Exact soliton solutions of the perturbed nonlinear Schrödinger equation with conformable fractional derivative by two distinct methods

This study investigates the perturbed nonlinear Schr & ouml;dinger equation (PNLSE) in its conformable time fractional form, a model critical for understanding nonlinear optical phenomena and wave propagation in dispersive media. By employing the B & auml;cklund transformation and complete polynomial discrimination system, we derive new exact soliton solutions in exponential, trigonometric, hyperbolic, and rational forms. These solutions are validated using symbolic computation, ensuring their mathematical and physical consistency. The inclusion of Kerr law nonlinearity enhances the model's applicability to optical fibers, where nonlinear effects such as self-phase modulation and dispersion play a crucial role. Graphical simulations (3D, contour, and density plots) are provided to visualize the dynamic evolution of the soliton solutions, offering insights into their physical interpretations. The methods and results presented here are novel and represent a significant contribution to the field of nonlinear physics.