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Exact solutions and instability analysis of a fifth-order nonlinear evolution equation exhibiting multi-scale wave
Wafy M Hasan1,2, Hamdy M Ahmed3, Ahmed M Ahmed4
1Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt. wafy.hasan@sut.edu.eg.
Scientific Reports
|July 13, 2026
Summary
Researchers analyzed a fifth-order nonlinear partial differential equation to find exact traveling wave solutions. This study reveals diverse analytical solutions, including solitons and periodic patterns, and examines their stability.
Area of Science:
- Nonlinear Partial Differential Equations
- Mathematical Physics
- Wave Phenomena
Background:
- Nonlinear partial differential equations (PDEs) model complex phenomena in various scientific fields.
- Higher-order PDEs often exhibit intricate wave behaviors like solitons and periodic patterns.
- Analytical methods are crucial for understanding the fundamental properties of these nonlinear systems.
Purpose of the Study:
- To construct exact traveling wave solutions for a specific fifth-order nonlinear PDE.
- To explore a unified approach for deriving diverse analytical solutions.
- To investigate the stability properties of the obtained wave solutions.
Main Methods:
- Modified extended mapping method applied to a fifth-order nonlinear PDE.
- Conversion of the nonlinear PDE into an equivalent ordinary differential equation.
- Linear perturbation framework used for stability analysis.
Main Results:
- Derivation of multiple classes of analytical solutions, including bright/dark solitons, singular patterns, and rational/elliptic function solutions.
- Identification of combined bright-dark soliton structures.
- Analysis of the dispersion relation to determine instability conditions and wave evolution.
Conclusions:
- The modified extended mapping method provides a unified framework for solving higher-order nonlinear PDEs.
- The study offers a comprehensive understanding of traveling wave solutions and their stability.
- This research lays the groundwork for investigating multi-scale wave interactions and stability in nonlinear systems.
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