Slow Schroedinger dynamics of gauged vortices

TL;DR

This paper investigates the slow dynamics of gauged vortices in a modified Landau-Ginzburg model, revealing Hamiltonian structures, integrability for three vortices, and stability properties, with comparisons to classical vortex systems.

Contribution

It demonstrates that vortex dynamics in this model are Hamiltonian on the moduli space, shows integrability for three vortices, and provides asymptotic and stability analyses.

Findings

Reduced vortex flow is Hamiltonian with respect to the L^2 Kaehler form.

The three-vortex flow is integrable in the Liouville sense.

Spectral stability of certain vortex polygons is analyzed.

Abstract

Multivortex dynamics in Manton's Schroedinger--Chern--Simons variant of the Landau-Ginzburg model of thin superconductors is studied within a moduli space approximation. It is shown that the reduced flow on M_N, the N vortex moduli space, is hamiltonian with respect to \omega_{L^2}, the L^2 Kaehler form on \M_N. A purely hamiltonian discussion of the conserved momenta associated with the euclidean symmetry of the model is given, and it is shown that the euclidean action on (M_N,\omega_{L^2}) is not hamiltonian. It is argued that the N=3 flow is integrable in the sense of Liouville. Asymptotic formulae for \omega_{L^2} and the reduced Hamiltonian for large intervortex separation are conjectured. Using these, a qualitative analysis of internal 3-vortex dynamics is given and a spectral stability analysis of certain rotating vortex polygons is performed. Comparison is made with the dynamics of classical fluid point vortices and geostrophic vortices.