  Dynamics of vector dark solitons in two-component Bose-Einstein condensates
is studied within the framework of the coupled one-dimensional nonlinear
Schr\"odinger (NLS) equations. We consider the small amplitude limit in which
the coupled NLS equations are reduced to the coupled Korteweg-de Vries (KdV)
equations. For a specific choice of the parameters the obtained coupled KdV
equations are exactly integrable. We find that there exist two branches of
(slow and fast) dark solitons corresponding to the two branches of the sound
waves. Slow solitons, corresponding to the lower branch of the acoustic wave
appear to be unstable and transform during the evolution into the stable fast
solitons (corresponding to the upper branch of the dispersion law). Vector dark
solitons of arbitrary depths are studied numerically. It is shown that
effectively different parabolic traps, to which the two components are
subjected, cause instability of the solitons leading to splitting of their
components and subsequent decay. Simple phenomenological theory, describing
oscillations of vector dark solitons in a magnetic trap is proposed.
