  We consider solutions of the defocusing nonlinear Schr\"odinger (NLS)
equation on the half-line whose Dirichlet and Neumann boundary values become
periodic for sufficiently large $t$. We prove a theorem which, modulo certain
assumptions, characterizes the pairs of periodic functions which can arise as
Dirichlet and Neumann values for large $t$ in this way. The theorem also
provides a constructive way of determining explicit solutions with the given
periodic boundary values. Hence our approach leads to a class of new exact
solutions of the defocusing NLS equation on the half-line.
