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

We consider the one-dimensional cubic nonlinear Schr\"odinger equation $$\ii\partial_tu+\frac12\partial_{xx}u=\la|u|^2u,\,\lambda=\pm 1$$ 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 (2\pi)^{-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.