Coefficient Determination for Non-Linear Schrödinger Equations on manifolds
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914932540309504 |
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| author | Lassas, Matti Oksanen, Lauri Sahoo, Suman Kumar Salo, Mikko Tetlow, Alexander |
| author_facet | Lassas, Matti Oksanen, Lauri Sahoo, Suman Kumar Salo, Mikko Tetlow, Alexander |
| contents | We consider an inverse problem of recovering the unknown coefficients $β(t,x)$ and $V(t,x)$ appearing in a time-dependent nonlinear Schrödinger equation $ (\mathrm{i} \partial_t +Δ+V)u + βu^2=0$ in $(0,T) \times M$, on Euclidean geometry as well as on Riemannian geometry. We consider measurements in $Ω\subset M$ that is a neighborhood of the boundary of $M$ and the source-to-solution map $ L_{β, V}$ that maps a source $f$ supported in $ Ω\times (0,T) $ to the restriction of the solution $u$ in $ Ω\times (0,T) $. We show that the map $L_{β, V}$ uniquely determines the time-dependent potential and the coefficient of the non-linearity, for the above non-linear Schrödinger equation and for the Gross-Pitaevskii equation, with a cubic non-linear term $β|u|^2 \, u$, that is encountered in quantum physics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_03699 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Coefficient Determination for Non-Linear Schrödinger Equations on manifolds Lassas, Matti Oksanen, Lauri Sahoo, Suman Kumar Salo, Mikko Tetlow, Alexander Analysis of PDEs 35R30, 31B20, 31B30, 35J40 We consider an inverse problem of recovering the unknown coefficients $β(t,x)$ and $V(t,x)$ appearing in a time-dependent nonlinear Schrödinger equation $ (\mathrm{i} \partial_t +Δ+V)u + βu^2=0$ in $(0,T) \times M$, on Euclidean geometry as well as on Riemannian geometry. We consider measurements in $Ω\subset M$ that is a neighborhood of the boundary of $M$ and the source-to-solution map $ L_{β, V}$ that maps a source $f$ supported in $ Ω\times (0,T) $ to the restriction of the solution $u$ in $ Ω\times (0,T) $. We show that the map $L_{β, V}$ uniquely determines the time-dependent potential and the coefficient of the non-linearity, for the above non-linear Schrödinger equation and for the Gross-Pitaevskii equation, with a cubic non-linear term $β|u|^2 \, u$, that is encountered in quantum physics. |
| title | Coefficient Determination for Non-Linear Schrödinger Equations on manifolds |
| topic | Analysis of PDEs 35R30, 31B20, 31B30, 35J40 |
| url | https://arxiv.org/abs/2201.03699 |