Stable and unstable vector dark solitons of coupled nonlinear   Schr\"odinger equations. Application to two-component Bose-Einstein   condensates
  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. 
V.A. Brazhnyi, V.V. Konotop