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.