Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition

We consider the focusing nonlinear Schrodinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic (ae(2i omega t)). The goal of this Note is to study the asymptotic behavior of the solution of this initial-boundary-value problem. The main tool is the asymptotic analysis of an associated matrix Riemann-Hilbert problem. We show that the solution of the IBV problem has different asymptotic behaviors in different regions. In the region x > 4bt (b = root(a(2)-omega)/2>0) the solution has the form of a Zakharov-Manakov vanishing asymptotics. In the region 4bt - 1/2a N log t < x < 4bt, where N is an integer, the solution behaves as a finite train of asymptotic solitons. In the region 4(b - a root 2)t < x < 4bt the solution is a modulated elliptic wave. Finally, in the sector 0 < x < 4(b - a root 2)t the solution is a plane wave.