https://doi.org/10.1140/epjs/s11734-025-01859-7
Regular Article
Analytical solutions of generalized (3 + 1)-dimensional KP equation via neural network-augmented bilinear framework
1
School of Mathematics and Statistics, Xiamen University of Technology, 361024, Xiamen, People’s Republic of China
2
College of Computer and Information Engineering, Xiamen University of Technology, 361024, Xiamen, People’s Republic of China
3
School of Automation and Software Engineering, Shanxi University, 030013, Taiyuan, People’s Republic of China
a
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b
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Received:
9
June
2025
Accepted:
8
August
2025
Published online:
18
August
2025
Abstract
This study establishes a neural network-enhanced bilinear framework for systematically deriving analytical solutions to the generalized (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation. Through innovative integration of symbolic computation with neural architectures, we develop two specialized network configurations (“4-3-1” and “4-2-2-1” topologies) that synergize with bilinear formulation principles. Our methodology yields several novel results: (1) first successful reconstruction of N-soliton solutions within neural network architecture; (2) comprehensive derivation of M-lump solutions; (3) coexistence of bright–dark soliton pairs; and (4) hybrid periodic wave patterns. The solutions spatiotemporal evolution is rigorously characterized through 3D surface visualizations, line plots, density plots, and temporal evolution graphs. These findings contribute to the mathematical modeling of nonlinear wave interactions in integrable systems by providing dynamic simulations of ocean surface gravity waves, analyses of shock wave structures in supersonic aircraft, soliton transmission optimizations in optical fiber communications, and investigations of nonlinear wave interactions in plasma physics.
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© The Author(s), under exclusive licence to EDP Sciences, 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.
