Optical soliton solutions of the M-fractional paraxial wave equation

This research used a modified and extended auxiliary mapping method to examine the optical soliton solutions of the truncated time M-fractional paraxial wave equation. We employed the truncated time M-fractional derivative to eliminate the fractional order in the governing model. The few optical wave examples of the paraxial wave condition can assume an insignificant part in depicting the elements of optical soliton arrangements in optics and photonics for the investigation of different actual cycles, including the engendering of light through optical frameworks like focal points, mirrors, and fiber optics. We identified the solution using a few free parameters regarding hyperbolic function form. We discovered periodic wave, bright and dark kink wave, bell wave, and singular soliton solution for the numerical values of the free parameters. To explain the behavior of various solutions, we have spoken the obtained solutions graphically for a physical explanation using MATLAB. The strategy introduced is fundamental and robust as a smart soliton solution for nonlinear partial differential equations, and it may play a crucial role in nonlinear optics, fiber optics, and communication systems.