We consider the focusing NLS equation on the quarter plane. Initial data are vanishing at infinity while boundary date are time-periodic (ae^[2i\omega t]). The main tool is the asymptotic analysis of the associated matrix Riemann-Hilbert problem. We will show that we obtain 4 different asymptotics in different regions:
region 1: a Zakharov-Manakov vanishing asymptotics region 2: a train of asymptotics solitons region 3: a modulated elliptic wave region 4: a plane wave.