A Nonrelativistic Chiral Soliton in One Dimension

Abstract

I analyze the one-dimensional, cubic Schr\"odinger equation, with nonlinearity constructed from the current density, rather than, as is usual, from the charge density. A soliton solution is found, where the soliton moves only in one direction. Relation to higher-dimensional Chern--Simons theory is indicated. The theory is quantized and results for the two-body quantum problem agree at weak coupling with those coming from a semiclassical quantization of the soliton.