https://doi.org/10.1140/epjp/s13360-025-06705-2
Regular Article
Dynamics of N-soliton waves, lump-breathers, and M-lump collision with improved bilinear neural network method
1
Department of Mathematics, Shanghai University and Newtouch Center for Mathematics of Shanghai University, 200444, Shanghai, China
2
Department of Mathematics, College of Science, University of Zakho, Zakho, Iraq
3
Department of Computer Science, College of Science, Knowledge University, 44001, Erbil, Iraq
4
Department of Mathematics and Statistics, The University of Lahore, Sargodha Campus, Sargodha, Pakistan
5
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, 602105, Chennai, Tamil Nadu, India
6
Department of Mathematics, Federal University Dutse, Dutse, Jigawa, Nigeria
7
Department of Mathematics, Firat University, Elazığ, Türkiye
a
yfp@shu.edu.cn
b
hajar.ismael@uoz.edu.krd
Received:
2
July
2025
Accepted:
30
July
2025
Published online:
2
September
2025
In this paper, we focus on the analytical study of the (3+1)-dimensional Geng equation by employing the improved bilinear neural network method which is quite effective in solving nonlinear equations. The Geng equation, which is used to describe the phenomena in the shallow water dynamics and nonlinear wave interactions, can be used to analyze the higher-dimensional wave system. We are thus able to construct a variety of solutions such as N-soliton waves, lump-breathers, and M-lump solutions under the condition of using multivariate test functions in the one-hidden-layer structure “4-3-1” and the two-hidden-layer structure “4-2-3-1” neural network model. These solutions capture intricate interactions and dynamic behaviors inherent to the Geng equation, offering deeper insights into its underlying structure. Additionally, in order to improve the interpretation of the obtained solutions, we offer a detailed graphical representation in the form of 3D, 2D and contour plots. The proposed methodology, thus, proves highest efficiency and reliability while solving high-dimensional nonlinear equations and opens the avenue for future study on physical and mathematical interpretation of such equations, especially in the context of shallow water wave modeling.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
