The nonlinear Schr\"odinger equation with $t$-periodic data: II. Perturbative results

We consider the nonlinear Schr\"odinger equation on the half-line with a given Dirichlet boundary datum which for large $t$ tends to a periodic function. We assume that this function is sufficiently small, namely that it can be expressed in the form $\alpha g_0^b(t)$, where $\alpha$ is a small constant. Assuming that the Neumann boundary value tends for large $t$ to the periodic function $g_1^b(t)$, we show that $g_1^b(t)$ can be expressed in terms of a perturbation series in $\alpha$ which can be constructed explicitly to any desired order. As an illustration, we compute $g_1^b(t)$ to order $\alpha^8$ for the particular case that $g_0^b(t)$ is the sum of two exponentials. We also show that there exist particular functions $g_0^b(t)$ for which the above series can be summed up, and therefore for these functions $g_1^b(t)$ can be obtained in closed form. The simplest such function is $\exp(i\omega t)$, where $\omega$ is a real constant.