Painlev\'e-type asymptotics for the defocusing Manakov system with nonzero boundary conditions

We investigate the long-time asymptotic behavior of a class of solutions to the defocusing Manakov system under nonzero boundary conditions. These solutions are characterized by a $3 \times 3$ matrix Riemann Hilbert problem. We find that they exhibit interesting asymptotic behavior within a narrow transition zone in the $x$-$t$ plane. We determine the leading-order asymptotic term and the error bound in this region, and we demonstrate that the leading term can be expressed in terms of the Hastings-McLeod solution of the Painlev\'e II equation. The proof is rigorously established by applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann Hilbert problem.