In this work, we investigated the (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation, which models long wave propagation in shallow water and plays a significant role in fluid mechanics and plasma physics. Using the improved simple equations method, we obtained various solutions, including dark, bright, and singular solitons, and combinations of singular periodic solutions and exponential rational solutions. Additionally, we performed a linear stability analysis to examine the stability properties of these wave solutions. To further illustrate their characteristics during propagation, we provided 3D and contour plots for some opted wave solutions.

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.