  The propagation of a wave-packet in a nonlinear disordered medium exhibits
interesting dynamics. Here, we present an analysis based on the nonlinear
Schr\"odinger equation (Gross-Pitaevskii equation). This problem is directly
connected to experiments on expanding Bose gases and to studies of transverse
localization in nonlinear optical media. In a nonlinear medium the energy of
the wave-packet is stored both in the kinetic and potential parts, and details
of its propagation are to a large extent determined by the transfer from one
form of energy to the other. A theory describing the evolution of the
wave-packet has been formulated in [G. Schwiete and A. Finkelstein, Phys. Rev.
Lett. 104, 103904 (2010)] in terms of a nonlinear kinetic equation. In this
paper, we present details of the derivation of the kinetic equation and of its
analysis. As an important new ingredient we study interparticle-collisions
induced by the nonlinearity and derive the corresponding collision integral. We
restrict ourselves to the weakly nonlinear limit, for which disorder scattering
is the dominant scattering mechanism. We find that in the special case of a
white noise impurity potential the mean squared radius in a two-dimensional
system scales linearly with t. This result has previously been obtained in the
collisionless limit, but it also holds in the presence of collisions. Finally,
we mention different mechanisms through which the nonlinearity may influence
localization of the expanding wave-packet.
