## Abstract

Motivated by the study of non-abelian Chern Simons vortices of non-topological type in Gauge Field Theory, see e.g. Gudnason (Nucl Phys B 821:151–169, 2009), Gudnason (Nucl Phys B 840:160–185, 2010) and Dunne (Lecture Notes in Physics, New Series, vol 36. Springer, Heidelberg, 1995), we analyse the solvability of the following (normalised) Liouville-type system in the presence of singular sources: (Formula presented.) with τ> 0 and N> 0. We identify necessary and sufficient conditions on the parameter τ and the “flux” pair: (β_{1}, β_{2}) , which ensure the radial solvability of (1) _{τ}. Since for τ=1/2, problem (1)_{τ} reduces to the (integrable) 2 × 2 Toda system, in particular we recover the existence result of Lin et al. (Invent Math 190(1):169–207, 2012) and Jost and Wang (Int Math Res Not 6:277–290, 2002), concerning this case. Our method relies on a blow-up analysis for solutions of (1)_{τ}, which (even in the radial setting) takes new turns compared to the single equation case. We mention that our approach also permits handling the non-symmetric case, where in each of the two equations in (1)_{τ}, the parameter τ is replaced by two different parameters τ_{1}> 0 and τ_{2}> 0 respectively, and also when the second equation in (1)_{τ} includes a Dirac measure supported at the origin.

Original language | English |
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Pages (from-to) | 223-270 |

Number of pages | 48 |

Journal | Communications in Mathematical Physics |

Volume | 347 |

Issue number | 1 |

DOIs | |

State | Published - 1 Oct 2016 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics