Perturbative and exact results on the Neumann value for the nonlinear Schr\"odinger equation on the half-line

The most challenging problem in the implementation of the so-called \textit{unified transform} to the analysis of the nonlinear Schr\"odinger equation on the half-line is the characterization of the unknown boundary value in terms of the given initial and boundary conditions. For the so-called \textit{linearizable} boundary conditions this problem can be solved explicitly. Furthermore, for non-linearizable boundary conditions which decay for large $t$, this problem can be largely bypassed in the sense that the unified transform yields useful asymptotic information for the large $t$ behavior of the solution. However, for the physically important case of periodic boundary conditions it is necessary to characterize the unknown boundary value. Here, we first present a perturbative scheme which can be used to compute explicitly the asymptotic form of the Neumann boundary value in terms of the given $\tau$-periodic Dirichlet datum to any given order in a perturbation expansion. We then discuss briefly an extension of the pioneering results of Boutet de Monvel and co-authors which suggests that if the Dirichlet datum belongs to a large class of particular $\tau$-periodic functions, which includes $\{a \exp(i \omega t) \, | \, a>0, \, \omega \geq a^2\}$, then the large $t$ behavior of the Neumann value is given by a $\tau$-periodic function which can be computed explicitly.