Dynamics of focusing nonlinear Schrödinger equation with partial harmonic confinement in higher dimensions
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910078376869888 |
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| author | Liu, Tianhao Ma, Zuyu Song, Yilin Zheng, Jiqiang |
| author_facet | Liu, Tianhao Ma, Zuyu Song, Yilin Zheng, Jiqiang |
| contents | We study the following focusing intercritical nonlinear Schrödinger equation with partial harmonic confinement: \begin{equation*} \begin{cases}
i\partial_t u+Δ_{z}u-y^2 u =- |u|^αu,\quad t\in \mathbb{R},\newline
u(0,z)= u_0(z), \ z=(x,y)\in \mathbb{R}^d\times \mathbb{R}, \end{cases}
\end{equation*} where $d \geq 1 $ is an integer and the exponent $α$ satisfies
\begin{equation}\label{assumption}
\frac{4}{d}< α<\begin{cases} \frac{4}{d-1}, \,\,\, \text{if} ~~ d\geq 2; \newline
+ \infty,\,\,\, \text{if} ~~ d=1. \end{cases}
\end{equation}
For this model, A. Ardia and R. Carles [Comm. Math. Sci. 19 (2021), 993-1032] established a sharp scattering result below the ground state threshold in dimensions $d \leq 4$ via the concentration-compactness and rigidity argument. However, their approach breaks down in higher dimensions due to the lack of smoothness in the nonlinearity. In this paper, we introduce a new strategy that removes this dimensional restriction and extend their results to higher dimensions by circumventing the concentration-compactness principle. The main ingredients of our work are the interaction Morawetz-Dodson-Murphy estimates and an alternative variational characterization of the ground state threshold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26372 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dynamics of focusing nonlinear Schrödinger equation with partial harmonic confinement in higher dimensions Liu, Tianhao Ma, Zuyu Song, Yilin Zheng, Jiqiang Analysis of PDEs We study the following focusing intercritical nonlinear Schrödinger equation with partial harmonic confinement: \begin{equation*} \begin{cases} i\partial_t u+Δ_{z}u-y^2 u =- |u|^αu,\quad t\in \mathbb{R},\newline u(0,z)= u_0(z), \ z=(x,y)\in \mathbb{R}^d\times \mathbb{R}, \end{cases} \end{equation*} where $d \geq 1 $ is an integer and the exponent $α$ satisfies \begin{equation}\label{assumption} \frac{4}{d}< α<\begin{cases} \frac{4}{d-1}, \,\,\, \text{if} ~~ d\geq 2; \newline + \infty,\,\,\, \text{if} ~~ d=1. \end{cases} \end{equation} For this model, A. Ardia and R. Carles [Comm. Math. Sci. 19 (2021), 993-1032] established a sharp scattering result below the ground state threshold in dimensions $d \leq 4$ via the concentration-compactness and rigidity argument. However, their approach breaks down in higher dimensions due to the lack of smoothness in the nonlinearity. In this paper, we introduce a new strategy that removes this dimensional restriction and extend their results to higher dimensions by circumventing the concentration-compactness principle. The main ingredients of our work are the interaction Morawetz-Dodson-Murphy estimates and an alternative variational characterization of the ground state threshold. |
| title | Dynamics of focusing nonlinear Schrödinger equation with partial harmonic confinement in higher dimensions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.26372 |