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

We consider the focusing nonlinear Schrödinger equation on the quarter plane. The initial data are vanishing at infinity while the boundary data are time- periodic, of the form ({a{rm e}^{ialpha} {rm e}^{2iomega 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 ({bmathop{:=} sqrt{(a^2-omega)/2} > 0}) , the solution takes the form of the Zakharov-Manakov vanishing asymptotics. In a region of type ({4bt-frac{N+1}{2a} {rm log} t < x < 4bt}) , where N is any integer, the solution is asymptotic to a train of asymptotic solitons. In the region ({4(b-asqrt2)t < x < 4bt}) , the solution takes the form of a modulated elliptic wave. In the region ({0 < x < 4(b-asqrt2)t}) , the solution takes the form of a plane wave.