Soliton propagation in optical metamaterials with nonlocal responses: A fractional calculus approach using (q,τ)-Mittag-Leffler functions

This work investigates soliton solutions of nonlinear wave equations modeling light propagation in optical metamaterials with nonlocal nonlinear responses, incorporating external optical potentials. The residual power series method (RPSM) is employed to construct enhanced analytical solutions, capturing both dispersive and memory effects effectively. In addition, this study investigates the propagation of solitons in optical metamaterials with nonlocal responses using (q,τ)-fractional calculus. This calculus is based on the generalization of the quantum gamma function ((q,τ)−Γ(.)). By employing (q,τ)-fractional derivatives in the form of the (q,τ)-Mittag-Leffler function, we explore the dynamics of soliton fields in these materials. The model considers key parameters such as the fractional order α, the generalized parameters q and τ, and the initial weight parameter β. The flexibility of these parameters allows for a more accurate description of optical metamaterials, capturing both classical soliton behavior and more complex nonlocal and memory effects. We compare fractional models with classical models and demonstrate the advantages of using fractional calculus to model memory effects and nonlocal interactions. Numerical simulations, including the residual series method, reveal the enhanced accuracy and insights provided by the fractional approach in optical metamaterials. The study provides a detailed framework for understanding soliton propagation in advanced optical materials, paving the way for the design of next-generation optical devices.