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Generalized Modified Unstable Nonlinear Schrödinger’s Equation: Optical Solitons and Modulation Instability

by Jamilu Sabi’u, Ibrahim Sani Ibrahim, Khomsan Neamprem, Surattana Sungnul and Sekson Sirisubtawee

Mathematics 2025, 13(12), 2032; https://doi.org/10.3390/math13122032 - 19 Jun 2025

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This paper proposes the generalized modified unstable nonlinear Schrödinger’s equation with applications in modulated wavetrain instabilities. The extended direct algebra and generalized Ricatti equation methods are applied to find innovative soliton solutions to the equation. The solutions are obtained in the form of elliptic, hyperbolic, and trigonometric functions. Moreover, a Galilean transformation is used to convert the problem into a dynamical system. We use the theory of planar dynamical systems to derive the equilibrium points of the dynamical system and analyze the Hamiltonian polynomial. We further investigate the bifurcation phase portrait of the system and study its chaotic behaviors when an external force is applied to the system. Graphical 2D and 3D plots are explored to support our mathematical analysis. A sensitivity analysis confirms that the variation in initial conditions has no substantial effect on the stability of the solutions. Furthermore, we give the modulation instability gain spectrum of the considered model and graphically indicate its dynamics using 2D plots. The reported results demonstrate not only the dynamics of the analyzed equation but are also conceptually relevant in establishing the temporal development of modest disturbances in stable or unstable media. These disturbances will be critical for anticipating, planning treatments, and creating novel mechanisms for modulated wavetrain instabilities. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations, 2nd Edition)

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15 pages, 3317 KiB

Open AccessArticle

Construction of Novel Bright-Dark Solitons and Breather Waves of Unstable Nonlinear Schrödinger Equations with Applications

by Ambreen Sarwar, Muhammad Arshad, Muhammad Farman, Ali Akgül, Iftikhar Ahmed, Mustafa Bayram, Shahram Rezapour and Manuel De la Sen

Symmetry 2023, 15(1), 99; https://doi.org/10.3390/sym15010099 - 30 Dec 2022

Cited by 9 | Viewed by 2513

Abstract

The unstable nonlinear Schrödinger equations (UNLSEs) are universal equations of the class of nonlinear integrable systems, which reveal the temporal changing of disruption in slightly stable and unstable media. In current paper, an improved auxiliary equation technique is proposed to obtain the wave results of UNLSE and modified UNLSE. Numerous varieties of results are generated in the mode of some special Jacobi elliptic functions and trigonometric and hyperbolic functions, many of which are distinctive and have significant applications such as pulse propagation in optical fibers. The exact soliton solutions also give information on the soliton interaction in unstable media. Furthermore, with the assistance of the suitable parameter values, various kinds of structures such as bright-dark, multi-wave structures, breather and kink-type solitons, and several periodic solitary waves are depicted that aid in the understanding of the physical interpretation of unstable nonlinear models. The various constructed solutions demonstrate the effectiveness of the suggested approach, which proves that the current technique may be applied to other nonlinear physical problems encountered in mathematical physics. Full article

(This article belongs to the Section Mathematics)

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