  Vortices in non-Abelian gauge field theory play important roles in
confinement mechanism and are governed by systems of nonlinear elliptic
equations of complicated structures. In this paper, we present a series of
existence and uniqueness theorems for multiple vortex solutions of the
non-Abelian BPS vortex equations over R^2 and on a doubly periodic domain. Our
methods are based on calculus of variations and a fixed-point argument. The
necessary and sufficient conditions for the existence of a unique solution in
the doubly periodic situation are explicitly expressed in terms of several
physical parameters involved.
