In this article, we explored the stochastic nonlinear reaction-diffusion (RD) equation under the influence of multiplicative white noise. To obtain novel soliton solutions, we employed two powerful analytical techniques: the unified Riccati equation expansion method and the modified Kudryashov method. These methods yield a diverse set of soliton solutions, including combo-dark solitons, dark solitons, singular solitons, combo-bright-singular solitons, and periodic wave solutions. We also performed a comprehensive stability analysis of the stochastic nonlinear RD equation with multiplicative white noise. The findings provide valuable insights into the behavior of solitons in stochastic nonlinear systems, with significant implications for fields such as mathematical physics, nonlinear science, and applied mathematics. These results hold particular relevance for soliton dynamics in optical physics, where they can be applied to improve understanding of wave propagation in noisy environments, signal transmission, and the design of robust optical communication systems.

This paper investigates the (3+1)-dimensional nonlinear Schrö dinger equation, incorporating cross-spatial dispersion and a generalized form of Kudryashov's self-phase modulation. Using the generalized Jacobi elliptic method, we systematically derive novel soliton solutions expressed in terms of Jacobi elliptic and Weierstrass elliptic functions, providing deeper insights into wave dynamics in nonlinear optical media. The obtained solutions exhibit diverse structural transformations governed by the parameter (n) known as full nonlinearity, encompassing optical bullet solutions, optical domain wall solutions, singular solitons, and periodic solutions. Furthermore, we discuss the potential experimental realization of these solitonic structures in ultrafast fiber lasers and nonlinear optical systems, drawing connections to recent experimental findings. To facilitate a comprehensive understanding of their physical properties, we present detailed three-dimensional (3D), two-dimensional (2D), and contour visualizations, highlighting the interplay among dispersion, nonlinearity, and self-modulation effects. These results offer new perspectives on soliton interactions and have significant implications for optical communication, signal processing, and nonlinear wave phenomena.

This study investigated the perturbed Gerdjikov-Ivanov equation (PGIE) affected by multiplicative noise in the Itô sense. The equation examined the influence of stochastic perturbations on its solitonic structures through an analytical and computational method. We found precise soliton solutions and examined their stability through the new Jacobi elliptic function expansion method. Subsequent to the mathematical study, graphical representations were executed. The results enhance the comprehensive understanding of nonlinear stochastic wave equations and their applications in optical fiber communications and many physical systems. Subsequently, the three-dimensional and two-dimensional model demonstrate the presence of bright and dark solitons, Jacobi-elliptic solutions, and periodic solutions for various values of $\sigma$.