  We establish the uniqueness of ground states of some coupled nonlinear
Schrodinger systems in the whole space. We firstly use Schwartz symmetrization
to obtain the existence of ground states for a more general case. To prove the
uniqueness of ground states, we use the radial symmetry of the ground states to
transform the systems into an ordinary differential system, and then we use the
integral forms of the system. More interestingly, as an application of our
uniqueness results, we derive a sharp vector-valued Gagliardo-Nirenberg
inequality.
