Novel analytical approaches and stability examination for soliton solutions in a dispersive perturbed Gardner model

Wafaa B. Rabie; Hamdy M. Ahmed; A. M. Abd-Alla; Khadiga A. Ismail; Ahmed Ramady; Wafaa B. Rabie; Hamdy M. Ahmed; A. M. Abd-Alla; Khadiga A. Ismail; Ahmed Ramady

doi:10.3934/math.20251213

AIMS Mathematics

2025, Volume 10, Issue 11: 27581-27607. doi: 10.3934/math.20251213

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Novel analytical approaches and stability examination for soliton solutions in a dispersive perturbed Gardner model

1.
Department of Mathematics, Faculty of Science Luxor University, Taiba, Luxor, Egypt
2.
Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El Shorouk Academy, Cairo, Egypt
3.
Department of Mathematics, Faculty of Science, University of Sohag, Sohag, Egypt
4.
Department of Clinical Laboratory Sciences, College of Applied Medical Sciences, Taif University, Taif 21944, Saudi Arabia
5.
GRC Department, The Applied College, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
6.
Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

Received: 25 August 2025 Revised: 08 November 2025 Accepted: 12 November 2025 Published: 26 November 2025
MSC : 35G20, 35C07, 35C08, 35C09

This study presented a rigorous analytical investigation of the perturbed Gardner equation, a fundamental model describing nonlinear wave propagation in plasma physics, fluid dynamics, and nonlinear optics. Using an advanced analytical framework, we derived exact solutions capturing diverse wave behaviors: Localized bright solitons representing stable energy packets, dark solitons manifesting as intensity voids, singular solitons with distinctive phase profiles, and periodic waves governed by elliptic functions. Linear stability analysis revealed that solution robustness emerged from a delicate balance between nonlinear focusing and dispersive spreading, with precise parameter windows identified for maintaining structural integrity. A key finding identified the dispersion triplet—higher-order dispersion coefficients—as crucial for controlling wave amplitude, spectral broadening, and nonlinearity-dispersion balance. Numerical validation confirmed mathematical consistency and physical feasibility, with strong agreement between theory and simulations. This established a framework for understanding nonlinear wave phenomena in photonic and plasma applications.
- perturbed Gardner equation,
- soliton solutions,
- dispersion effects,
- stability analysis,
- nonlinear wave dynamics,
- analytical methods
Citation: Wafaa B. Rabie, Hamdy M. Ahmed, A. M. Abd-Alla, Khadiga A. Ismail, Ahmed Ramady. Novel analytical approaches and stability examination for soliton solutions in a dispersive perturbed Gardner model[J]. AIMS Mathematics, 2025, 10(11): 27581-27607. doi: 10.3934/math.20251213

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Abstract

This study presented a rigorous analytical investigation of the perturbed Gardner equation, a fundamental model describing nonlinear wave propagation in plasma physics, fluid dynamics, and nonlinear optics. Using an advanced analytical framework, we derived exact solutions capturing diverse wave behaviors: Localized bright solitons representing stable energy packets, dark solitons manifesting as intensity voids, singular solitons with distinctive phase profiles, and periodic waves governed by elliptic functions. Linear stability analysis revealed that solution robustness emerged from a delicate balance between nonlinear focusing and dispersive spreading, with precise parameter windows identified for maintaining structural integrity. A key finding identified the dispersion triplet—higher-order dispersion coefficients—as crucial for controlling wave amplitude, spectral broadening, and nonlinearity-dispersion balance. Numerical validation confirmed mathematical consistency and physical feasibility, with strong agreement between theory and simulations. This established a framework for understanding nonlinear wave phenomena in photonic and plasma applications.

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