A Nonrelativistic Chiral Soliton in One Dimension

TL;DR

This paper investigates a one-dimensional cubic Schrödinger equation with current-based nonlinearity, discovering a unidirectional soliton solution, exploring its relation to Chern--Simons theory, and quantizing the model to compare quantum results.

Contribution

It introduces a novel current-based nonlinearity in the Schrödinger equation and analyzes the resulting soliton and its quantum properties, connecting to higher-dimensional topological theories.

Findings

Found a unidirectional soliton solution.

Quantized the theory and matched quantum results with semiclassical analysis.

Indicated a relation to higher-dimensional Chern--Simons theory.

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.