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Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation. (English) Zbl 1018.58008
Summary: In this article, we construct self-dual \(N\)-vortex solutions with a large magnetic flux \(\Phi\) of \((2+1)\)-dimensional relativistic Chern-Simons model, provided that the coupling constant \(\kappa\) is small and the sites of vorticity \(\{p_1,\ldots,p_n\}\) satisfies \[ \sum_{k\neq j} \log (|p_j-p_k|)\text{ is independent of }j.\tag{1} \] Our solutions exhibit the bubbling phenomenon at each \(p_j\). Near each vortex \(p_j\), solutions are locally asymptotically symmetric with respect to \(p_j\), and the curvature \(F_{12}\) tends to a sum of Dirac measures as \(\kappa\) tends to zero. By a heuristic argument, it is shown that (1) is also a necessary condition for the existence of multi-vortex solutions which has a locally asymptotically symmetric vortex at \(p_j\), \(j=1,2,\ldots,N\).

MSC:
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
81T13 Yang-Mills and other gauge theories in quantum field theory
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