  We study the propagation of ultra-short short solitons in a cubic nonlinear
medium modeled by nonlinear Maxwell's equations with stochastic variations of
media. We consider three cases: variations of (a) the dispersion, (b) the phase
velocity, (c) the nonlinear coefficient. Using a modified multi-scale expansion
for stochastic systems, we derive new stochastic generalizations of the short
pulse equation that approximate the solutions of stochastic nonlinear Maxwell's
equations. Numerical simulations show that soliton solutions of the short pulse
equation propagate stably in stochastic nonlinear Maxwell's equations and that
the generalized stochastic short pulse equations approximate the solutions to
the stochastic Maxwell's equations over the distances under consideration. This
holds for both a pathwise comparison of the stochastic equations as well as for
a comparison of the resulting probability densities.
