Analysis of time-fractional Schrödinger equation with group velocity dispersion coefficients and second-order spatiotemporal effects: a new Kudryashov approach

This study investigates the application of the novel Kudryashov approach to a time-fractional nonlinear Schrödinger model featuring second-order spatiotemporal and group velocity dispersion coefficients. Various exact solutions for this model in optical fibers are established, utilizing hyperbolic and exponential functions. These solutions encompass diverse optical solitons, such as bright, singular, bell-shaped, mixed dark-bright, dark-bright, and wave solitons. To assess the significance of the time-fractional nonlinear Schrödinger model and illustrate the different forms of these innovative optical solutions, contour plots, three-dimensional plots, and two-dimensional plots are presented. Furthermore, the influence of the conformable fractional order derivative on a specific category of the new optical solutions is explored through illustrative graphs, emphasizing the impact of fractional parameters. The primary objective of this paper is to elucidate the significant influence of the conformable fractional derivative parameter on the Schrödinger equation, particularly in shaping various physical aspects of signal propagation in optical fiber. Understanding and manipulating this parameter provide opportunities for optimizing optical fiber systems for specific applications. Moreover, the proposed technique demonstrates its reliability as a tool for examining analytical solutions of fractional differential equations. The introduced Schrödinger model holds potential applications in the transmission of ultra-fast pulses through optical fibers.