Optical Soliton Solutions of the (3+1)-Dimensional Stochastic Nonlinear Schr{\"o}dinger Equation via the Sine-Gordon Equation Method

This paper addresses the $(3+1)$-dimensional nonlinear Schrödinger equation in the presence of multiplicative white noise, which models the impact of stochastic perturbations on the propagation of nonlinear optical waves in dispersive media. The incorporation of noise is essential to realistically capture environmental fluctuations and inherent uncertainties in practical optical systems. To obtain exact soliton solutions under this framework, we employ the sine-Gordon equation algorithm, a powerful analytical technique known for its efficiency in transforming complex nonlinear partial differential equations into integrable forms. This methodological approach allows us to systematically derive a variety of soliton structures, including bright, dark, and singular optical solitons, under the influence of stochastic modulation. The results reveal how multiplicative white noise alters the profile, stability, and evolution of the soliton solutions, offering new perspectives on soliton dynamics in higher-dimensional stochastic environments. Moreover, the study provides explicit conditions under which stable soliton propagation is maintained despite the presence of noise. These findings have significant implications for the design and control of nonlinear optical systems, particularly in high-dimensional waveguides and optical fibers where random disturbances are unavoidable. The proposed framework advances the understanding of noise-induced phenomena in nonlinear wave theory and provides a more robust methodology for modeling real-world optical signal transmission