Dual-Soliton Structures and Stability Analysis in a Nonlinear Fractional Schrodinger Equation with Dual-Mode Dispersion Using Modified Extended Mapping Method

This work addresses the critical challenge of obtaining exact solutions for the β-fractional dual-mode nonlinear Schrödinger equation (NLSE). Understanding this model is fundamentally important as it governs wave propagation in complex media exhibiting simultaneous nonlinearity, dispersion, and fractional order effects, with direct applications in nonlinear optics and quantum physics. To tackle this problem, we employ the powerful and systematic modified extended mapping method (MEMM). Furthermore, we conduct a comprehensive modulation instability analysis to rigorously assess the dynamical stability of the governing model. Our investigation yields a novel and distinct family of exact solutions. Key results include the derivation of bright, dark, singular, and combined bright-dark soliton solutions, alongside a spectrum of other wave patterns such as Jacobi elliptic, hyperbolic, and periodic functions. Graphical analysis reveals the profound and controllable influence of the fractional-order derivative (β) on the amplitude, width, and propagation dynamics of these waves. The stability analysis confirms the robustness of the model under small perturbations. We conclusively demonstrate that the β-fractional dualmode NLSE supports a wide variety of stable, exact wave solutions. The fractional parameter β acts as a crucial tuning knob for wave modulation. These validated solutions provide a reliable theoretical foundation for predicting and engineering wave behaviors in fractional physical systems. The novelty of this work lies in the first successful application of the MEMM to the β-fractional dual-mode NLSE, revealing previously unreported solution classes. It significantly extends the literature by moving beyond standard solitons to uncover combined structures and fractional-dependent dynamics, thereby offering deeper analytical insight and enhanced control over nonlinear wave phenomena for advanced technological applications.