An analytical investigation of nonlinear time-fractional Schrödinger and coupled Schrödinger–KdV equations

This study applies the Fractional Reduced Differential Transform Method (FRDTM) to solve two nonlinear fractional equations: the time-fractional Schrödinger equation (TFSE) and the coupled Schrödinger–KdV (Sch–KdV) equation, which are prominent in quantum mechanics, plasma physics, and wave propagation studies. These equations model various wave phenomena, including dust-acoustic waves, electromagnetic waves, and Langmuir waves, in plasma physics. Using the Liouville–Caputo fractional derivative, FRDTM effectively incorporates nonlocality and memory effects. The solutions are derived as rapidly convergent infinite series with terms that are straightforward to compute, ensuring both accuracy and efficiency. Numerical examples highlight the method’s ability to closely approximate exact solutions, with graphical and tabular comparisons illustrating the effect of the fractional order on solution behavior and validating the approach through computed absolute errors. Additionally, discussions on the modulation instability of the models reinforce the robustness of FRDTM as a powerful tool for solving complex nonlinear fractional systems.