Global well-posedness and orbital stability of the Gross--Pitaevskii branch in a coupled defocusing Schrödinger system
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.