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New analytical wave solutions for the gardner equation via the Riccati-modified extended simple equation method
Yousef Jawarneh1, Musawa Yahya Almusawa2, Izatmand Haleemzai3
1Department of Mathematics, College of Science, University of Ha'il, Ha'il, 2440, Kingdom of Saudi Arabia.
Scientific Reports
|April 18, 2026
Summary
This study presents new analytical solutions for the Gardner equation, crucial for understanding nonlinear waves in fluid, plasma, and optical systems. The findings offer deeper insights into wave dynamics and phenomena like soliton interactions.
Area of Science:
- Nonlinear Dynamics
- Mathematical Physics
- Wave Phenomena
Background:
- The Gardner equation models complex nonlinear phenomena, including internal waves, ion-acoustic waves, and signal propagation.
- It integrates aspects of the Korteweg-de Vries (KdV) and modified KdV equations, highlighting its significance in diverse physical systems.
Purpose of the Study:
- To derive a comprehensive set of precise analytical solutions for the Gardner equation.
- To explore the nonlinear dynamics and physical relevance of these solutions.
Main Methods:
- Utilized the Riccati-based Modified Extended Simple Equation Method (RM-ESEM).
- Developed a unified analytical framework to encompass various solution families.
Main Results:
- Generated diverse analytical solutions including trigonometric, hyperbolic, rational, exponential, and mixed forms.
- Demonstrated the stability, periodicity, and transient nature of these solutions through graphical analysis.
- Revealed complex nonlinear behaviors such as soliton interactions and parameter-dependent wave dynamics.
Conclusions:
- The study provides an enlarged solution space for the Gardner equation, enhancing understanding of nonlinear wave phenomena.
- Findings have significant theoretical and practical applications in fluid physics, nonlinear optics, and plasma physics.
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