Scattering and localized states for defocusing nonlinear Schrödinger equations with potential
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911328402145280 |
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| author | Soffer, Avy Stewart, Gavin |
| author_facet | Soffer, Avy Stewart, Gavin |
| contents | We study the large-time behavior of global energy class ($H^1$) solutions of the one-dimensional nonlinear Schrödinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction Morawetz estimate localized to an exterior region, we prove that these solutions decompose into a free wave and a weakly localized part which is asymptotically orthogonal to any fixed free wave. We further show that the $L^2$ norm of this weakly localized part is concentrated in the region $|x| \leq t^{1/2+}$, and that the energy ($\dot{H}^1$) norm is concentrated in $|x| \leq t^{1/3+}$. Our results hold for solutions with arbitrarily large initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11366 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scattering and localized states for defocusing nonlinear Schrödinger equations with potential Soffer, Avy Stewart, Gavin Analysis of PDEs Mathematical Physics 35Q55, 35Q41 (Primary) 35B40 (Secondary) We study the large-time behavior of global energy class ($H^1$) solutions of the one-dimensional nonlinear Schrödinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction Morawetz estimate localized to an exterior region, we prove that these solutions decompose into a free wave and a weakly localized part which is asymptotically orthogonal to any fixed free wave. We further show that the $L^2$ norm of this weakly localized part is concentrated in the region $|x| \leq t^{1/2+}$, and that the energy ($\dot{H}^1$) norm is concentrated in $|x| \leq t^{1/3+}$. Our results hold for solutions with arbitrarily large initial data. |
| title | Scattering and localized states for defocusing nonlinear Schrödinger equations with potential |
| topic | Analysis of PDEs Mathematical Physics 35Q55, 35Q41 (Primary) 35B40 (Secondary) |
| url | https://arxiv.org/abs/2402.11366 |