On the final-state problem for the 1D cubic NLS

We consider the one-dimensional cubic nonlinear Schrödinger equation $$\text{ii}{\mathrm{\partial }}_{t}u+\frac{1}{2}{\mathrm{\partial }}_{xx}u=\text{la}\mathrm{\mid }u{\mathrm{\mid }}^{2}u,\lambda =\pm 1$$ and solve the final-state (modified wave operator) problem for small asymptotic data. More precisely, given a small $W(\xi )$, we construct a solution $u$ such that \begin{equation*} u\rightarrow (2π)^{-1/2}(\ii t)^{-1/2}e^{\ii x^2/(2t)}, W!\Big(\frac{x}{t}\Big)\exp(-\ii\la|W(x/t)|^2\log t). \end{equation*} Crucially, we design a contraction map, so that we can run the analysis in the spirit of Kato–Pusateri \cite{KP} for $w$ with a forcing term depending {\it only} on the final data $W$. This scheme is easy to adapt to solving final state problems with a complete theory for the forward problems.