Solitary wave solutions, bifurcation theory, and sensitivity analysis in the modified unstable nonlinear Schrödinger equation

This study examines the modified unstable nonlinear Schr & ouml;dinger model, a fundamental nonlinear physical model vital for depicting optical solitary wave solutions and their evolution in dynamic fiber optics. The propagation of waves in nonlinear dispersive media is of great significance, presenting new opportunities for data processing in communication systems and generating ultrafast light pulses. Applying a wave transformation reduces the nonlinear Schr & ouml;dinger equation to an ordinary differential equation, and an extended direct algebraic method is used to derive various soliton solutions. The novelty lies in using this method to uncover various soliton forms, which are then explored through graphical representations in 2D, 3D, and contour plots. The phase portraits and bifurcation theory also qualitatively assess the undisturbed planar system. Sensitivity analysis under different initial conditions indicates how the soliton solutions adapt to changes in system parameters. The outcomes confirm that the presented approach effectively evaluates soliton solutions across diverse nonlinear models, contributing to the understanding of wave propagation in slightly stable and unstable media.