Exact solitary waves in pre-compressed granular media

Sarfaraz Ahmed, Ujala Rehman, Jianbo Fei, Muhammad Irslan Khalid and Xiangsheng Chen

Chaos, Solitons & Fractals, 2026, vol. 210, issue P2

Abstract: In this study, we propose a newly developed method that combines neural network models and symbolic computing to quickly find exact solutions to nonlinear partial differential equations. By combining the high accuracy of symbolic computing with the strong adaptability of neural networks, this method significantly improves the efficiency and accuracy of decomputation. As an application, this paper proposes a neural network symbol calculation method, for constructing exact solitary wave solutions in a pre-compressed granular chain of elastic spheres. The nonlinear wave dynamics, governed by Hertzian contact interactions and pre-compression induced stiffness, are first reduced from a discrete granular model to a long wave continuum equation incorporating dispersion and nonlinearity. A family of neural symbolic architectures is then designed to represent parameter-dependent solitary wave solutions in closed analytical form. By embedding heuristic activation functions and systematically enforcing the governing equation, the proposed framework transforms the nonlinear partial differential equation into an algebraic constraint system on network parameters, enabling multiple sets of new analytical solutions without using the traditional bilinear transformations. Several single layer and multi-layer neural network configurations are explored, leading to a rich class of parameter-dependent complex hybrid wave solutions expressed in terms of polynomial, hyperbolic, exponential, elliptic, and Weierstrass functions. The results demonstrate that the proposed method provides a flexible and efficient tool for discovering exact solutions of nonlinear wave equations in granular media and related nonlinear phononic systems.

Keywords: Neural–symbolic computation; Exact solutions; Solitary waves; Granular chains; Nonlinear wave dynamics (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:210:y:2026:i:p2:s0960077926008593

DOI: 10.1016/j.chaos.2026.118718

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