Long-Time Asymptotics for the Focusing NLS Equation with Time-Periodic Boundary Condition on the Half-Line

We consider the focusing nonlinear Schrodinger equation on the quarter plane. The initial data are vanishing at infinity while the boundary data are time-periodic, of the form ae(i alpha)e(2i omega t). The goal of this paper 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 for omega < -3a(2) the solution of the IBV problem has different asymptotic behaviors in different regions. In the region x > 4bt, where b := root(a(2) - omega)/2 > 0, the solution takes the form of the Zakharov-Manakov vanishing asymptotics. In a region of type 4bt - N+1/2a log t < x < 4bt, where N is any integer, the solution is asymptotic to a train of asymptotic solitons. In the region 4(b - a root 2)t < x < 4bt, the solution takes the form of a modulated elliptic wave. In the region 0 < x < 4(b - a root 2)t, the solution takes the form of a plane wave.