Soliton solutions and sensitive analysis to nonlinear wave model arising in optics

Abstract In this study, we use analytical algorithms, specifically the auxiliary equation (AE) approach, the improved F -expansion method, and the modified Sardar sub-equation (MSSE) method to investigate complex wave structures for plentiful solutions associated with the fractional perturbed Gerdjikov-Ivanov (PGI) model with the M -fractional operator. The investigated model is a well-established mathematical model used to represent a variety of physical events in nonlinear dynamics and mathematical physics. By using the aforementioned techniques, we scrutinize some new optical wave solutions in the framework of dark, bright, periodic, combo, W -shaped, M -shape, V -shape, kink type, singular rational, exponential, trigonometric, and hyperbolic solutions. The acquired solutions address a wide range of optical solutions in the form of 3D plots, contour plots, and 2D plots, declaring the free parameters of such optical soliton solutions and comprehending their dynamic behavior. Also, the sensitive analysis of the selected model is analyzed. The main contribution of this study is to extract diverse solitary wave solutions of the adopted model. Some of the solutions are similar and some diverge from the previous solutions which justifies the novelty of the study. Finally, we discovered that the current technique provides a reliable instrument for investigating the analytic solutions of fractional differential equations. The proposed PGI model can be used to transmit ultra-fast pulses across optical fibers. This research goes beyond to the advancement of mathematical techniques for solving fractional differential equations and broadens their application to a wide range of real-world scientific and engineering problems.