Dynamics of vector dark soliton of coupled nonlinear
Schrödinger equations. Application to
two-component Bose-Einstein condensates
We report recent studies of dynamics of dark solitons in two-component Bose-Einstein condensates. In the case of a cigar-shaped condensate with relatively low density the system of coupled 1D Gross-Pitaevskii equations is reduced to the system of effective 1D coupled nonlinear Schrödinger (CNLS) equations. As first step, we study the small amplitude limit of the CNLS which is reduced to the coupled Korteweg-de Vries (KdV) equations. For a specific choice of the parameters when the CNLS are reduces to the integrable Manakov system we obtain integrable coupled KdV equations. We find that there exist two branches of dark solitons corresponding to two branches of the sound waves. Slow solitons, however appear to be unstable and transform during the evolution to the stable upper branch solitons. Oscillations of solitons in a parabolic trap are studied numerically.