  Stochastically perturbed Korteweg-de Vries (KdV) equations are widely used to
describe the effect of random perturbations on coherent solitary waves. We
present a collective coordinate approach to describe the effect on coherent
solitary waves in stochastically perturbed KdV equations. The collective
coordinate approach allows one to reduce the infinite-dimensional stochastic
partial differential equation (SPDE) to a finite-dimensional stochastic
differential equation for the amplitude, width and location of the solitary
wave. The reduction provides a remarkably good quantitative description of the
shape of the solitary waves and its location. Moreover, the collective
coordinate framework can be used to estimate the time-scale of validity of
stochastically perturbed KdV equations for which they can be used to describe
coherent solitary waves. We describe loss of coherence by blow-up as well as by
radiation into linear waves. We corroborate our analytical results with
numerical simulations of the full SPDE.
