Optical solutions with Kudryashov’s arbitrary type of generalized non-local nonlinearity and refractive index via the new Kudryashov approach

This paper studies the new Kudryashov approach for deriving optical solutions for a time-fractional nonlinear Schrödinger equation. The equation incorporates Kudryashov’s arbitrary refractive index and two distinct nonlocal nonlinearities. This incorporation adds complexity and potentially broadens the applicability of the model compared to previous studies that may not have considered these factors simultaneously. The complex wave transformations are inserted into the existing time-fractional nonlinear Schrödinger equations, and a nonlinear ordinary differential equation is derived. Through the application of this approach to the nonlinear ordinary differential equation, we obtain a system of nonlinear equations in polynomial form. By solving this system, we generate multiple solution sets, each characterized by distinct values for the parameters of the analyzed equation. The resulting optical solutions are expressed in exponential and hyperbolic functions, encompassing mixed dark–bright, bell-shaped, bright, wave, and wave solitons. The graphical representations of these solutions are presented through two-dimensional, three-dimensional, and contour plots to highlight their characteristics. Furthermore, the behavior of these novel optical solutions is illustrated through graphs that depict variations in the time parameter and fractional order derivative. The proposed methodology is asserted to be a reliable approach for investigating optical solutions across a spectrum of Schrödinger equations with both fractional and integer orders. The present model provides increased versatility, heightened nonlinearity, novel soliton solutions, and enhanced predictive capabilities when compared to conventional nonlinear Schrödinger equations.