Abstract
A statistical approach of the propagation of solitons in media with spatially random dispersive perturbations is developed. Applying the inverse scattering transform several regimes are put into evidence which are determined by the mass and the velocity of the incoming soliton and also by the correlation length of the perturbation. Namely, the mass of the soliton is almost conserved if it is initially large. If the initial mass is too small, then the mass decays with the length of the system. The decay rate is exponential in case of a white noise perturbation, but the mass will decrease as the inverse of the square root of the length if the central wave number of the soliton lies in the tail of the spectrum of the perturbation.
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