https://doi.org/10.1140/epjp/s13360-025-06781-4
Regular Article
Bilinear neural network solutions for nonlinear waves in the Sawada–Kotera model studied in heat transfer
1
Department of Mathematics, COMSATS University Islamabad, Lahore, Pakistan
2
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
3
Mathematics Department, Faculty of Science, Taibah University, 41411, Al-Madinah Al-Munawarah, Kingdom of Saudi Arabia
4
Basic Sciences Research Center (BSRC), Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, 13318, Saudi Arabia
Received:
16
July
2025
Accepted:
21
August
2025
Published online:
5
September
2025
In this study, the bilinear neural network method (BNNM) is employed to obtain exact analytical solutions of the (2 + 1)-dimensional bidirectional Sawada–Kotera (bSK) equation. BSK describes the evolution and interaction of nonlinear wave structures propagating simultaneously in opposite spatial directions within a multidimensional medium. BNNM approach begins with the generalized Hirota bilinear method to transform the equation into bilinear form, after which the BNNM-integrating classical bilinear theory with neural network (NN) architectures is applied to derive closed-form solutions. In this framework, the trial functions are modeled through layered NNs, where activation functions and weight matrices determine the solution structures; the “3-2-1” and “3-3-1” architectures are used to generate diverse waveforms, including lump solutions, lump-kink interactions, rogue waves, breather–lump solitons, and periodic waves. The results are visualized through contour plots, and 3D surface graphs to capture the waves geometrical and dynamical attributes. This work is significant as it demonstrates that BNNM not only recovers known wave patterns but also generates new classes of test functions, offering a flexible and computationally efficient framework for exploring multidimensional nonlinear wave phenomena. The method has broad applicability in nonlinear optics, plasma physics, fluid dynamics, geophysical flows, and acoustics, where understanding the formation, interaction, and stability of localized and periodic structures is of practical importance.
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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.
