Soliton solutions of the time-fractional nonlinear Schrödinger equation with refractive index and generalized nonlocal nonlinearity via two integration methods

This study explores the use of [Formula: see text]-expansion and [Formula: see text]-expansion methods to derive optical solutions for a conformable nonlinear Schrödinger equation. This equation features an arbitrary refractive index, as proposed by Kudryashov, and includes two different types of nonlocal nonlinearities. Several optical soliton solutions are constructed using the proposed techniques. The new soliton solutions exhibit certain physical characteristics, which are illustrated using three-dimensional and two-dimensional. Further, the behavior of these optical solutions is analyzed through illustrative graphs that account for variations in the time parameter and the conformable order derivative. The methods introduced in this paper provide accurate approaches for analyzing optical solutions in various formulations of nonlinear Schrödinger equations, contributing to the understanding of light behavior in complex optical systems. This research enhances our knowledge of nonlinear optics and has potential applications in areas like telecommunications, laser technology, and optical signal processing.