Modulation instability, stochastic soliton dynamics, and analytical solutions in the Fokas-Lenells equation with quadratic-cubic nonlinearity

This study explores the stochastic dynamics of optical solitons in the Fokas-Lenells equation (FLE) with nonlinear chromatic dispersion and generalized quadratic-cubic self-phase modulation (SPM). As a non-integrable extension of the nonlinear Schrödinger equation (NLSE), the FLE provides a robust framework for modeling ultrashort pulse propagation in optical fibers, where higher-order nonlinearities, stochastic perturbations, and modulation instability (MI) play pivotal roles. We analytically solve this system using the modified extended direct algebraic method (MEDAM), which systematically generates a hierarchy of exact solutions under stochastic noise, an inherent feature of real-world fiber-optic environments. Our results encompass bright, dark, and singular solitons, as well as singular periodic waves, exponential solutions, periodic waves, Jacobi elliptic solutions, Weierstrass elliptic functions, and hyperbolic solutions, revealing the interplay between nonlinearity, dispersion, and noise. Key advancements include modulation instability analysis, where we derive the MI gain spectrum and stability criteria, identifying parameter regimes where stochastic noise amplifies or suppresses MI-driven soliton formation. Noise-resistant solitons, for which the MEDAM demonstrates exceptional efficacy in navigating the mathematical intricacies of stochastic NLSE-type equations, preserve physical constraints while ensuring analytical tractability. Stability, for which we delineate critical parameter regimes governing the existence of each solution class, offers novel perspectives on noise-resistant soliton propagation and MI-induced energy localization in next-generation photonic technologies. This work bridges theoretical rigor with practical insights, advancing the understanding of stochastic soliton dynamics and MI control in nonlinear optical systems.