Global well-posedness and orbital stability of the Gross--Pitaevskii branch in a coupled defocusing Schrödinger system
Abstract
We consider a one-dimensional coupled defocusing Schrödinger system with a nonzero background in its first component and finite mass in its second component. We prove global well-posedness in the full energy space, together with conservation of energy, bright mass, and total untwisted momentum. For all nonnegative coupling coefficients, every subsonic Gross--Pitaevskii soliton with vanishing second component is orbitally stable against perturbations of both components, including the zero-speed black soliton. At nonzero speed the proof uses a fixed energy--momentum functional augmented by the conserved bright mass. At zero speed it uses weighted coercivity and a local real lift of the momentum. We also identify the sharp threshold for positivity of the unaugmented bright energy and explain why this threshold is not a necessary condition for orbital stability.
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BibTeX
@misc{kowalczyk2026global,
title = {{Global well-posedness and orbital stability of the Gross--Pitaevskii branch in a coupled defocusing Schrödinger system}},
author = {Michał Kowalczyk and Jarosław Mederski and Isidora Miranda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24261},
eprint = {2610.24261},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24261},
}
RIS
TY - UNPB
TI - Global well-posedness and orbital stability of the Gross--Pitaevskii branch in a coupled defocusing Schrödinger system
AU - Michał Kowalczyk
AU - Jarosław Mederski
AU - Isidora Miranda
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24261
UR - https://arxiv.org/abs/2610.24261
AB - We consider a one-dimensional coupled defocusing Schrödinger system with a nonzero background in its first component and finite mass in its second component. We prove global well-posedness in the full energy space, together with conservation of energy, bright mass, and total untwisted momentum. For all nonnegative coupling coefficients, every subsonic Gross--Pitaevskii soliton with vanishing second component is orbitally stable against perturbations of both components, including the zero-speed black soliton. At nonzero speed the proof uses a fixed energy--momentum functional augmented by the conserved bright mass. At zero speed it uses weighted coercivity and a local real lift of the momentum. We also identify the sharp threshold for positivity of the unaugmented bright energy and explain why this threshold is not a necessary condition for orbital stability.
ER -