equations by a New exp(φ(ξ))-Expansion Method. International Journal of Theoretical and Applied Mathematics, 11(2), 26-33. https://doi.org/10.11648/j.ijtam.20251102.11
ACS Style
AL-Amry, M. S. A.; AL-Abdali, E. F. A. Exact Soliton Solutions for New (4+1)-Dimensional Nonlinear Partial Differential Equations by a New exp(φ(ξ))-Expansion Method. Int. J. Theor. Appl. Math. 2025, 11(2), 26-33. doi: 10.11648/j.ijtam.20251102.11
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Download@article{10.11648/j.ijtam.20251102.11,
author = {Mohammed Salem Ahmed AL-Amry and Eman Fadhl Abdullah AL-Abdali},
title = {Exact Soliton Solutions for New (4+1)-Dimensional Nonlinear Partial Differential Equations by a New exp(φ(ξ))-Expansion Method
},
journal = {International Journal of Theoretical and Applied Mathematics},
volume = {11},
number = {2},
pages = {26-33},
doi = {10.11648/j.ijtam.20251102.11},
url = {https://doi.org/10.11648/j.ijtam.20251102.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20251102.11},
abstract = {In this paper, we present a new two equations. The first equation is the (4 + 1)-dimensional Generalized Nonlinear Boussinesq Equation (G-NBE), and the second is the (4+1)-dimensional Generalized Camassa–Holm Kadomtsev–Petviashvili Equation (G-CH-KPE). We use a new exp(φ(ξ))-expansion method for solve our new equations. We determine a variety of exact solutions for each equation and expressed in terms of hyperbolic functions, trigonometric functions, exponential functions and rational functions.
},
year = {2025}
}
TY - JOUR T1 - Exact Soliton Solutions for New (4+1)-Dimensional Nonlinear Partial Differential Equations by a New exp(φ(ξ))-Expansion Method AU - Mohammed Salem Ahmed AL-Amry AU - Eman Fadhl Abdullah AL-Abdali Y1 - 2025/07/14 PY - 2025 N1 - https://doi.org/10.11648/j.ijtam.20251102.11 DO - 10.11648/j.ijtam.20251102.11 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 26 EP - 33 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20251102.11 AB - In this paper, we present a new two equations. The first equation is the (4 + 1)-dimensional Generalized Nonlinear Boussinesq Equation (G-NBE), and the second is the (4+1)-dimensional Generalized Camassa–Holm Kadomtsev–Petviashvili Equation (G-CH-KPE). We use a new exp(φ(ξ))-expansion method for solve our new equations. We determine a variety of exact solutions for each equation and expressed in terms of hyperbolic functions, trigonometric functions, exponential functions and rational functions. VL - 11 IS - 2 ER -
Department of Mathematics, Faculty of Education Aden, Aden University, Aden, Yemen
Department of Mathematics, Faculty of Education Aden, Aden University, Aden, Yemen
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