Brownian motion of spatial solitons

Spatial solitons are localized excitations of electromagnetic field. In some cases, for example in optical cavity, the position of spatial soliton is affected by phase inhomogeneties. The trajectories of spatial soliton are disturbed by noisy kicks:

Spatiotemporal evolution of spatial solitons is governed by generalized Ginzburg - Landau equation on random background:

The phase inhomogeneties are supposed to be Rayleigh random field :