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

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  • Jamilu Sabi’u

    (Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
    Department of Mathematics, Northwest University, Kano, Nigeria)

  • Ibrahim Sani Ibrahim

    (Department of Mathematics, Northwest University, Kano, Nigeria)

  • Khomsan Neamprem

    (Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
    Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand)

  • Surattana Sungnul

    (Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
    Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand)

  • Sekson Sirisubtawee

    (Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
    Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand)

Abstract

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.

Suggested Citation

  • Jamilu Sabi’u & Ibrahim Sani Ibrahim & Khomsan Neamprem & Surattana Sungnul & Sekson Sirisubtawee, 2025. "Generalized Modified Unstable Nonlinear Schrödinger’s Equation: Optical Solitons and Modulation Instability," Mathematics, MDPI, vol. 13(12), pages 1-25, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:12:p:2032-:d:1683005
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    References listed on IDEAS

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    1. Rizvi, Syed T.R. & Seadawy, Aly R. & Ahmed, Sarfaraz & Younis, Muhammad & Ali, Kashif, 2021. "Study of multiple lump and rogue waves to the generalized unstable space time fractional nonlinear Schrödinger equation," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
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