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The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory. (English) Zbl 1002.58015

The $\left(2+1\right)$-dimensional relativistic Chern-Simons equations form a nonlinear system of partial differential equations for a gauge field ${A}_{\mu }$ and a Higgs field $\phi$ defined on ${ℝ}^{3}$ with standard Lorentzian metric. The self-dual solutions absolutely minimize the energy. There are two possible boundary conditions $|\phi \left(x\right)|\to 1$ or $|\phi \left(x\right)|\to 0$ as ${ℝ}^{2}\ni x\to \infty$ consistent with finite energy. Solutions with $|\phi \left(x\right)|\to 1$ have been dubbed ‘topological’ and were shown to exist by R. Wang [Commun. Math. Phys. 137, No. 3, 587-597 (1991; Zbl 0733.58009)].

In this article, the authors consider the existence of self-dual ‘non-topological’ solutions, i.e. with boundary condition $|\phi \left(x\right)|\to 0$. 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.

##### MSC:
 58E50 Applications of variational methods in infinite-dimensional spaces 81T13 Yang-Mills and other gauge theories 35J60 Nonlinear elliptic equations
##### Keywords:
Chern-Simons theory; self-dual solutions; Higgs field