On the existence of hylomorphic vortices in the nonlinear Klein-Gordon equation

In this paper we prove the existence of vortices, namely standing waves with non null angular momentum, for the nonlinear Klein-Gordon equation in dimension $N\geq 3$. We show with variational methods that the existence of these kind of solutions, that we have called \emph{hylomorphic vortices}, depends on a suitable energy-charge ratio. Our variational approach turns out to be useful for numerical investigations as well. In particular, some results in dimension $N=2$ are reported, namely exemplificative vortex profiles by varying charge and angular momentum, together with relevant trends for vortex frequency and energy-charge ratio. The stability problem for hylomorphic vortices is also addressed. In the absence of conclusive analytical results, vortex evolution is numerically investigated: the obtained results suggest that, contrarily to solitons with null angular momentum, vortex are unstable.