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.