  Bright solitary waves in a Bose-Einstein condensate contain thousands of
identical atoms held together despite their only weakly attractive contact
interactions. They nonetheless behave like a compound object, staying whole in
collisions, with their collision properties strongly affected by inter-soliton
quantum coherence. We show that separate solitary waves decohere due to phase
diffusion, dependent on their effective ambient temperature, after which their
initial mean-field relative phases are no longer well defined or relevant for
collisions. In this situation, collisions occur predominantly repulsively and
can no longer be described within mean field theory. When considering the
time-scales involved in recent solitary wave experiments where non-equilibrium
phenomena play an important role, these features could explain the
predominantly repulsive collision dynamics observed in most condensate soliton
train experiments.
