Abstract Since multi-scale processes and anomalous diffusion phenomena are important in various scientific fields, including biology, materials science, and physics, this work analytically examines the stochastic reaction-diffusion model. The coupled diffusion equation is solved using both analytical methods and numerical simulations. These approaches reveal emerging phenomena and provide insights into the dynamics of complex diffusing entities. The results help improve understanding in domains such as plasma physics and optical fibers by clarifying the range of solutions to this equation. A modified exponential rational function technique is used to obtain exact analytical results. This technique produces a variety of traveling wave solutions, including trigonometric, exponential, rational, and hyperbolic forms. The approach also enables the exploration of several solutions from significant physical perspectives, including soliton solution, periodic solitons, kink solitons, singular and rational solution as well as their noise term effects on Brownian motion, based on the Ito sense. The effects of the noise term on solitons are illustrated using two-dimensional (2D) and three-dimensional (3D) graphics. The suggested method is simple and effective for solving various nonlinear equations in mathematical physics. The properties of some solutions under multiplicative temporal noise are displayed using various charts.
Yasin et al. (Thu,) studied this question.