This study investigates the stability properties of solutions to time–fractional nonlinear Schrödinger equations, which model the time evolution of disturbances in complex media. The primary objective is to establish Hyers–Ulam and Hyers–Ulam–Rassias stability criteria for solutions to these equations by incorporating the time–fractional \(\beta \) –time derivative. A key research question explored in this work is how the inclusion of the time–fractional \(\beta \) –time derivative influences the stability behavior of solutions to these time–fractional Schrödinger equations. To address this, we employ the fixed–point method, a powerful analytical tool, to derive novel stability results. The theoretical findings are validated through numerical simulations and concrete examples, demonstrating their applicability and effectiveness in real–world scenarios. The results indicate that the proposed framework not only extends existing stability analyses but also provides deeper insights into the behavior of disturbances in fractional–order nonlinear systems. Incorporating the time–fractional \(\beta \) –time derivative introduces new stability characteristics, refining the theoretical understanding of these time–fractional Schrödinger equations. This work represents the first extension of stability analysis to solutions of time–fractional nonlinear Schrödinger equations incorporating the time–fractional \(\beta \) –time derivative, addressing a critical gap in the literature. The findings have significant implications for applications in quantum mechanics, optical solitons, signal processing, and other fields where fractional–order models are crucial for capturing memory and hereditary effects.
