Existence of non-Abelian vortices with product gauge groups

In this paper we establish several sharp existence and uniqueness theorems for some non-Abelian vortex models arising in supersymmetric gauge field theories. We prove these results by studying a family of systems of elliptic equations with exponential nonlinear terms in both doubly periodic-domain and planar cases. In the doubly periodic-domain case we obtain some necessary and sufficient conditions, each explicitly expressed in terms of a single inequality interestingly relating the vortex numbers, to coupling parameters and size of the domain, for the existence of solutions to these systems. In the planar case we establish the existence results for any vortex numbers and coupling parameters. Sharp decay estimates for the planar solutions are also obtained. Furthermore, the solutions are unique, which give rise to the quantized integrals in all cases.