On the vortex solutions of some nonlinear scalar field equations (Q808332)

The author considers the vortex solutions of the form U(r)exp(in\(\theta\)) for the nonlinear Schrödinger, Klein-Gordon, and heat equation in \({\mathbb{R}}^ 2\). These vortex solutions are known to have an infinite energy due to a logarithmic divergence of the solutions. Subtracing an asymptotic form of the solutions the author builds a finite relative energy functional and gives a variational characterization of vortex solutions as minimisers of this functional. Employing the relative energy functional as the Lyapunov function the author discusses stability of the vortex solutions.