  Motivated by recent experimental observations, we study theoretically
multiple bright solitary waves of trapped Bose-Einstein condensates. Through
variational and numerical analyses, we determine the threshold for collapse of
these states. Under $\pi$-phase differences between adjacent waves, we show
that the experimental states lie consistently at the threshold for collapse,
where the corresponding in-phase states are highly unstable. Following the
observation of two long-lived solitary waves in a trap, we perform detailed
three-dimensional simulations which confirm that in-phase waves undergo
collapse while a $\pi$-phase difference preserves the long-lived dynamics and
gives excellent quantitative agreement with experiment. Furthermore,
intermediate phase differences lead to the growth of population asymmetries
between the waves, which ultimately triggers collapse.
