A WAS neural network framework for computing and analyzing solutions of a generalized $ (3+1) $-dimensional nonlinear Wave equation: Stability analysis

Waseem Razzaq; Asim Zafar; Ahmed Al Nuaim; Abdullah Al Nuaim; Waseem Razzaq; Asim Zafar; Ahmed Al Nuaim; Abdullah Al Nuaim

doi:10.3934/math.2026440

AIMS Mathematics

2026, Volume 11, Issue 4: 10694-10715. doi: 10.3934/math.2026440

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A WAS neural network framework for computing and analyzing solutions of a generalized $ (3+1) $-dimensional nonlinear Wave equation: Stability analysis

1.
Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Pakistan
2.
Department of Management Information Systems, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received: 09 March 2026 Revised: 05 April 2026 Accepted: 13 April 2026 Published: 20 April 2026
MSC : 35Q55, 35C08, 37K10, 35A20

This paper discusses the generalized non-linear $ (3+1) $-dimensional wave equation by modeling and analyzing the dynamics of multi-dimensional nonlinear waves with the WAS-neural network technique. The suggested framework accurately models various wave forms such as bright, singular, and bright-dark solitons. Insofar as we know such neural network based solutions of this model are not reported before. To ensure the reliability and proficiency of the WAS neural network technique. The gain solutions are stable or not by executing the stability analysis on them. The graphical visualization in three-dimensional surface and two-dimensional plots are used. The findings validate that the WAS neural network technique is an efficient and strong alternative to classical techniques of higher-dimensional nonlinear wave equations, and has applications in fluid mechanics and engineering systems that have to deal with gas liquid interactions.
- the $ (3+1) $-dimensional wave equation,
- WAS neural network method,
- analytical method,
- neural network,
- soliton solutions,
- stability analysis
Citation: Waseem Razzaq, Asim Zafar, Ahmed Al Nuaim, Abdullah Al Nuaim. A WAS neural network framework for computing and analyzing solutions of a generalized $ (3+1) $-dimensional nonlinear Wave equation: Stability analysis[J]. AIMS Mathematics, 2026, 11(4): 10694-10715. doi: 10.3934/math.2026440

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Abstract

This paper discusses the generalized non-linear $ (3+1) $-dimensional wave equation by modeling and analyzing the dynamics of multi-dimensional nonlinear waves with the WAS-neural network technique. The suggested framework accurately models various wave forms such as bright, singular, and bright-dark solitons. Insofar as we know such neural network based solutions of this model are not reported before. To ensure the reliability and proficiency of the WAS neural network technique. The gain solutions are stable or not by executing the stability analysis on them. The graphical visualization in three-dimensional surface and two-dimensional plots are used. The findings validate that the WAS neural network technique is an efficient and strong alternative to classical techniques of higher-dimensional nonlinear wave equations, and has applications in fluid mechanics and engineering systems that have to deal with gas liquid interactions.

References

[1]	M. A. S. Murad, M. A. Mustafa, U. Younas, H. Emadifar, A. S. Khalifa, W. W. Mohammed, et al., Soliton solutions to the generalized derivative nonlinear Schrödinger equation under the effect of multiplicative white noise and conformable derivative, Sci. Rep., 15 (2025), 19599. https://doi.org/10.1038/s41598-025-04981-7 doi: 10.1038/s41598-025-04981-7
[2]	K. Wang, New computational approaches to the fractional coupled nonlinear Helmholtz equation, Eng. Computation., 41 (2024), 1285–1300. https://doi.org/10.1108/EC-08-2023-0501 doi: 10.1108/EC-08-2023-0501
[3]	Z. Aydın, F. Taşcan, Application of new Kudryashov method to Sawada–Kotera and Kaup–Kupershmidt equations, Comput. Methods Diffe., 13 (2025), 608–617. https://doi.org/10.22034/CMDE.2024.60001.2558 doi: 10.22034/CMDE.2024.60001.2558
[4]	M. Rahioui, E. H. E. Kinani, A. Ouhadan, Bäcklund transformations, consistent tanh expansion solvability, interaction solutions, and some exact solitary waves of a (3+1)-dimensional P-type evolution equation, Int. J. Theor. Phys., 64 (2025), 187. https://doi.org/10.1007/s10773-025-06054-x doi: 10.1007/s10773-025-06054-x
[5]	W. Razzaq, A. Zafar, A. Nazir, H. Ahmad, A. A. Zaagan, Dark, bright and singular optical solitons for Biswas–Milovic equation with Kerr and dual power law nonlinearities, Mod. Phys. Lett. B, 39 (2025), 2450443. https://doi.org/10.1142/S0217984924504438 doi: 10.1142/S0217984924504438
[6]	Z. Li, Optical solutions of the nonlinear Kodama equation with the M-truncated derivative via the extended $(G'/G)$-expansion method, Fractal Fract., 9 (2025), 300. https://doi.org/10.3390/fractalfract9050300 doi: 10.3390/fractalfract9050300
[7]	M. S. Ullah, M. Z. Ali, H. O. Roshid, Stability analysis, model expansion method, and diverse chaos-detecting tools for the DSKP model, Sci. Rep., 15 (2025), 13658. https://doi.org/10.1038/s41598-025-98275-7 doi: 10.1038/s41598-025-98275-7
[8]	A. Mumtaz, M. Shakeel, A. Manan, N. A. Shah, S. F. Ahmed, A comparative study of new traveling wave solutions for the (2+1)-dimensional fractional Wazwaz Kaur Boussinesq equation using novel modified $(G'/G^{2})$-expansion method, AIP Adv., 15 (2025), 035204. https://doi.org/10.1063/5.0253219 doi: 10.1063/5.0253219
[9]	S. Hossain, M. M. Rahman, M. H. Bashar, S. Biswas, M. M. Roshid, Dynamical structure of the soliton solution of M-fractional (2+1)-dimensional Heisenberg ferromagnetic spin chain model, J. Appl. Math., 2025 (2025), 5535543. https://doi.org/10.1155/jama/5535543 doi: 10.1155/jama/5535543
[10]	M. I. Khan, A. Farooq, K. S. Nisar, N. A. Shah, Unveiling new exact solutions of the unstable nonlinear Schrödinger equation using the improved modified Sardar sub-equation method, Results Phys., 59 (2024), 107593. https://doi.org/10.1016/j.rinp.2024.107593 doi: 10.1016/j.rinp.2024.107593
[11]	J. Muhammad, A. H. Tedjani, E. Hussain, U. Younas, Exploring the exact solutions to the nonlinear systems with neural networks method, Sci. Rep., 15 (2025), 36818. https://doi.org/10.1038/s41598-025-21095-2 doi: 10.1038/s41598-025-21095-2
[12]	A. M. Saxe, J. L. McClelland, S. Ganguli, Exact solutions to the nonlinear dynamics of learning in deep linear neural networks, arXiv Preprint, 2013. https://doi.org/10.48550/arXiv.1312.6120
[13]	X. R. Xie, R. F. Zhang, Neural network-based symbolic calculation approach for solving the Korteweg–de Vries equation, Chaos Soliton. Fract., 194 (2025), 116232. https://doi.org/10.1016/j.chaos.2025.116232 doi: 10.1016/j.chaos.2025.116232
[14]	Z. Zou, G. E. Karniadakis, Multi-head physics-informed neural networks for learning functional priors and uncertainty quantification, J. Comput. Phys., 531 (2025), 113947. https://doi.org/10.1016/j.jcp.2025.113947 doi: 10.1016/j.jcp.2025.113947
[15]	G. Panichi, S. Corli, E. Prati, Quantum physics-informed neural networks for multivariable partial differential equations, Phys. Rev. Appl., 25 (2026), 014001. https://doi.org/10.1103/6nh4-yh2y doi: 10.1103/6nh4-yh2y
[16]	G. U. Urazboev, M. M. Khasanov, On exact finite-gap solutions of the negative-order Korteweg–de Vries equation, Theor. Math. Phys., 226 (2026), 87–96. https://doi.org/10.1134/S0040577926010058 doi: 10.1134/S0040577926010058
[17]	A. Zafar, W. Razzaq, A. Nazir, M. A. Alomair, A. S. Al Naim, A. Alomair, Bifurcation analysis and solitons dynamics of the fractional Biswas–Arshed equation via analytical method, Mathematics, 13 (2025), 3147. https://doi.org/10.3390/math13193147 doi: 10.3390/math13193147
[18]	H. Shang, C. Guo, Y. Wu, Z. Li, J. Yang, Solving the many-electron Schrödinger equation with a transformer-based framework, Nat. Commun., 16 (2025), 8464. https://doi.org/10.1038/s41467-025-63219-2 doi: 10.1038/s41467-025-63219-2
[19]	N. M. A. Alsafri, Solitonic behaviors in the coupled Drinfeld–Sokolov–Wilson system with fractional dynamics, AIMS Math., 10 (2025), 4747–4774. https://doi.org/10.3934/math.2025218 doi: 10.3934/math.2025218
[20]	K. Farooq, F. S. Alshammari, Z. Li, E. Hussain, Soliton dynamics and stability in the Boussinesq equation for shallow water applications, Front. Phys., 13 (2025), 1637491. https://doi.org/10.3389/fphy.2025.1637491 doi: 10.3389/fphy.2025.1637491
[21]	K. K. Ahmed, H. H. Hussein, H. M. Ahmed, W. B. Rabie, W. Alexan, Analysis of the dynamical behaviors for the generalized Bogoyavlensky–Konopelchenko equation, Ain Shams Eng. J., 15 (2024), 103000. https://doi.org/10.1016/j.asej.2024.103000 doi: 10.1016/j.asej.2024.103000
[22]	A. S. Khalifa, N. M. Badra, H. M. Ahmed, W. B. Rabie, H. Emadifar, K. K. Ahmed, Building novel solitary wave solutions for the generalized nonlinear (3+1)-dimensional wave equation, Part. Differ. Equ. Appl. Math., 16 (2025), 101272. https://doi.org/10.1016/j.padiff.2025.101272 doi: 10.1016/j.padiff.2025.101272
[23]	X. Y. Gao, In an ocean or a river: Bilinear auto-Bäcklund transformations and similarity reductions on a (3+1)-dimensional shallow water wave equation, China Ocean Eng., 39 (2025), 160–165. https://doi.org/10.1007/s13344-025-0012-y doi: 10.1007/s13344-025-0012-y
[24]	K. J. Wang, F. Shi, S. Li, G. Li, P. Xu, Resonant Y-type soliton and interaction wave solutions to the (3+1)-dimensional shallow water wave equation, J. Math. Anal. Appl., 542 (2025), 128792. https://doi.org/10.1016/j.jmaa.2024.128792 doi: 10.1016/j.jmaa.2024.128792
[25]	M. Şenol, Abundant solitary wave solutions to a (3+1)-dimensional nonlinear evolution equation arising in fluid dynamics, Mod. Phys. Lett. B, 39 (2025), 2450475. https://doi.org/10.1142/S021798492450475X doi: 10.1142/S021798492450475X
[26]	X. T. Gao, B. Tian, In a river or an ocean: Similarity reduction on a (3+1)-dimensional extended shallow water wave equation, Appl. Math. Lett., 160 (2025), 109310. https://doi.org/10.1016/j.aml.2024.109310 doi: 10.1016/j.aml.2024.109310
[27]	C. H. Feng, B. Tian, X. T. Gao, Bilinear Bäcklund ransformations, as well as N-Soliton, Breather, Fission/Fusion and Hybrid solutions for a (3+1)-dimensional integrable wave equation in a fluid, Qual. Theor. Dyn. Syst., 24 (2025), 100. https://doi.org/10.1007/s12346-025-01241-x doi: 10.1007/s12346-025-01241-x
[28]	W. Razzaq, A. Zafar, Stability analysis of soliton solutions for the Kairat-X equation via a WAS neural network method, Z. Angew. Math. Phys., 77 (2026), 13. https://doi.org/10.1007/s00033-025-02659-8 doi: 10.1007/s00033-025-02659-8
[29]	A. Akbulut, W. Razzaq, F. Taşcan, Exact solutions for the propagation of pulses in optical fibers, Commun. Theor. Phys., 76 (2024), 105003. https://doi.org/10.1088/1572-9494/ad526c doi: 10.1088/1572-9494/ad526c
[30]	W. Ma, S. Bilige, Novel interaction solutions to the (3+1)-dimensional Hirota bilinear equation by neural network method, Mod. Phys. Lett. B, 38 (2024), 2450240. https://doi.org/10.1142/S0217984924502403 doi: 10.1142/S0217984924502403

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