Doubly periodic vortices for a Chern-Simons model

In this paper we establish the existence of vortex solutions for a relativistic self dual U(1) x U(1) Chern-Simons system over a doubly periodic domain. We use a constrained minimization approach to obtain the existence result, the system admits a solution when the Chern-Simons coupling parameter k > 0 is sufficiently small; while no solution exists when k > 0 is not sufficiently small. And we also get the quantized integral. (C) 2017 Published by Elsevier Inc.