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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

A Variety of Optical Soliton Solutions of DNLSE via Complete Discrimination System for Polynomial Method

Journal of Applied Nonlinear Dynamics 14(4) (2025) 757--776 | DOI:10.5890/JAND.2025.12.001

Syed Tahir Raza Rizvi$^1$, Kashif Ali$^1$, Noor Aziz$^1$, Aly. R. Seadawy$^{2\dagger}$

$^1$ Department of Mathematics, COMSATS University Islamabad Lahore Campus, Lahore, 1.5 KM Defence Road, Off Raiwind Road, Lahore, Pakistan

$^2$ Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, 41411, Kingdom of Saudi Arabia

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Abstract

The aim of this research is to demonstrate that bifurcation, topological properties and critical conditions of nonlinear dynamics can be clearly understood using complete discrimination system for polynomial method (CDSPM). Our conclusion is supported by an example of derivative nonlinear Schr\"{o}dinger equation (DNLSE) with quintic nonlinearity, which describes how femtosecond pulses travel across nonlinear optical fibre. The results show that CDSPM is not only implemented to achieve the quantitative results such as classification of travelling waves, but also to carry out qualitative analysis of nonlinear differential equations. Jacobian elliptic solutions, hyperbolic function solutions, rational solutions are a few of the optical soliton solutions of DNLSE that are obtained by CDSPM. The qualitative analysis highlights the equilibrium points and the phase portraits of the system. Phase portraits are graphical representations applied in dynamical structures to depict the periodic behavior and the stability of the system. Our findings have broad applications in nonlinear optics, communication physics and other related domains. These chirps either amplify or compress the wave signals in optical fibers and nonlinear electrical transmission lines. The newly discovered chirped soliton solutions of DNLSE are useful in comprehending phenomena when waves are controlled by this kind of equation.

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