  We investigate the interplay between coherent effects characteristic of the
propagation of linear waves, the non-linear effects due to interactions, and
the quantum manifestations of classical chaos due to geometrical confinement,
as they arise in the context of the transport of Bose-Einstein condensates. We
specifically show that, extending standard methods for non-interacting systems,
the body of the statistical distribution of intensities for scattering states
solving the Gross-Pitaevskii equation is very well described by a local
Gaussian ansatz with a position-dependent variance. We propose a semiclassical
approach based on interfering classical paths to fix the single parameter
describing the universal deviations from a global Gaussian distribution. Being
tail effects, rare events like rogue waves characteristic of non-linear field
equations do not affect our results.
