The traveling wave solutions to a variant of the Boussinesq equation

Authors

Muhammad Nadeem School of Mathematics and Statistics, Qujing Normal University, Qujing 655011, China
Loredana Florentina Iambor Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania

https://doi.org/10.61383/ejam.20231347

Keywords:

the extended trial equation method, B( n+1,1,n ) equations, soliton solution, elliptic solutions

Abstract

In this study, we study a variant of the Boussinesq equation called as $B( n+1,\,1,\,n )$ equation, and construct some traveling wave solutions by using an effective approach called the extended trial equation method. Thus, the soliton solutions, rational function solutions, elliptic function solutions and Jacobi elliptic function solutions, which show the existence of various mathematical and physical structures and events in the fundamental equation considered, have been constructured. In order to make a more detailed examination of the physical behavior of these solutions, two- and three-dimensional graphs of some solution functions were drawn with the help of the Mathematica package program. In the section of Discussion, we suggest a more general version of the trial equation method for nonlinear differential equations.

References

Khalida Faisal, Souleymanou Abbagari, Arash Pashrashid, Alphonse Houwe, Shao-Wen Yao, and Hijaz Ahmad, Pure-cubic optical solitons to the Schr"odinger equation with three forms of nonlinearities by Sardar subequation method, Results Phys. 48 (2023), 106412. DOI: https://doi.org/10.1016/j.rinp.2023.106412

Kang-jia Wang, and Jing-Hua Liu, On abundant wave structures of the unsteady korteweg-de vries equation arising in shallow water, J. Ocean Eng. Sci. 8 (2023), 595-601. DOI: https://doi.org/10.1016/j.joes.2022.04.024

Mohammed Borg, Niveen M. Badra, Hamdy M. Ahmed, and Wafaa B. Rabie, Derivation of optical solitons in magneto-optical waveguides to coupled BMEs having Kudryashov's nonlinearity, Opt. Quantum Electron. 55 (2023), 672. DOI: https://doi.org/10.1007/s11082-023-04935-1

Jesmin Akter, and M. Ali Akbar, Exact solutions to the Benney-Luke equation and the Phi-4 equations by using modified simple equation method, Results Phys. 5 (2015), 125-130. DOI: https://doi.org/10.1016/j.rinp.2015.01.008

V. Senthil Kumar, Hadi Rezazadeh, Mostafa Eslami, Franoosh Izadi, and M. S Osman, Jacobi elliptic function expansion method for solving KdV equation with conformable derivative and dual-power law nonlinearity, Int. J. Appl. Comput. Math. 5 (2019), 127. DOI: https://doi.org/10.1007/s40819-019-0710-3

M.S. Osman, Multiwave solutions of time-fractional (2+1)-dimensional Nizhnik-Novikov-Veselov equations, Pramana-J. Phys. 88 (2017). 67. DOI: https://doi.org/10.1007/s12043-017-1374-3

Tugba Aydemir, Application of the generalized unified method to solve (2+1)-dimensional Kundu-Mukherjee-Naskar equation, Opt. Quantum Electron. 55 (2023), 534. DOI: https://doi.org/10.1007/s11082-023-04807-8

Behzad Ghanbari, and J.F. Gomez-Aguilar, The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with $beta$-conformable time derivative, Rev. Mex. Fis. 65 (2019), no. 5, 503-518. DOI: https://doi.org/10.31349/RevMexFis.65.503

B. Gunay, Chun-Ku Kuo, and Wen-Xiu Ma, An application of the exponential rational function method to exact solutions to the Drinfeld-Sokolov system, Results Phys. 29 (2021), 104733. DOI: https://doi.org/10.1016/j.rinp.2021.104733

Melike Kaplan, Rubayyi T. Alqahtani, and Nadiyah Hussain Alharthi, Wave propagation and stability analysis for Ostrovsky and symmetric regularized long-wave equations, Mathematics 11 (2023), no. 19, 4030. DOI: https://doi.org/10.3390/math11194030

Dipankar Kumar, Chun-Ku Kuo, Gour Chandra Paul, Jui Saha, and Israt Jahan, Wave propagation of resonance multi-stripes, complexitons, and lump and its variety interaction solutions to the (2+1)-dimensional pKP equation, Commun. Nonlinear Sci. Numer. Simul. 100 (2021), 105853. DOI: https://doi.org/10.1016/j.cnsns.2021.105853

Chun-Ku Kuo, Novel resonant multi-soliton solutions and inelastic interactions to the (3+1)-and (4+1)-dimensional Boiti-Leon-Manna-Pempinelli equations via the simplified linear superposition principle, Eur. Phys. J. Plus. 136 (2021), no. 1, 1-11. DOI: https://doi.org/10.1140/epjp/s13360-020-01062-8

Cheng-shi Liu, A new trial equation method and its applications, Commun. Theor. Phys. 45 (2006), 395-397. DOI: https://doi.org/10.1088/0253-6102/45/3/003

Cheng-shi Liu, Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations, Comput. Phys. Commun. 181 (2010), 317-324. DOI: https://doi.org/10.1016/j.cpc.2009.10.006

Cheng Yan-jun, Classification of traveling wave solutions to the Vakhnenko equations, Comput. Math. Appl. 62 (2011), 3987-3996. DOI: https://doi.org/10.1016/j.camwa.2011.09.060

Cheng Yan-jun, Classification of traveling wave solutions to the modified form of the Degasperis-Procesi equation, Math. Comput. Model. 56 (2012), 43-48. DOI: https://doi.org/10.1016/j.mcm.2011.12.021

Usman Afzal, Nauman Raza, and Isma Ghulam Murtaza, On soliton solutions of time fractional form of Sawada-Kotera equation, Nonlinear Dyn. 95 (2019), 391-405. DOI: https://doi.org/10.1007/s11071-018-4571-9

Damin Cao, and Lijuan Du, The classification of the single traveling wave solutions to (1+1)-dimensional Gardner equation with variable coefficients, Adv. Differ. Equ. 2019 (2019), 121. DOI: https://doi.org/10.1186/s13662-019-2061-0

Yang Liu, and Xin Wang, The construction of solutions to Zakharov-Kuznetsov equation with fractional power nonlinear terms, Adv. Differ. Equ. 2019 (2019), 134. DOI: https://doi.org/10.1186/s13662-019-2063-y

Tao Liu, Exact solutions to time-fractional fifth order KdV equation by trial equation method based on symmetry, Symmetry 11 (2019), 742. DOI: https://doi.org/10.3390/sym11060742

Yusuf Pandir, Yusuf Gurefe, Ugur Kadak, Emine Misirli, Classifications of exact solutions for some nonlinear partial differential equations with generalized evolution, Abstr. Appl. Anal. 2012 (2012), 478531. DOI: https://doi.org/10.1155/2012/478531

Yusuf Gurefe, Emine Misirli, Abdullah Sonmezoglu, Mehmet Ekici, Extended trial equation method to generalized nonlinear partial differential equations, Appl. Math. Comput. 219 (2013), 5253-5260. DOI: https://doi.org/10.1016/j.amc.2012.11.046

Yusuf Pandir, Yusuf Gurefe, Emine Misirli, Classification of exact solutions to the generalized Kadomtsev-Petviashvili equation, Phys. Scr. 87 (2013), 1-12. DOI: https://doi.org/10.1088/0031-8949/87/02/025003

Abdul-Majid Wazwaz, The sine-cosine and the tanh methods: Reliable tools for analytic treatment of nonlinear dispersive equations, Appl. Math. Comput. 173 (2006), 150-164. DOI: https://doi.org/10.1016/j.amc.2005.02.047

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Published

2023 Dec 24

How to Cite

[1]

M. Nadeem and L. Florentina Iambor, “The traveling wave solutions to a variant of the Boussinesq equation”, Electron. J. Appl. Math., vol. 1, no. 3, pp. 26–37, Dec. 2023.