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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 methodis that it requires the values of more boundary data than given for awell-posed problem. In the case of cubic NLS, knowledge of the Dirichet datasuffices to make the problem well-posed but the Fokas method also requiresknowledge of the values of Neumann data. The study of the Dirichlet to Neumannmap is thus necessary before the application of the `Fokas transform'. In thispaper, we provide a rigorous study of this map for a large class of decayingDirichlet data. We show that the Neumann data are also sufficiently decayingand hence that the Fokas method can be applied.  This is an addendum to a previous paper. We consider the case of non-zeroinitial data, thus completing the discussion of the previous paper.

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