Nuclear Physics B 846 [PM] (2011) 650–676 www.elsevier.com/locate/nuclphysb Sharp existence and uniqueness theorems for non-Abelian multiple vortex solutions Chang-Shou Lin a , Yisong Yang b,∗ a Department of Mathematics, National Taiwan University,Taipei, Taiwan 10617, ROC b Department of Mathematics, Polytechnic Institute of New York University, Brooklyn, NY 11201, USA Received 29 December 2010; accepted 19 January 2011 Available online 22 January 2011 Abstract Vortices in non-Abelian gauge field theory play essential roles in the mechanism of color confinement and are governed by systems of nonlinear elliptic equations of complicated structure. In this paper, we present a series of sharp existence and uniqueness theorems for multiple vortex solutions of the non-Abelian BPS equations over R 2 and on a doubly periodic domain. Our methods are based on calculus of variations which may be used to analyze more extended problems. The necessary and sufficient conditions for the existence of a unique solution in the doubly periodic situation are expressed in terms of physical parameters involved explicitly.  2011 Elsevier B.V. All rights reserved. Keywords: Non-Abelian gauge field theory; BPS vortices; Confinement; Higgs condensed solitons; Existence and uniqueness 1. Introduction Vortices have important applications in many fundamental areas of physics including super- conductivity [1,15], particle physics [14], optics [5], and cosmology [29]. The first and also the best-known rigorous multiple vortex construction in gauge field theory is due to Taubes [15,26, 27] regarding the existence and uniqueness of static solutions of the Abelian Higgs model or the Ginzburg–Landau model [11] governed by the energy density * Corresponding author. E-mail address: yisongyang@gmail.com (Y. Yang). 0550-3213/$ – see front matter  2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2011.01.019