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

Abstract

We establish the orbital stability of the black soliton, or kink solution, v 0 (x) = th(), to the one-dimensional Gross-Pitaevskii equation, with respect to perturbations in the energy space.

Stability in the energy space for chains of solitons of the Landau–Lifshitz equation

“…In the context of the Gross-Pitaevskii equation, the orbital stability of the vanishing soliton was proved in [2,13]. Part of the analysis in this further context certainly extends to the soliton u 0 for the Landau-Lifshitz equation.…”

“…Similarly, we decompose the derivative ∂ c v c = αu c + r c , with r c satisfying (2.5). 2 We next compute…”

Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems

“…In one dimension, the orbital stability (or instability) of traveling waves has been proved in [Lin02] for cubic-quintic nonlinearities, respectively in [BGSS08] and [GZ08] for the Gross-Pitaevskii nonlinearity. Even in one dimension, no asymptotic stability result seems to be available.…”

Asymptotic stability of the black soliton for the Gross–Pitaevskii equation

“…This issue was mostly solved in a series of recent papers. The orbital stability of dark solitons was derived in [28] (see also [1]), whereas the case of the black soliton was solved in [3,22]. More recently, the asymptotic stability of dark solitons was proved in [5].…”