On Non-Topological Solutions for Planar Liouville Systems of Toda-Type

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: (β₁, β₂) , 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 τ₁> 0 and τ₂> 0 respectively, and also when the second equation in (1)_τ includes a Dirac measure supported at the origin.