Perturbative and Exact Results on the Neumann Value for the Nonlinear Schrodinger on the Half-line

The most challenging problem in the implementation of the so-called unified transform to the analysis of the nonlinear Schrodinger 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 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(iwt)vertical bar a > 0, omega >= a(2)}, then the large t behavior of the Neumann value is given by a tau-periodic function which can be computed explicitly.