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

TL;DR

This paper rigorously analyzes the Dirichlet to Neumann map for the 1D cubic NLS on the half-line with non-zero initial data, ensuring the applicability of the Fokas method for solving the problem.

Contribution

It extends previous work by studying the Dirichlet to Neumann map with non-zero initial data, confirming decay properties necessary for the Fokas transform.

Findings

Neumann data decay sufficiently for a large class of Dirichlet data

The Fokas method is applicable to non-zero initial data cases

Completes the analysis of the Dirichlet to Neumann map for cubic NLS

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. This is an addendum to a previous paper. We consider the case of non-zero initial data, thus completing the discussion of the previous paper.