Distinct solitary wave solutions for the (3+1)-dimensional integrable pKP–BKP equation using the modified extended direct algebraic technique

Elsonbaty, Nivan M.; Ahmed, Hamdy M.; Badra, Niveen M.; Rabie, Wafaa B.; Eslami, Mostafa

doi:10.22034/cmde.2025.65398.3004

Distinct solitary wave solutions for the (3+1)-dimensional integrable pKP–BKP equation using the modified extended direct algebraic technique

Document Type : Research Paper

Authors

¹ 1. Department of Physics and Mathematics Engineering, Faculty of Engineering, Ain Shams University, Cairo, Egypt. 2. Basic Sciences Department, Faculty of Engineering, The British University in Egypt, Cairo, Egypt.

² Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, Egypt.

³ Department of Physics and Mathematics Engineering, Faculty of Engineering, Ain Shams University, Cairo, Egypt.

⁴ Department of Engineering Mathematics and Physics, Higher Institute of Engineering and Technology, Tanta, Egypt.

⁵ Department of Applied Mathematics, Faculty of Mathematical Sciences University of Mazandaran, Babolsar, Iran.

10.22034/cmde.2025.65398.3004

Abstract

This study aims to derive solitons and other traveling wave solutions for the pKP-BKP equation, which integrates the potential Kadomtsev–Petviashvili (pKP) and B-type Kadomtsev–Petviashvili (BKP) equations in three spatial dimensions. This equation is used to describe long water waves in oceans, impoundments, and estuaries, as well as to predict tsunamis, analyze river, tidal, and irrigation flows, and simulate weather patterns. The modified extended direct algebraic approach is employed to obtain various types of exact solutions, including dark solitons, combo dark-singular solitons, singular solitons, hyperbolic solutions, singular periodic solutions, exponential solutions, rational solutions, and Jacobi elliptic solutions. The derived solutions are visualized using Mathematica software, with contour, 2D, and 3D graphical representations to illustrate their dynamic behavior.

Keywords

Main Subjects

Applications to the sciences

References

[1] A. R. Adem and C. M. Khalique, New exact solutions and conservation laws of a coupled Kadomtsev–Petviashvili system, Computers & Fluids, 81 (2013), 10–16.
[2] K. K. Ahmed, N. M. Badra, H. M. Ahmed, and W. B. Rabie, Soliton solutions of generalized Kundu-Eckhaus equation with an extra-dispersion via improved modified extended tanh-function technique, Optical and Quantum Electronics, 55(4) (2023), 299.
[3] M. H. Ali, H. M. Ahmed, H. M. El-Owaidy, A. A. El-Deeb, and I. Samir, FPGA Implementations for Data Encryption and Decryption via Concurrent and Parallel Computation: A Review, Qubahan Academic Journal, 1(2) (2021), 8-16.
[4] M. H. Ali, H. M. Ahmed, H. M. El-Owaidy, A. A. El-Deeb, and I. Samir, New analytic wave solutions to (2+ 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation using the modified extended mapping method, Optical and Quantum Electronics, 56(3) (2024), 320.
[5] A. Q. Al-Shetwi and M. Z. Sujod, Modeling and simulation of photovoltaic module with enhanced perturb and observe MPPT algorithm using MATLAB/Simulink, ARPN Journal of Engineering and Applied Sciences, 11(20) (2016).
[6] F. Bouchaala, M. Y. Ali, and J. Matsushima, Estimation of compressional seismic wave attenuation of carbonate rocks in Abu Dhabi, United Arab Emirates, Comptes Rendus Geoscience, 346(7-8) (2014), 169-178.
[7] F. Bouchaala, M. Y. Ali, and J. Matsushima, Attenuation modes from vertical seismic profiling and sonic waveform in a carbonate reservoir, Abu Dhabi, United Arab Emirates, Geophysical Prospecting, 64(4) (2016), 1030-1047.
[8] F. Bouchaala, M. Y. Ali, and J. Matsushima, Estimation of compressional seismic wave attenuation of carbonate rocks in Abu Dhabi, United Arab Emirates, Comptes Rendus Geoscience, 346(7-8) (2014), 169-178.
[9] O. El-shamy, R. El-barkoki, H. M. Ahmed, W. Abbas, and I. Samir, Exploration of new solitons in optical medium with higher-order dispersive and nonlinear effects via improved modified extended tanh function method, Alexandria Engineering Journal, 68 (2023), 611-618.
[10] Y. Feng and S. Bilige, Resonant multi-soliton, M-breather, M-lump and hybrid solutions of a combined pKP-BKP equation, Journal of Geometry and Physics, 169 (2021), 104-322.
[11] B. Feng, J. Manafian, O. Alp, A. M. Roa, A. H. Agadi, Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He’s variational direct methods for the KP–BBM equation, International Journal of Modern Physics B, 35(27) (2021), 215-0275.
[12] M. S. Ghayad, H. M. Ahmed, N. M. Badra, H. Rezazadeh, M. A. Hosseinzadeh, and W. B. Rabie, New analytical wave structures for generalized B-type Kadomtsev–Petviashvili equation by improved modified extended tanh function method, Physica Scripta, 99(12) (2024), 125-224.
[13] R. Hirota, The Direct Method in Soliton Theory, Cambridge University, Pr., Cambridge, (2004).
[14] R. Hirota, A new form of Bäcklund transformations and its relation to the inverse scattering problem, Progress of Theoretical Physics, 52(5) (1974), 1498–1512.
[15] A. S. Khalifa, H. M. Ahmed, N. M. Badra, J. Manafian, K. H. Mahmoud, K. S. Nisar, and W. B. Rabie, Derivation of some solitary wave solutions for the (3+1)-dimensional pKP-BKP equation via the IME tanh function method, AIMS Mathematics, 9(10) (2024), 27704-27720.
[16] C. M. Khalique and M. Y. T. Lephoko, Conserved vectors and solutions of the two-dimensional potential KP equation, Open Physics, 21(1), (2023), 2023-0103.
[17] M. Lakestani, J. Manafian, A. R. Najafizadeh, and M. Partohaghighi, Some new soliton solutions for the nonlinear fifth-order integrable equations, Computational Methods for Differential Equations, 8(2) (2020), 123-135.
[18] M. Lakestani, J. Manafian, A. R. Najafizadeh, and M. Partohaghighi, Modeling Unmanned Submarine by an Adaptive Control to Follow an Ellipse Path by Considering Disturbances on the System, Journal of Research in Science, Engineering and Technology, 10(4) (2022), 173-187.
[19] W. X. Ma, Bilinear approach and soliton solutions for a higher-dimensional integrable system, Journal of Geometry and Physics, 154 (2020), 103-598.
[20] W. X. Ma, N-soliton solution of a combined pKP–BKP equation, Journal of Geometry and Physics, 165 (2021), 104-191.
[21] H. Ma, A. Yuey, S. Deng, Lump and interaction solutions for a (2 + 1)-dimensional combined pKP–BKP equation in fluids, Modern Phys Lett B, 13(36) (2022), 2250-069.
[22] Z. Y. Ma, J. X. Fei, W. P. Cao, and H. L. Wu, The explicit solution and its soliton molecules in the (2+1)-dimensional pKP-BKP equation, Results Phys, 35 (2022), 105–363.
[23] J. Manafian and M. Lakestani, Interaction among a lump, periodic waves, and kink solutions to the fractional generalized CBS-BK equation, Mathematical Methods in the Applied Sciences, 44(1) (2021), 1052-1070.
[24] J. Manafian, New exact multi-soliton solutions and interaction structures of a higher-order nonlinear equation, Partial Differential Equations in Applied Mathematics, 6 (2023), 100-600.
[25] S. A. Memon, H. M. Ahmed, and M. A. M. Kadhim, IoT and Cloud Computing Issues, Challenges and Opportunities: A Review, Qubahan Academic Journal, 1(2) (2021), 1-7.
[26] W. B. Rabie and H. M. Ahmed, Optimizing the Maintenance and Repair Program based on Condition Monitoring in Combined Cycle Power Plants, Journal of Research in Science, Engineering and Technology, 10(3) (2022), 157-172.
[27] W. B. Rabie, H. M. Ahmed, and H. H. Hussein, Novel analytic solutions of strain wave model in micro-structured solids, Nonlinear Engineering, 13(1) (2024), 2022-0293.
[28] W. B. Rabie and H. M. Ahmed, Optical solitons for multiple-core couplers with polynomial law of nonlinearity using the modified extended direct algebraic method, Optik, 258 (2022), 168-848.
[29] W. B. Rabie, T. A. Khalil, N. Badra, H. M. Ahmed, M. Mirzazadeh, and M. S. Hashemi, Soliton solutions and other solutions to the (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili equation using modified extended mapping method, Qualitative Theory of Dynamical Systems, 23(2) (2024), 87.
[30] W. B. Rabie, H. M. Ahmed, T. A. Nofal, and E. M. Mohamed, Novel analytical superposed nonlinear wave structures for the eighth-order (3+1)-dimensional Kac-Wakimoto equation using improved modified extended tanh function method, AIMS Mathematics, 9 (2024), 33386-33400.
[31] S. Saifullah, S. Ahmad, M. A. Khan, M. Rahman, Multiple solitons with fission and multi-waves interaction solutions of a (3+ 1)-dimensional combined pKP-BKP integrable equation, Physica Scripta, 99(6) (2024), 065-242.
[32] K. U. Tariq, A. M. Wazwaz, and R. Tufail, Lump, periodic and travelling wave solutions to the (2+1)-dimensional pKP–BKP model, Eur Phys J Plus, 137(10) (2022), 1100.
[33] A. M. Wazwaz, Two kinds of multiple wave solutions for the potential YTSF equation and a potential YTSF-type equation, J Appl Nonlinear Dyn, 1(1) (2012), 51–8.
[34] A. M. Wazwaz and S. A. El-Tantawy, Solving the (3+ 1)-dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method, Nonlinear Dynamics 88(4) (2017), 3017–3021.
[35] A. M. Wazwaz, New Painlev´e integrable (3+ 1)-dimensional combined pKP-BKP equation: lump and multiple soliton solutions, Chinese Physics Letters, 12(40) (2023), 120-501.
[36] A. M. Wazwaz, Breather wave solutions for an integrable (3+ 1)-dimensional combined pKP–BKP equation, Chaos, Solitons & Fractals, 182 (2024), 114-886.
[37] L. Zhang, Z. Zhao, and Y. Zhang, Dynamics of transformed nonlinear waves for the (2+ 1)-dimensional pKP-BKP equation: interactions and molecular waves, Physica Scripta, 99(7) (2024), 075-220.