Mathematical modeling of influence of multiplicative white noise on dynamical soliton solutions in the KdV equation

Abstract

This study investigates the impact of multiplicative white noise on soliton solutions of the Korteweg–de Vries (KdV) equation, a classical model for nonlinear wave propagation. While the deterministic KdV framework effectively captures soliton dynamics in idealized settings, real-world systems often experience stochastic fluctuations that can significantly affect wave behavior. By introducing space–time-dependent multiplicative noise, we formulate a stochastic KdV (SKdV) equation and derive analytical expressions for both single and multi-soliton solutions using a novel wave transformation in conjunction with the Hirota bilinear method. Numerical simulations across varying noise intensities reveal that stochastic perturbations lead to soliton deformation, amplitude modulation, phase shifts, and disrupted coherence. These effects become more pronounced with stronger noise, altering soliton interactions and long-term stability. The findings emphasize the importance of incorporating stochastic effects into non-linear wave models, with implications for applications in fluid dynamics, plasma physics, and optical communications.