Mathematical Analysis of Stochastic Solitons in the Kaup–Newell Equation for Birefringent Fibers

In this study, we investigate the stochastic Kaup–Newell equation, which models various optical phenomena, including soliton dynamics, four-wave mixing, self-phase modulation, wave propagation in nonlinear optical fibers, and stimulated Raman scattering. To analyze this model, we employ the Improved Nonlinear Riccati Equation Method (INREM) and Wang’s direct mapping method, deriving soliton solutions and identifying key constraint conditions for their existence. Graphical representations of selected cases are provided using Mathematica to generate 3D surface plots, contour diagrams, and 2D plots, facilitating a clearer visualization of solution behaviors and the effects of stochasticity. The accuracy of the derived solutions is further verified using computational sotware. The results demonstrate the effectiveness of the proposed methods in systematically obtaining soliton, trigonometric, hyperbolic, exponential solutions and establish a robust mathematical framework for addressing a wide range of nonlinear partial differential equations in applied science and engineering. These findings reveal novel behaviors not previously reported in the literature.