The nonlinear Schrödinger equation on the half-line with homogeneous Robin boundary conditions.

Abstract

We consider the nonlinear Schrödinger equation on the half-line $x⩾0$ with a Robin boundary condition at $x=0$ and with initial data in the weighted Sobolev space $H^{1,1}\left(R_{+}\right)$ . We prove that there exists a global weak solution of this initial-boundary value problem and provide a representation for the solution in terms of the solution of a Riemann-Hilbert problem. Using this representation, we obtain asymptotic formulas for the long-time behavior of the solution. In particular, by restricting our asymptotic result to solutions whose initial data are close to the initial profile of the stationary one-soliton, we obtain results on the asymptotic stability of the stationary one-soliton under any small perturbation in $H^{1,1}\left(R_{+}\right)$ . In the focusing case, such a result was already established by Deift and Park using different methods, and our work provides an alternative approach to obtain such results. We treat both the focusing and the defocusing versions of the equation.