Soliton solutions for time-fractional (3+1)-dimensional nonlinear Schrödinger equation with cubic-quintic nonlinearity terms

This paper investigates the time-fractional extended (3+1)-dimensional nonlinear conformable Schr & ouml;dinger equation in a dispersion and nonlinearity managed fiber laser. By utilizing the new direct mapping method and the unified Riccati equation technique, we derive various optical soliton solutions for the nonlinear Schr & ouml;dinger equation with conformable derivative. The importance of these newly constructed soliton solutions is demonstrated through contour, three-dimensional, and two-dimensional graphs, demonstrating dark, bell-shaped, bright, and wave soliton formation. Further, the effect of the fractional order parameter and the temporal parameter on these solutions is analyzed, offering valuable insights into the conformable nonlinear Schr & ouml;dinger model. The algorithms developed in this study hold promise for broader application across various nonlinear Schr & ouml;dinger equations in fields such as nonlinear optics and applied mathematics. Ultimately, this study deepens our understanding of nonlinear optics by uncovering new soliton dynamics and their governing mechanisms within dispersion and nonlinearity-managed fiber laser systems.