Einstein-Infeld-Hoffman method and soliton dynamics in a parity noninvariant system

TL;DR

This paper analyzes the dynamics of vortices in a nonlinear Schrödinger system coupled with Chern-Simons gauge fields under magnetic fields, revealing cyclotron motion as the primary nontrivial solution.

Contribution

It introduces a novel approach to vortex dynamics by treating velocities and accelerations equally, deriving the vortex motion as a perturbation akin to magnetic translation.

Findings

Vortex motion corresponds to cyclotron orbits.

Linearized field equations determine vortex trajectories.

Static solutions are trivial without considering acceleration.

Abstract

We consider slow motion of a pointlike topological defect (vortex) in the nonlinear Schrodinger equation minimally coupled to Chern-Simons gauge field and subject to external uniform magnetic field. It turns out that a formal expansion of fields in powers of defect velocity yields only the trivial static solution. To obtain a nontrivial solution one has to treat velocities and accelerations as being of the same order. We assume that acceleration is a linear form of velocity. The field equations linearized in velocity uniquely determine the linear relation. It turns out that the only nontrivial solution is the cyclotron motion of the vortex together with the whole condensate. This solution is a perturbative approximation to the center of mass motion known from the theory of magnetic translations.