Orbital stability in the cubic defocusing NLS equation: II. The black soliton

TL;DR

This paper proves the orbital stability of the black soliton in the cubic defocusing NLS equation by using a novel variational approach involving conserved quantities, providing a simpler proof of stability.

Contribution

It introduces an elementary variational characterization combining energy and higher-order conserved quantities to establish the black soliton's stability.

Findings

Black soliton is a local minimizer of a conserved quantity.

The stability proof is simplified using this variational approach.

The black soliton is orbitally stable in H^2(R).

Abstract

Combining the usual energy functional with a higher-order conserved quantity originating from integrability theory, we show that the black soliton is a local minimizer of a quantity that is conserved along the flow of the cubic defocusing NLS equation in one space dimension. This unconstrained variational characterization gives an elementary proof of the orbital stability of the black soliton with respect to perturbations in $H^2(\mathbb{R})$.