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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2605.23063 |
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| _version_ | 1866916038010994688 |
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| author | Chen, Gong Qiang, Yongyu |
| author_facet | Chen, Gong Qiang, Yongyu |
| contents | We consider the one-dimensional cubic nonlinear Schrödinger equation $$ \ii\partial_tu+\frac12\partial_{xx}u=\la|u|^2u,\,λ=\pm 1 $$ and solve the final-state (modified wave operator) problem for small asymptotic data. More precisely, given a small $W(ξ)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23063 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the final-state problem for the 1D cubic NLS Chen, Gong Qiang, Yongyu Analysis of PDEs We consider the one-dimensional cubic nonlinear Schrödinger equation $$ \ii\partial_tu+\frac12\partial_{xx}u=\la|u|^2u,\,λ=\pm 1 $$ and solve the final-state (modified wave operator) problem for small asymptotic data. More precisely, given a small $W(ξ)$, 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. |
| title | On the final-state problem for the 1D cubic NLS |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.23063 |