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

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.