Perturbation of the nonlinear Schrödinger equation by a localized nonlinearity
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912539272544256 |
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| author | Chen, Gong Liu, Jiaqi Tian, Yuanhong |
| author_facet | Chen, Gong Liu, Jiaqi Tian, Yuanhong |
| contents | We revisit the perturbative theory of infinite dimensional integrable systems developed by P. Deift and X. Zhou \cite{DZ-2}, aiming to provide new and simpler proofs of some key $L^\infty$ bounds and $L^p$ \emph{\textit{a priori}} estimates. Our proofs emphasizes a further step towards understanding focussing problems and extends the applicability to other integrable models. As a concrete application, we examine the perturbation of the one-dimensional defocussing cubic nonlinear Schrödinger equation by a localized higher-order term. We introduce improved estimates to control the power of the perturbative term and demonstrate that the perturbed equation exhibits the same long-time behavior as the completely integrable nonlinear Schrödinger equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11463 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perturbation of the nonlinear Schrödinger equation by a localized nonlinearity Chen, Gong Liu, Jiaqi Tian, Yuanhong Analysis of PDEs We revisit the perturbative theory of infinite dimensional integrable systems developed by P. Deift and X. Zhou \cite{DZ-2}, aiming to provide new and simpler proofs of some key $L^\infty$ bounds and $L^p$ \emph{\textit{a priori}} estimates. Our proofs emphasizes a further step towards understanding focussing problems and extends the applicability to other integrable models. As a concrete application, we examine the perturbation of the one-dimensional defocussing cubic nonlinear Schrödinger equation by a localized higher-order term. We introduce improved estimates to control the power of the perturbative term and demonstrate that the perturbed equation exhibits the same long-time behavior as the completely integrable nonlinear Schrödinger equation. |
| title | Perturbation of the nonlinear Schrödinger equation by a localized nonlinearity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.11463 |