Bifurcation analysis and the modulation instability in a nonlinear silica optical fibers

Maha Alammari; Muhammad Abuzar; Solomon Manukure; Maha Alammari; Muhammad Abuzar; Solomon Manukure

doi:10.3934/math.2025749

AIMS Mathematics

2025, Volume 10, Issue 7: 16692-16719. doi: 10.3934/math.2025749

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Bifurcation analysis and the modulation instability in a nonlinear silica optical fibers

1.
Department of Mathematics, College of Science, King Saud University, P.O. Box 22452 Riyadh 11495, Saudi Arabia
2.
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Yunyan 550003, China
3.
Department of Mathematics, Florida A&M University, Tallahassee, FL 32307, USA

Received: 19 May 2025 Revised: 05 July 2025 Accepted: 17 July 2025 Published: 24 July 2025
MSC : 35C05, 35Q53

The Schäfer-Wayne equation (SWE), a crucial model for ultrashort pulse propagation in nonlinear silicon optical fibers, is investigated using the $ F $-expansion method and enhanced modified extended tanh expansion method (EMETEM). We derive diverse solitary wave solutions, including dark, bright, periodic, multi-peak periodic, and breather-like periodic solutions, visualized through $ 2D $ and $ 3D $ graphics. Novel contributions include comprehensive bifurcation analysis via planar dynamical systems revealing phase portrait classifications, modulation instability analysis for solution stability evaluation, and sensitivity analysis assessing parameter dependence and initial condition effects. The diverse solitary wave solutions represent a new advancement in understanding SWE dynamics. The study demonstrates the methods' robustness in examining nonlinear wave dynamics with applications in optics, engineering, and telecommunications.
- short pulse equation,
- analytical techniques,
- nonlinear waves,
- phase portraits,
- modulation instability,
- sensitive analysis,
- Schäfer-Wayne equation
Citation: Maha Alammari, Muhammad Abuzar, Solomon Manukure. Bifurcation analysis and the modulation instability in a nonlinear silica optical fibers[J]. AIMS Mathematics, 2025, 10(7): 16692-16719. doi: 10.3934/math.2025749

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Abstract

The Schäfer-Wayne equation (SWE), a crucial model for ultrashort pulse propagation in nonlinear silicon optical fibers, is investigated using the $ F $-expansion method and enhanced modified extended tanh expansion method (EMETEM). We derive diverse solitary wave solutions, including dark, bright, periodic, multi-peak periodic, and breather-like periodic solutions, visualized through $ 2D $ and $ 3D $ graphics. Novel contributions include comprehensive bifurcation analysis via planar dynamical systems revealing phase portrait classifications, modulation instability analysis for solution stability evaluation, and sensitivity analysis assessing parameter dependence and initial condition effects. The diverse solitary wave solutions represent a new advancement in understanding SWE dynamics. The study demonstrates the methods' robustness in examining nonlinear wave dynamics with applications in optics, engineering, and telecommunications.

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