Modulation instability and soliton families of the complex Ginzburg-Landau equation having the parabolic with nonlocal law of self-phase modulation

Wael W. Mohammed; Neslihan Ozdemir; Aydın Secer; Muslum Ozisik; Mustafa Bayram; Taha Radwan; Karim K. Ahmed; Wael W. Mohammed; Neslihan Ozdemir; Aydın Secer; Muslum Ozisik; Mustafa Bayram; Taha Radwan; Karim K. Ahmed

doi:10.3934/math.2025818

AIMS Mathematics

2025, Volume 10, Issue 8: 18321-18336. doi: 10.3934/math.2025818

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Modulation instability and soliton families of the complex Ginzburg-Landau equation having the parabolic with nonlocal law of self-phase modulation

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, Yildiz Technical University, Istanbul-Turkey
3.
Department of Mathematical Engineering, Yildiz Technical University, Istanbul-Turkey
4.
Department of Computer Engineering, Biruni University, Istanbul-Turkey
5.
Department of Mathematics, Faculty of Engineering and Natural Sciences, Istinye University, Istanbul-Turkey
6.
Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia
7.
Department of Mathematics, Faculty of Engineering, German International University (GIU), New Administrative Capital, Cairo, Egypt

Received: 05 July 2025 Revised: 01 August 2025 Accepted: 06 August 2025 Published: 13 August 2025
MSC : 35Q51, 35C08, 35Q56

This paper acquaints the adopted parabolic with the nonlocal law of self-phase modulation form of the complex Ginzburg–Landau model, which regularizes the evolution of specific amplitudes of instability pulses in diverse dissipative systems. We employ the new Kudryashov and Sinh-Gordon equation expansion schemes to obtain bright and dark soliton families under particular conditions on the parameters of the physical model. Furthermore, the effect of diverse model parameters such as the chromatic dispersion, the parabolic law, and the nonlocal nonlinearity terms on the behaviors of bright and dark soliton solutions is also explored. We also search for modulation instability analysis for the model. The primary contribution of this study is the examination of a different version of the complex Ginzburg–Landau model, which is not yet available in the literature, along with the first comprehensive analysis of its modulation instability. Thus, this study highlights the practical and prompt results received by the Sinh-Gordon equation expansion scheme. Thus, this study is expected to offer meaningful implications for ongoing and future research within the framework of this model.
- the complex Ginzburg–Landau equation,
- nonlocal nonlinearity,
- soliton dynamics,
- partial differential equations,
- modulation instability
Citation: Wael W. Mohammed, Neslihan Ozdemir, Aydın Secer, Muslum Ozisik, Mustafa Bayram, Taha Radwan, Karim K. Ahmed. Modulation instability and soliton families of the complex Ginzburg-Landau equation having the parabolic with nonlocal law of self-phase modulation[J]. AIMS Mathematics, 2025, 10(8): 18321-18336. doi: 10.3934/math.2025818

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Abstract

This paper acquaints the adopted parabolic with the nonlocal law of self-phase modulation form of the complex Ginzburg–Landau model, which regularizes the evolution of specific amplitudes of instability pulses in diverse dissipative systems. We employ the new Kudryashov and Sinh-Gordon equation expansion schemes to obtain bright and dark soliton families under particular conditions on the parameters of the physical model. Furthermore, the effect of diverse model parameters such as the chromatic dispersion, the parabolic law, and the nonlocal nonlinearity terms on the behaviors of bright and dark soliton solutions is also explored. We also search for modulation instability analysis for the model. The primary contribution of this study is the examination of a different version of the complex Ginzburg–Landau model, which is not yet available in the literature, along with the first comprehensive analysis of its modulation instability. Thus, this study highlights the practical and prompt results received by the Sinh-Gordon equation expansion scheme. Thus, this study is expected to offer meaningful implications for ongoing and future research within the framework of this model.

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