Large-time asymptotics for the defocusing Manakov system on a nonzero background

The Manakov system is a two-component nonlinear Schr\"odinger equation. In this paper, we derive a long-time asymptotic formula for the solution of the defocusing Manakov system with nonzero boundary conditions and provide a detailed proof. We first formulate the inverse problem as a $3\times3$ matrix Riemann--Hilbert problem. We then carry out the Deift--Zhou steepest descent analysis for this Riemann--Hilbert problem and obtain the long-time asymptotics in the space-time soliton region. In this region, the leading order of the solution takes the form of a modulated multisoliton. Apart from the error term, we also discover that the defocusing Manakov system has a dispersive correction term of order $t^{-1/2}$, but this term does not exist in the scalar case, and we provide the explicit expression for this dispersion term.