Collisions between solitary waves of three-dimensional Bose-Einstein   condensates
  We study bright solitary waves of three dimensional trapped Bose-Einstein condensates and their collisions. For a single solitary wave, in addition to an upper critical number, we also find a {\em lower} cut-off, below which no stable state can be found. Collisions between solitary waves can be elastic, inelastic with either reduced or increased outgoing speed, or completely unstable due to a collapse instability. A $\pi$-phase difference between the waves promotes elastic collisions, and gives excellent agreement with recent experimental results over long timescales. 
N.G. Parker, A. M. Martin, S. L. Cornish and C. S. Adams