Admissible boundary values for the defocusing nonlinear Schr\"odinger equation with asymptotically time-periodic data

Abstract

We consider solutions of the defocusing nonlinear Schr\"odinger equation in the quarter plane whose Dirichlet boundary data approach a single exponential $\alpha e^{i\omega t}$ as $t \to \infty$. In order to determine the long time asymptotics of the solution, it is necessary to first characterize the asymptotic behavior of the Neumann value in terms of the given data. Assuming that the initial data decay as $x \to \infty$, we derive necessary conditions for the Neumann value to asymptote towards a single exponential of the form $ce^{i\omega t}$. Since our approach yields expressions which relate $\alpha$, $\omega$, and $c$, the result can be viewed as a characterization of the large $t$ behavior of the Dirichlet to Neumann map for single exponential profiles.