Non-topological solutions in the generalized self-dual Chern-Simons-Higgs theory. (Q1809946)

In an earlier article \textit{D. Chae} and \textit{O.Yu. Imanuvilov} [Commun.\ Math.\ Phys. 215, 119--142 (2000; Zbl 1002.58015)], the authors showed the existence of non-topological multivortex solutions to the \((2+1)\)-dimensional relativistic Chern-Simons equations. Here, `non-topological' means that the Higgs field tends to zero at spatial infinity and `multivortex' pertains in the strong sense that the zeroes of the Higgs field can be arbitrarily prescribed. In this article the authors extend their earlier argument to cover the generalized self-dual Chern-Higgs-Simons equations. These are natural quasilinear elliptic partial differential equations for a complex-valued Higgs field \(\phi\) and a gauge field \(A_\mu\) this time in two dimensions: defined on~\({\mathbb{R}}^2\) and having finite energy. Non-topological again means that there is no topology out at infinity: the field \(\phi\) and its covariant derivative with respect to \(A_\mu\) both tend to zero. The construction is obtained by perturbation about explicit solutions of the Liouville equation. Again, the zeroes of \(\phi\) may be arbitrarily prescribed.