Fractional Stochastic Biswas–arshed Equation for Understanding Stochastic Soliton Phenomena in Optical Propagation

Stochastic partial differential equations (SPDEs) have become key tools for modeling randomly disturbed systems in a broad spectrum of scientific and engineering disciplines. Among them, the optical soliton propagation in birefringent fibers has been of great interest because of the natural stochastic effects present in real optical communication systems. Although some research has been conducted on soliton dynamics in deterministic and integer-order models, the literature has few studies focusing on fractional stochastic frameworks coupled with M-truncated fractional operators. This paper fills this gap by proposing and studying the fractional stochastic Biswas–Arshed equation (FSBAE) that includes multiplicative white noise to represent random perturbations during optical signal propagation. We employ the Formula: see text method with Itô calculus and M-truncated fractional derivatives: in order to arrive at three different classes of solutions: stochastic optical breather solitons, M-shaped solitons, and singular solitons. The first contribution of this work consists in the combination of M-truncated fractional calculus — a not extensively developed operator in stochastic nonlinear optics — within the FSBAE, for the first time applied to this model. Comparative graphical analysis under conditions of different levels of white noise and fractional orders also identifies the robustness and stability of the solutions in close proximity to the zero-noise limit. This research adds to the development of soliton theory in that it has proposed a new mathematical model that can capture both nonlocal memory effect and stochastic variability, which are essential in real optical systems. The efficiency, simplicity, and flexibility of the approach make it an effective tool in solving a wider class of nonlinear stochastic issues in optical engineering and other fields dealing with stochastic partial differential equations.