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

Abstract

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 ${a{\rm e}^{\i\alpha} {\rm e}^{2\i\omega t}}$ . 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 ${b\mathop{:=} \sqrt{(a^2-\omega)/2} > 0}$ , 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 ${4(b-a\sqrt2)t < x < 4bt}$ , the solution takes the form of a modulated elliptic wave. In the region ${0 < x < 4(b-a\sqrt2)t}$ , the solution takes the form of a plane wave.