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

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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:

$$(1)_\tau \begin{cases}-\Delta u_1 = e^{u_1} - \tau e^{u_2} - 4N \pi \, \delta_0,\\-\Delta u_2 = e^{u_2} - \tau e^{u_1},\\ \beta_1 = \frac{1}{2\pi} \int_{\mathbb{R}^{2}} e^{u_1} \, {\rm and } \, \beta_2 = \frac{1}{2\pi} \int_{\mathbb{R}^{2}} e^{u_2},\end{cases}$$

with \({\tau > 0}\) and \({N > 0}\).

We identify necessary and sufficient conditions on the parameter \({\tau}\) and the “flux” pair: \({(\beta_1, \beta_2),}\) which ensure the radial solvability of \({(1)_\tau.}\)

Since for \({\tau=\frac{1}{2},}\) problem \({(1)_\tau}\) reduces to the (integrable) 2 \({\times}\) 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 \({(1)_\tau}\), 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 \({(1)_\tau}\), the parameter \({\tau}\) is replaced by two different parameters \({\tau_1 > 0}\) and \({\tau_2 > 0}\) respectively, and also when the second equation in \({(1)_\tau}\) includes a Dirac measure supported at the origin.

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Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, 84105, Israel

    Arkady Poliakovsky

  2. Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133, Rome, Italy

    Gabriella Tarantello

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  1. Arkady Poliakovsky
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Correspondence to Gabriella Tarantello.

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Communicated by H.-T. Yau

A. Poliakovsky was supported by the Israel Science Foundation, Grant No. 999/13”.

G. Tarantello was supported by PRIN09 project: Nonlinear elliptic problems in the study of vortices and related topics, PRIN12 project: Variational and Perturbative Aspects of Nonlinear Differential Problems and FIRB project: Analysis and Beyond.

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Poliakovsky, A., Tarantello, G. On Non-Topological Solutions for Planar Liouville Systems of Toda-Type. Commun. Math. Phys. 347, 223–270 (2016). https://doi.org/10.1007/s00220-016-2662-3

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