Edge solitons in the QHE

TL;DR

This paper explores edge solitons in the quantum Hall effect by deriving a modified nonlinear Schrödinger equation from Chern-Simons theory, analyzing its solutions, and demonstrating how additional potentials lead to integrability.

Contribution

It introduces a modified NLS derived from Chern-Simons theory, analyzes its solutions, and shows how adding a six-order potential makes the equation integrable.

Findings

Linear phase solutions correspond to ordinary NLS with time-dependent coefficients.

Only traveling solitons are consistent with the modified NLS.

Adding a six-order potential renders the equation integrable.

Abstract

The spacelike reduction of the Chern-Simons Lagrangian yields a modified Nonlinear Schr\"odinger Equation (jNLS) where in the non-linearity the particle density is replaced by current. When the phase is linear in the position, this latter is an ordinary NLS with time-dependent coefficients which admits interesting solutions, whose arisal is explained by the conformal properties of non-relativistic spacetime. However, only the usual travelling soliton is consistent with the jNLS. Adding a six-order potential converts jnLS into an integrable equation.