Painlev\'e analysis and integrability properties of a $2+1$ nonrelativistic field theory.

TL;DR

This paper applies Painlevé analysis to a 2+1 dimensional nonrelativistic field theory with a Chern-Simons term, finding that the equations are generally non-integrable but useful for studying reductions to integrable cases.

Contribution

It adapts Painlevé analysis to gauge-independent forms of the equations, providing insights into their (non-)integrability and highlighting specific integrable reductions.

Findings

Resonance values are all integers.

Compatibility conditions are generally not satisfied.

The equations are mostly non-integrable, but certain reductions are integrable.

Abstract

A model for planar phenomena introduced by Jackiw and Pi and described by a Lagrangian including a Chern-Simons term is considered. The associated equations of motion, among which a 2+1 gauged nonlinear Schr\"odinger equation, are rewritten into a gauge independent form involving the modulus of the matter field. Application of a Painlev\'e analysis, as adapted to partial differential equations by Weiss, Tabor and Carnevale, shows up resonance values that are all integer. However, compatibility conditions need be considered which cannot be satisfied consistently in general. Such a result suggests that the examined equations are not integrable, but provides tools for the investigation of the integrability of different reductions. This in particular puts forward the familiar integrable Liouville and 1+1 nonlinear Schr\"odinger equations.