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

TL;DR

This paper proves the existence of hylomorphic vortices, standing waves with angular momentum, in the nonlinear Klein-Gordon equation for dimensions three and higher, and explores their properties and stability through variational and numerical methods.

Contribution

It introduces the concept of hylomorphic vortices in the nonlinear Klein-Gordon equation and demonstrates their existence using variational techniques, also providing numerical insights into their profiles and stability.

Findings

Hylomorphic vortices exist in dimensions N≥3.

Numerical results show vortex profiles vary with charge and angular momentum.

Vortices are suggested to be unstable based on numerical evolution studies.

Abstract

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.