Advanced Analytical and Numerical Studies on Coupled Schrödinger Equations with Fractional Order Damping

This paper embarks on a thorough analytical and numerical exploration of coupled Schrödinger equations under the influence of fractional order damping mechanisms. By integrating fractional damping, which introduces memory effects and non-local dissipative interactions, into the coupled Schrödinger framework, we aim to dissect and understand the nuanced dynamics that govern these complex quantum systems. The research delves into the mathematical underpinnings, stability characteristics, and the dynamical behaviors that emerge from the intricate balance between quantum coupling and fractional damping effects. Through a blend of analytical rigor and sophisticated numerical simulations, this study unveils new insights into the complex interplay among quantum entanglement, dissipation, and non-linear dynamics, offering potential implications for quantum computing, optical systems, and beyond.