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

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