Presentation of Soliton Solutions and Modulation Instability of a Nonlinear Schrödinger Equation

Şeyma Firdevs Hizal; Abdullahi Yusuf; Hasan Bulut

doi:10.47000/tjmcs.1848499

Presentation of Soliton Solutions and Modulation Instability of a Nonlinear Schrödinger Equation

Abstract

In this study, the nonlinear Schrödinger equation (NLSE) is analytically investigated by employing the Extended Rational Sine–Cosine method, the Extended Rational Sinh–Cosh method, and the Modified F-Expansion Method (MFEM). Through the application of these techniques, a broad class of exact travelling-wave solutions with rational–trigonometric and rational–hyperbolic structures is systematically constructed. The obtained solutions include bright solitons, dark solitons, periodic waves, singular solutions, and hybrid composite wave profiles. All solutions are verified by direct substitution into the governing equation, thereby ensuring their mathematical consistency. The spatiotemporal behavior of the solutions is illustrated in detail through two- and three-dimensional graphical representations. In addition, a modulation instability (MI) analysis is carried out for the continuous-wave background solution of the equation, and the corresponding MI gain spectrum is explicitly derived. The results reveal the influence of dispersion and nonlinear parameters on the modulational behavior of the system and demonstrate that the continuous-wave solution remains stable within the considered parameter regime. Overall, the findings extend the solution space of the NLSE and provide deeper physical insight into nonlinear wave dynamics in optical media.

Keywords

References

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Details

Primary Language

English

Subjects

Applied Mathematics (Other)

Journal Section

Research Article

Authors

Şeyma Firdevs Hizal
0000-0002-7165-8462
Türkiye

Abdullahi Yusuf
0000-0002-8308-7943
Türkiye

Hasan Bulut ^*
0000-0002-6089-1517
Türkiye

Publication Date

October 13, 2026

Submission Date

December 26, 2025

Acceptance Date

March 13, 2026

Published in Issue

Year 2026 Volume: 18 Number: 3

DOI

https://doi.org/10.47000/tjmcs.1848499

IZ

https://izlik.org/JA67ZZ35UM

Cite

RIS / Bibtex

APA

Hizal, Ş. F., Yusuf, A., & Bulut, H. (2026). Presentation of Soliton Solutions and Modulation Instability of a Nonlinear Schrödinger Equation. Turkish Journal of Mathematics and Computer Science, 18(3), 683-699. https://doi.org/10.47000/tjmcs.1848499

AMA

1.Hizal ŞF, Yusuf A, Bulut H. Presentation of Soliton Solutions and Modulation Instability of a Nonlinear Schrödinger Equation. TJMCS. 2026;18(3):683-699. doi:10.47000/tjmcs.1848499

Chicago

Hizal, Şeyma Firdevs, Abdullahi Yusuf, and Hasan Bulut. 2026. “Presentation of Soliton Solutions and Modulation Instability of a Nonlinear Schrödinger Equation”. Turkish Journal of Mathematics and Computer Science 18 (3): 683-99. https://doi.org/10.47000/tjmcs.1848499.

EndNote

Hizal ŞF, Yusuf A, Bulut H (October 1, 2026) Presentation of Soliton Solutions and Modulation Instability of a Nonlinear Schrödinger Equation. Turkish Journal of Mathematics and Computer Science 18 3 683–699.

IEEE

[1]Ş. F. Hizal, A. Yusuf, and H. Bulut, “Presentation of Soliton Solutions and Modulation Instability of a Nonlinear Schrödinger Equation”, TJMCS, vol. 18, no. 3, pp. 683–699, Oct. 2026, doi: 10.47000/tjmcs.1848499.

ISNAD

Hizal, Şeyma Firdevs - Yusuf, Abdullahi - Bulut, Hasan. “Presentation of Soliton Solutions and Modulation Instability of a Nonlinear Schrödinger Equation”. Turkish Journal of Mathematics and Computer Science 18/3 (October 1, 2026): 683-699. https://doi.org/10.47000/tjmcs.1848499.

JAMA

1.Hizal ŞF, Yusuf A, Bulut H. Presentation of Soliton Solutions and Modulation Instability of a Nonlinear Schrödinger Equation. TJMCS. 2026;18:683–699.

MLA

Hizal, Şeyma Firdevs, et al. “Presentation of Soliton Solutions and Modulation Instability of a Nonlinear Schrödinger Equation”. Turkish Journal of Mathematics and Computer Science, vol. 18, no. 3, Oct. 2026, pp. 683-99, doi:10.47000/tjmcs.1848499.

Vancouver

1.Şeyma Firdevs Hizal, Abdullahi Yusuf, Hasan Bulut. Presentation of Soliton Solutions and Modulation Instability of a Nonlinear Schrödinger Equation. TJMCS. 2026 Oct. 1;18(3):683-99. doi:10.47000/tjmcs.1848499