The defocusing nonlinear Schrödinger equation with step-like oscillatory initial data

We study the Cauchy problem for the defocusing nonlinear Schrödinger (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 $t\ge0$ 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 $x \ge 0$, we find a class of solutions with asymptotically time-periodic boundary values previously sought for in the context of the NLS half-line problem.