On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS equation on the Half-Line; Zero Initial Data

Abstract

Initial-boundary value problems for 1-dimensional `completely integrable' equations can be solved via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a well-posed problem. In the case of cubic NLS, knowledge of the Dirichet data suffices to make the problem well-posed but the Fokas method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the `Fokas transform'. In this paper, we provide a rigorous study of this map for a large class of decaying Dirichlet data. We show that the Neumann data are also sufficiently decaying and hence that the Fokas method can be applied. For simplicity we considered here the case of zero initial data. An addendum will follow, discussing the case of non-zero initial data.