Orbital stability of the black soliton to the Gross-Pitaevskii equation

TL;DR

This paper proves the orbital stability of the black soliton solution to the one-dimensional Gross-Pitaevskii equation within the energy space, confirming its robustness under perturbations.

Contribution

It establishes the first rigorous proof of orbital stability for the black soliton in the 1D Gross-Pitaevskii equation, a fundamental nonlinear wave model.

Findings

Black soliton is orbitally stable in the energy space.

Perturbations do not destabilize the black soliton.

The stability result applies to the kink solution of the equation.

Abstract

We establish the orbital stability of the black soliton, or kink solution, $\v_0(x) = \th \big(\frac{x}{\sqrt{2}} \big)$, to the one-dimensional Gross-Pitaevskii equation, with respect to perturbations in the energy space.