Optical soliton solutions of the stochastic perturbed Fokas–Lenells equation having the parabolic law of self-phase modulation in the presence of spatio-temporal dispersion with multiplicative white noise

In this study, we examine the perturbed Fokas–Lenells equation, incorporating the parabolic law of self-phase modulation along with spatio-temporal dispersion, under the influence of multiplicative white noise using Itô calculus The analysis unfolds in multiple stages. Initially, the nonlinear ordinary differential equation form is derived through a complex wave transformation. Next, we employ the new Kudryashov method (nKM) and the unified Riccati equation expansion method (UREEM) to extract bright, kink, and dark optical solitons. Subsequently, we explore how noise strength affects the dynamics of the soliton for each obtained soliton type. To reinforce our findings, we present solution functions through effective graphical simulations. Observations regarding the influence of stochastic term parameters are discussed in the relevant section. The validity of our results is confirmed by demonstrating their consistency with the governing equation. Notably, these methodologies have not been previously applied to the Fokas–Lenells equation incorporating a stochastic function to assess noise effects. The distinctive nature of the problem has led to the discovery of numerous novel solutions and their behavior under noise influence.