Decaying Long-Time Asymptotics for the Focusing NLS Equation with Periodic Boundary Condition

We consider the initial boundary value problem for the focusing nonlinear Schrodinger equation in the quarter plane x > 0, t > 0 in the case of decaying initial data (for t = 0, as x -> +infinity) and Dirichlet boundary data (for x = 0) approaching a periodic (single-frequency) background ae(2i omega t) as t -> +infinity. We first provide admissibility conditions for the normal derivative of the solution on the boundary, under the assumption that it behaves asymptotically in a similar (single-frequency) manner. We then show that for the range omega >= a(2)/2, the long-time asymptotics of the solution inside the quarter plane exhibits decaying oscillations of Zakharov-Manakov type.