Soliton wave phenomena of the complex hyperbolic nonlinear Schrödinger equation via emerging WAS-Exp neural network method and qualitative analysis

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

doi:10.3934/math.20251352

AIMS Mathematics

2025, Volume 10, Issue 12: 30806-30827. doi: 10.3934/math.20251352

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Research article

Soliton wave phenomena of the complex hyperbolic nonlinear Schrödinger equation via emerging WAS-Exp neural network method and qualitative analysis

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: 22 October 2025 Revised: 18 December 2025 Accepted: 23 December 2025 Published: 30 December 2025
MSC : 35A20, 35C08, 35Q55, 37K10

The main goal of this work was to develop the new analytical neural network technique known as WAS-Exp neural network method. A mathematical analytical technique that could produce solitary, hyperbolic, trigonometric, and rational wave structures in a single framework to build accurate solutions. An essential model for explaining nonlinear wave propagation in dispersive media like optical fibers and plasma channels is the complex hyperbolic nonlinear Schrödinger dynamical equation, which we examined in this work. In contrast to its classical version, the model improves capture memory and non locality and anomalous dispersion by introducing hyperbolic dispersion. We execute the newly proposed technique on this model and a variety of analytical solutions are obtained using this procedure. The resultant solutions provided new insights into the dynamics of nonlinear systems and their possible applications in plasma physics, optical communications, and related domains by demonstrating the wave behavior through the 3D and 2D surfaces. Furthermore, machine learning analysis was executed on the obtained solution for examining the wave dynamic behavior of the actual and predicted outcomes. Lastly, we plotted the x-asymptotic, y-asymptotic and t-asymptotic of the gain solutions through the 2D surfaces.
Citation: Waseem Razzaq, Asim Zafar, Ahmed Al Nuaim, Naif Almusallam. Soliton wave phenomena of the complex hyperbolic nonlinear Schrödinger equation via emerging WAS-Exp neural network method and qualitative analysis[J]. AIMS Mathematics, 2025, 10(12): 30806-30827. doi: 10.3934/math.20251352

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Abstract

The main goal of this work was to develop the new analytical neural network technique known as WAS-Exp neural network method. A mathematical analytical technique that could produce solitary, hyperbolic, trigonometric, and rational wave structures in a single framework to build accurate solutions. An essential model for explaining nonlinear wave propagation in dispersive media like optical fibers and plasma channels is the complex hyperbolic nonlinear Schrödinger dynamical equation, which we examined in this work. In contrast to its classical version, the model improves capture memory and non locality and anomalous dispersion by introducing hyperbolic dispersion. We execute the newly proposed technique on this model and a variety of analytical solutions are obtained using this procedure. The resultant solutions provided new insights into the dynamics of nonlinear systems and their possible applications in plasma physics, optical communications, and related domains by demonstrating the wave behavior through the 3D and 2D surfaces. Furthermore, machine learning analysis was executed on the obtained solution for examining the wave dynamic behavior of the actual and predicted outcomes. Lastly, we plotted the x-asymptotic, y-asymptotic and t-asymptotic of the gain solutions through the 2D surfaces.

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