Influence of the $\beta$-fractional derivative on optical soliton solutions of the pure-quartic nonlinear Schrödinger equation with weak nonlocality

This study investigated the dynamics of a pure-quartic nonlinear Schrödinger equation incorporating a $\beta$-fractional derivative and weak nonlocal effects prevalent in optical fiber systems. Using the improved modified extended tanh-function method, we obtained a diverse array of soliton solutions, including bright, dark, and singular solitons, as well as hyperbolic, trigonometric, and Jacobi elliptic solutions. The main goal was to clarify how fractional derivatives, defined by the parameter $\beta$, affect the characteristics and behavior of these soliton solutions. The key outcomes indicate that variations in the parameter $\beta$ lead to substantial changes in soliton amplitude, shape, and propagation patterns. Graphical illustrations clearly depict these transformations, highlighting how fractional derivatives have a major impact on the properties of solitons. Crucially, for certain fractional orders, the localization and stability of solitons are enhanced, which is essential for accurate modeling of nonlocal and dispersive effects in optical fibers. This work not only enhances fundamental understanding of nonlinear wave phenomena within optical communication systems but also offers valuable insights into using fractional calculus for designing and optimizing advanced photonic devices.