Stochastic solutions of the geophysical KdV equation: Numerical simulations and white noise impact

This research explored stochastic soliton and periodic wave (SPW) solutions for the geophysical Korteweg-de Vries (KdV) equation with variable coefficients, incorporating the effects of Earth's rotation, fluid stratification, and topographical variations. The classical KdV equation, widely used to model nonlinear wave propagation, was extended to describe geophysical wave dynamics in atmospheric and oceanic systems. Exact solutions for both the deterministic and Wick-type stochastic (W-TS) forms of the geophysical KdV equation were obtained using white noise (WN) theory, the Hermite transform (HT), and the exp-function method. By employing the HT, the stochastic equation was transformed into a deterministic counterpart, facilitating the derivation of novel SPW solutions expressed as rational functions involving exponential terms. The inverse HT is then applied to retrieve stochastic SPW solutions under Gaussian WN conditions. Numerical analysis highlights the influence of Brownian motion (B-M) on the formation and behavior of SPWs in geophysical settings. Additionally, numerical simulations illustrate how random fluctuations affect wave stability and evolution, offering deeper insights into nonlinear wave interactions in oceanic and atmospheric environments.