Random Exact Solutions for the Stochastic Korteweg-de Vries Equation

Abstract

This paper considers the stochastic Korteweg-de--de Vries (SKdV) equation perturbed by multiplicative Brownian motion, which is an important model reflecting the nonlinear science. After a systematic change and a rescaling, the SKdV equation is exactly recast into a deterministic KdV equation with random variable coefficients (KdV-RVCs). By using the Jacobi elliptic equation method and the generalized Riccati equation mapping approach, we obtain new exact solutions (rational, hyperbolic, trigonometric, and elliptic) for the KdV-RVCs. The latter are then used to form stochastic solutions for the SKdV equation. Of practical interest, these results are related to specific physical systems: magnetized plasmas in astrophysics and in 1D/2D fusion, soliton propagation in fiber optical communication, and surface-wave dynamics in fluid mechanics. For example, the resulting solutions explain how noise-induced perturbations change soliton propagation in optical fibers and stabilize wave patterns in Turbulence. To give some (visual) impression of how multiplicative noise influences the solution behavior, we use pictures of the probability density distributions and ensemble-averaged trajectories as examples. These findings show that multiplicative Brownian motion has a stabilizing effect on the SKdV solutions by keeping their variations more or less near zero. This connection between stochastic modeling and experimental observations in plasma turbulence, nonlinear optical signal processing, and fluid wave dynamics has implications for the development of predictive theories for noise-driven nonlinear systems.