In recent years, stochastic traveling waves have become a major area of interest in the field of stochastic PDEs. Various approaches have been to introduced to study the effects of noise on traveling waves, mainly in the setting of Reaction-Diffusion equations. Of particular interest is the notion of a stochastic wave position and its dynamics. This talk focusses on solitary waves in the Korteweg-de Vries equation. Due to a scaling symmetry, this dispersive PDE supports a solitary wave family of various amplitudes and velocities. We introduce stochastic processes that track the amplitude and position of solitons under the influence of multiplicative noise over long time-scales. Our method is based on a rescaled frame and stability properties of the solitary waves. We formulate expansions for the stochastic soliton amplitude and position, and compare their leading-order dynamics with numerical simulations.

This is joint work with Prof. H.J. Hupkes