Exact analytical solutions of the (2+1)-dimensional Sawada-Kotera equation: A study employing the bilinear neural network method

Zhiyuan Ma; Yuanlin Liu; Zhimin Ma; Zhao Li; Zhiyuan Ma; Yuanlin Liu; Zhimin Ma; Zhao Li

doi:10.3934/math.20251196

AIMS Mathematics

2025, Volume 10, Issue 11: 27217-27234. doi: 10.3934/math.20251196

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Research article

Exact analytical solutions of the (2+1)-dimensional Sawada-Kotera equation: A study employing the bilinear neural network method

1.
The Engineering and Technical College of Chengdu University of Technology, Leshan, China
2.
Jiamusi University, Jiamusi, China
3.
College of Computer Science, Chengdu University, Chengdu, China

Received: 23 September 2025 Revised: 25 October 2025 Accepted: 12 November 2025 Published: 21 November 2025
MSC : 9B62, 33B10, 39A12

The (2+1)-dimensional Sawada-Kotera equation is closely related to nonlinear wave applications in higher-dimensional spaces. In this study, we introduce, for the first time, the application of the bilinear neural network method (BNNM) to derive exact analytical solutions for this important equation. Moving beyond traditional approaches, this method allows us to systematically construct a rich family of solutions by designing specific neural network architectures. Specifically, we build single-layer bilinear neural network models to successfully obtain a variety of localized wave solutions. These include lump solutions, which are localized in all directions, breather solutions, which exhibit periodic oscillations in time, and intriguing hybrid lump-soliton solutions. To further explore the equation's complexity, we designed two distinct network architectures, namely [3-2-2-1] and [3-2-3-1], which effectively yielded novel interaction solutions between different wave types and periodic solutions. The characteristics and dynamic behaviors of all these solutions are then thoroughly investigated through detailed graphical representations, including three-dimensional surface plots, contour maps, and density plots, providing vivid insights into their evolution. The success of this work underscores the power of the BNNM framework. Its key advantage lies in its ability to incorporate and generalize numerous classical test functions used in bilinear theory, thereby demonstrating remarkable universality and potential as a unified method for tackling a broad class of nonlinear evolution equations. Our results not only enrich the solution set of the (2+1)-dimensional SK equation, but also establish BNNM as a promising and innovative tool in the field of mathematical physics.
- (2+1)-dimensional Sawada-Kotera equation,
- bilinear neural network method,
- lump solution,
- breather solutions,
- lump-soliton solutions,
- interaction solutions,
- periodic solution
Citation: Zhiyuan Ma, Yuanlin Liu, Zhimin Ma, Zhao Li. Exact analytical solutions of the (2+1)-dimensional Sawada-Kotera equation: A study employing the bilinear neural network method[J]. AIMS Mathematics, 2025, 10(11): 27217-27234. doi: 10.3934/math.20251196

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Abstract

The (2+1)-dimensional Sawada-Kotera equation is closely related to nonlinear wave applications in higher-dimensional spaces. In this study, we introduce, for the first time, the application of the bilinear neural network method (BNNM) to derive exact analytical solutions for this important equation. Moving beyond traditional approaches, this method allows us to systematically construct a rich family of solutions by designing specific neural network architectures. Specifically, we build single-layer bilinear neural network models to successfully obtain a variety of localized wave solutions. These include lump solutions, which are localized in all directions, breather solutions, which exhibit periodic oscillations in time, and intriguing hybrid lump-soliton solutions. To further explore the equation's complexity, we designed two distinct network architectures, namely [3-2-2-1] and [3-2-3-1], which effectively yielded novel interaction solutions between different wave types and periodic solutions. The characteristics and dynamic behaviors of all these solutions are then thoroughly investigated through detailed graphical representations, including three-dimensional surface plots, contour maps, and density plots, providing vivid insights into their evolution. The success of this work underscores the power of the BNNM framework. Its key advantage lies in its ability to incorporate and generalize numerous classical test functions used in bilinear theory, thereby demonstrating remarkable universality and potential as a unified method for tackling a broad class of nonlinear evolution equations. Our results not only enrich the solution set of the (2+1)-dimensional SK equation, but also establish BNNM as a promising and innovative tool in the field of mathematical physics.

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