Dynamics of Soliton Solutions to Nonlinear Dynamical Equations in Mathematical Physics: Application of Neural Network-Based Symbolic Methods

Jan Muhammad, Aljethi Reem Abdullah, Fengping Yao and Usman Younas ()
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Jan Muhammad: Department of Mathematics, Shanghai University, No. 99 Shangda Road, Shanghai 200444, China
Aljethi Reem Abdullah: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
Fengping Yao: Department of Mathematics, Shanghai University and Newtouch Center for Mathematics of Shanghai University, Shanghai 200444, China
Usman Younas: Department of Mathematics, Shanghai University and Newtouch Center for Mathematics of Shanghai University, Shanghai 200444, China

Mathematics, 2025, vol. 13, issue 21, 1-25

Abstract: While recent advances have successfully integrated neural networks with physical models to derive numerical solutions, there remains a compelling need to obtain exact analytical solutions. The ability to extract closed-form expressions from these models would provide deeper theoretical insights and enhanced predictive capabilities, complementing existing computational techniques. In this paper, we study the nonlinear Gardner equation and the (2+1)-dimensional Zabolotskaya–Khokhlov model, both of which are fundamental nonlinear wave equations with broad applications in various physical contexts. The proposed models have applications in fluid dynamics, describing shallow water waves, internal waves in stratified fluids, and the propagation of nonlinear acoustic beams. This study integrates a modified generalized Riccati equation mapping approach and a novel generalized G ′ G -expansion method with neural networks for obtaining exact solutions for the suggested nonlinear models. Researchers are currently investigating potential applications of these neural networks to enhance our understanding of complex physical processes and to develop new analytical techniques. The proposed strategies incorporate the solutions of the Riccati problem into neural networks. Neural networks are multi-layer computing approaches including activation and weight functions among neurons in input, hidden, and output layers. Here, the solutions of the Riccati equation are allocated to each neuron in the first hidden layer; thus, new trial functions are established. We evaluate the suggested models, which lead to the construction of exact solutions in different forms, such as kink, dark, bright, singular, and combined solitons, as well as hyperbolic and periodic solutions, in order to verify the mathematical framework of the applied methods. The dynamic properties of certain wave-related solutions have been shown using various three-dimensional, two-dimensional, and contour visualizations. This paper introduces a novel framework for addressing nonlinear partial differential equations, with significant potential applications in various scientific and engineering domains.

Keywords: neural networks methods; Gardner’s equation; analytical solutions; (2+1)-dimensional Zabolotskaya–Khokhlov model (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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