  We provide a theoretical explanation of the results on the intensity
distributions and correlation functions obtained from a random beam speckle
field in nonlinear bulk waveguides reported in the recent publication Y.
Bromberg et.al., Nat. Photonics \textbf{4}, 721 (2010). We study both focusing
and defocusing case and in the limit of small speckle size (short correlated
disordered beam) provide analytical asymptotes for the intensity probability
distributions at the output facet. Additionally we provide a simple relation
between the speckle sizes at the input and output of a focusing nonlinear
waveguide. The results are of practical significance for the nonlinear Hanbury
Brown and Twiss interferometry both in optical waveguides and Bose-Einstein
condensates.
