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

TL;DR

This paper rigorously analyzes the Dirichlet to Neumann map for the 1D cubic NLS on the half-line with zero initial data, establishing decay properties necessary for applying the Fokas method to solve the boundary value problem.

Contribution

It provides a detailed study of the Dirichlet to Neumann map for decaying data, enabling the use of the Fokas transform for the cubic NLS with zero initial conditions.

Findings

Neumann data decay sufficiently for the Fokas method

The Dirichlet to Neumann map is rigorously characterized

Applicable to a large class of decaying Dirichlet 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.