  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.
