  We study the Cauchy problem for the defocusing nonlinear Schr\"odinger (NLS)
equation under the assumption that the solution vanishes as $x \to + \infty$
and approaches an oscillatory plane wave as $x \to -\infty$. We first develop
an inverse scattering transform formalism for solutions satisfying such
step-like boundary conditions. Using this formalism, we prove that there exists
a global solution of the corresponding Cauchy problem and establish a
representation for this solution in terms of the solution of a Riemann-Hilbert
problem. By performing a steepest descent analysis of this Riemann-Hilbert
problem, we identify three asymptotic sectors in the half-plane $t \geq 0$ of
the $xt$-plane and derive asymptotic formulas for the solution in each of these
sectors. Finally, by restricting the constructed solutions to the half-line $x
\geq 0$, we find a class of solutions with asymptotically time-periodic
boundary values previously sought for in the context of the NLS half-line
problem.

Analysis of PDEs