The final state problem for the nonlinear Schrodinger equation in dimensions 1, 2 and 3
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917148937420800 |
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| author | Hassell, Andrew Jia, Qiuye |
| author_facet | Hassell, Andrew Jia, Qiuye |
| contents | In this article we consider the defocusing nonlinear Schrödinger equation, with time-dependent potential, in space dimensions $n=1, 2$ and $3$, with nonlinearity $|u|^{p-1} u$, $p$ an odd integer, satisfying $p \geq 5$ in dimension $1$, $p \geq 3$ in dimension $2$ and $p=3$ in dimension $3$. We also allow a metric perturbation, assumed to be compactly supported in spacetime, and nontrapping. We work with module regularity spaces, which are defined by regularity of order $k \geq 2$ under the action of certain vector fields generating symmetries of the free Schrödinger equation. We solve the large data final state problem, with final state in a module regularity space, and show convergence of the solution to the final state. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16090 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The final state problem for the nonlinear Schrodinger equation in dimensions 1, 2 and 3 Hassell, Andrew Jia, Qiuye Analysis of PDEs Mathematical Physics 35Q55 (primary) 35Q41, 35S05 (secondary) In this article we consider the defocusing nonlinear Schrödinger equation, with time-dependent potential, in space dimensions $n=1, 2$ and $3$, with nonlinearity $|u|^{p-1} u$, $p$ an odd integer, satisfying $p \geq 5$ in dimension $1$, $p \geq 3$ in dimension $2$ and $p=3$ in dimension $3$. We also allow a metric perturbation, assumed to be compactly supported in spacetime, and nontrapping. We work with module regularity spaces, which are defined by regularity of order $k \geq 2$ under the action of certain vector fields generating symmetries of the free Schrödinger equation. We solve the large data final state problem, with final state in a module regularity space, and show convergence of the solution to the final state. |
| title | The final state problem for the nonlinear Schrodinger equation in dimensions 1, 2 and 3 |
| topic | Analysis of PDEs Mathematical Physics 35Q55 (primary) 35Q41, 35S05 (secondary) |
| url | https://arxiv.org/abs/2411.16090 |