Asymptotic analysis of soliton solutions of nonlinear Granular equation via WAS-Exp neural network technique

Waseem Razzaq; Asim Zafar; Naif Almusallam; Ahmed Al Nuaim; Waseem Razzaq; Asim Zafar; Naif Almusallam; Ahmed Al Nuaim

doi:10.3934/math.20261199

AIMS Mathematics

2026, Volume 11, Issue 9: 30260-30279. doi: 10.3934/math.20261199

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Asymptotic analysis of soliton solutions of nonlinear Granular equation via WAS-Exp neural network technique

1.
Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Pakistan
2.
Department of Management Information Systems, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received: 04 June 2026 Revised: 27 August 2026 Accepted: 14 September 2026 Published: 17 September 2026
MSC : 35Q55, 35C08, 37K10, 35A20

This paper presented an analytical method called the WAS-Exp neural network method to develop precise solutions of nonlinear evolution equations. The novelty of WAS-Exp lies in its unified neural-network formulation, in which an exponential-function mechanism is embedded directly into the solution construction rather than used as a separate ansatz. Consequently, distinct wave families can be derived systematically from a single framework, reducing the dependence on separately designed solution forms. In order to illustrate its efficacy, the technique was used on the pre-compressed one-dimensional Granular crystal equation, which formed a paradigm for the modeling of nonlinear wave propagation in dispersive media like the optical fibers and plasma channels. The application of the WAS-Exp neural network technique produced an extensive family of high-precision solutions of the analytic, unfolding nonlinear waves. Further, asymptotic analysis was employed on the solutions to check the x-asymptotic and t-asymptotic behavior of the solutions. Physical properties and propagation of these solutions were further illustrated by the use of two- and three-dimensional plots, which showed the diversity and stability of their structure. The findings revealed that the suggested approach was an efficient tool to study nonlinear systems with complex nature that could be used in the fields of physical sciences, fluid dynamics, and optical communications technologies.
- the WAS-Exp neural network method,
- pre-compressed one-dimensional Granular crystal model,
- solitons,
- asymptotic analysis
Citation: Waseem Razzaq, Asim Zafar, Naif Almusallam, Ahmed Al Nuaim. Asymptotic analysis of soliton solutions of nonlinear Granular equation via WAS-Exp neural network technique[J]. AIMS Mathematics, 2026, 11(9): 30260-30279. doi: 10.3934/math.20261199

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Abstract

This paper presented an analytical method called the WAS-Exp neural network method to develop precise solutions of nonlinear evolution equations. The novelty of WAS-Exp lies in its unified neural-network formulation, in which an exponential-function mechanism is embedded directly into the solution construction rather than used as a separate ansatz. Consequently, distinct wave families can be derived systematically from a single framework, reducing the dependence on separately designed solution forms. In order to illustrate its efficacy, the technique was used on the pre-compressed one-dimensional Granular crystal equation, which formed a paradigm for the modeling of nonlinear wave propagation in dispersive media like the optical fibers and plasma channels. The application of the WAS-Exp neural network technique produced an extensive family of high-precision solutions of the analytic, unfolding nonlinear waves. Further, asymptotic analysis was employed on the solutions to check the x-asymptotic and t-asymptotic behavior of the solutions. Physical properties and propagation of these solutions were further illustrated by the use of two- and three-dimensional plots, which showed the diversity and stability of their structure. The findings revealed that the suggested approach was an efficient tool to study nonlinear systems with complex nature that could be used in the fields of physical sciences, fluid dynamics, and optical communications technologies.

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