Modulation instability and comparative observation of the effect of fractional parameters on new optical soliton solutions of the paraxial wave model

This work focuses on the study of the paraxial wave model with a space–time fractional form. This model has more importance for describing light propagation in nonlinear optical fibers and telecommunication lines. The main aim of this work is to observe the effect of the fractional parameters and compare the truncated M -fraction with the beta-fraction, conformable fraction form, and classical form of the PW model. For this observation, we applied the Simplest equation technique to acquire analytical solutions to the space–time (spatiotemporal) M -fractional paraxial wave model. We are able to acquire several new optical soliton solutions, including periodic waves, kink-type waves, rogue-type waves, and several novel periodic waves, by providing the appropriate fractional parametric values. These solutions have significance for shedding light on a number of physical phenomena in the realms of optical fiber and communication sciences. The diverse values of fractional parameters and the three-dimensional and contour plot graphs of certain chosen solutions are depicted, which are the most accurate physical characterizations of the outcomes. We also sketch the comparative graph of diverse fractional forms and the classical form of the paraxial wave equation in two-dimensional plots. Consequently, our findings represent an important breakthrough in this complex area and help further develop our comprehension of the behavior of solitons.