This article introduces new optical soliton solutions for a fractional version of the quadratic–cubic nonlinear Schrödinger equation that describes the transmission of optical pulses in fiber optic systems with superfast fibers. The solutions are obtained using the modified Sardar sub-equation method and the \((\frac{1}{\varphi (\zeta )},\frac{\varphi ^{'}(\zeta )}{\varphi (\zeta )})\) method. The model is transformed into a non-linear fractional partial differential equation with a non-integer order using the conformable derivative, which is an efficient fractional derivative. The proposed methods work by adding a new variable to the equation to convert its form into a non-linear equation with ordinary derivatives. A comparative analysis of the solutions is carried out, and the effect of varying the fractional parameter values on the behavior of the obtained solutions is investigated. The paper’s novelty lies in the fact that no previous articles have identified the new solutions obtained through the application of these two analytical methods. The acquired solutions include kink, bright, periodic, dark-bright, singular and dark-singular wave solutions, which are illustrated using several 3D and 2D graphs. Furthermore, to effectively validates the exactness of analytical solutions with an elevated precision, a numerical method known as differential transform method is carried out. The adopted approaches demonstrate notable performance and are suitable for solving other nonlinear partial differential equations that arise in the natural sciences.
