Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866915034159906816 |
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| author | Cheng, Xing Guo, Chang-Yu Guo, Zihua Liao, Xian Shen, Jia |
| author_facet | Cheng, Xing Guo, Chang-Yu Guo, Zihua Liao, Xian Shen, Jia |
| contents | In this paper, we consider the following three dimensional defocusing cubic nonlinear Schrödinger equation (NLS) with partial harmonic potential
\begin{equation*}\tag{NLS}
i\partial_t u + \left(Δ_{\mathbb{R}^3 }-x^2 \right) u = |u|^2 u, \quad u|_{t=0} = u_0.
\end{equation*}
Our main result shows that the solution $u$ scatters for any given initial data $u_0$ with finite mass and energy.
The main new ingredient in our approach is to approximate (NLS) in the large-scale case by a relevant dispersive continuous resonant (DCR) system. The proof of global well-posedness and scattering of the new (DCR) system is greatly inspired by the fundamental works of Dodson \cite{D3,D1,D2} in his study of scattering for the mass-critical nonlinear Schrödinger equation. The analysis of (DCR) system allows us to utilize the additional regularity of the smooth nonlinear profile so that the celebrated concentration-compactness/rigidity argument of Kenig and Merle applies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_02515 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials Cheng, Xing Guo, Chang-Yu Guo, Zihua Liao, Xian Shen, Jia Analysis of PDEs Mathematical Physics In this paper, we consider the following three dimensional defocusing cubic nonlinear Schrödinger equation (NLS) with partial harmonic potential \begin{equation*}\tag{NLS} i\partial_t u + \left(Δ_{\mathbb{R}^3 }-x^2 \right) u = |u|^2 u, \quad u|_{t=0} = u_0. \end{equation*} Our main result shows that the solution $u$ scatters for any given initial data $u_0$ with finite mass and energy. The main new ingredient in our approach is to approximate (NLS) in the large-scale case by a relevant dispersive continuous resonant (DCR) system. The proof of global well-posedness and scattering of the new (DCR) system is greatly inspired by the fundamental works of Dodson \cite{D3,D1,D2} in his study of scattering for the mass-critical nonlinear Schrödinger equation. The analysis of (DCR) system allows us to utilize the additional regularity of the smooth nonlinear profile so that the celebrated concentration-compactness/rigidity argument of Kenig and Merle applies. |
| title | Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2105.02515 |