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Zbl 1002.58015
Chae, Dongho; Imanuvilov, Oleg Yu.
The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory. (English)
[J] Commun. Math. Phys. 215, No.1, 119-142 (2000). ISSN 0010-3616; ISSN 1432-0916/e

The $(2+1)$-dimensional relativistic Chern-Simons equations form a nonlinear system of partial differential equations for a gauge field $A_\mu$ and a Higgs field $\varphi$ defined on ${\Bbb R}^3$ with standard Lorentzian metric. The self-dual solutions absolutely minimize the energy. There are two possible boundary conditions $|\varphi(x)|\to 1$ or $|\varphi(x)|\to 0$ as ${\Bbb R}^2\ni x\to\infty$ consistent with finite energy. Solutions with $|\varphi(x)|\to 1$ have been dubbed `topological' and were shown to exist by {\it R. Wang} [Commun. Math. Phys. 137, No. 3, 587-597 (1991; Zbl 0733.58009)]. \par In this article, the authors consider the existence of self-dual `non-topological' solutions, i.e. with boundary condition $|\varphi(x)|\to 0$. They prove the existence of solutions with arbitrarily prescribed zeroes for the Higgs field and other good properties. In particular, these solutions are not in any way symmetric. The construction is obtained by perturbation about explicit solutions of the Liouville equation.
[M.G.Eastwood (Adelaide)]

MSC 2000:: *58E50 Appl. of variational methods in infinite-dimensional spaces
81T13 Gauge theories
35J60 Nonlinear elliptic equations

Keywords: Chern-Simons theory; self-dual solutions; Higgs field

Citations: Zbl 0733.58009

Cited in: Zbl 1116.58012 Zbl 1080.35021

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