Partially concentrating standing waves for weakly coupled Schrödinger systems
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916125750591488 |
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| author | Pellacci, Benedetta Pistoia, Angela Vaira, Giusi Verzini, Gianmaria |
| author_facet | Pellacci, Benedetta Pistoia, Angela Vaira, Giusi Verzini, Gianmaria |
| contents | We study the existence of standing waves for the following weakly coupled system of two Schrödinger equations in $\mathbb{R}^N$, $N=2,3$, \[ \begin{cases} i \hslash \partial_{t}ψ_{1}=-\frac{\hslash^2}{2m_{1}}Δψ_{1}+ {V_1}(x)ψ_{1}-μ_{1}|ψ_{1}|^{2}ψ_{1}-β|ψ_{2}|^{2}ψ_{1} & \\ i \hslash \partial_{t}ψ_{2}=-\frac{\hslash^2}{2m_{2}}Δψ_{2}+ {V_2}(x)ψ_{2}-μ_{2}|ψ_{2}|^{2}ψ_{2}-β|ψ_{1}|^{2}ψ_{2},& \end{cases} \] where $V_1$ and $V_2$ are radial potentials bounded from below. We address the case $m_{1}\sim \hslash^2\to0$, $m_2$ constant, and prove the existence of a standing wave solution with both nontrivial components satisfying a prescribed asymptotic profile. In particular, the second component of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11648 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Partially concentrating standing waves for weakly coupled Schrödinger systems Pellacci, Benedetta Pistoia, Angela Vaira, Giusi Verzini, Gianmaria Analysis of PDEs We study the existence of standing waves for the following weakly coupled system of two Schrödinger equations in $\mathbb{R}^N$, $N=2,3$, \[ \begin{cases} i \hslash \partial_{t}ψ_{1}=-\frac{\hslash^2}{2m_{1}}Δψ_{1}+ {V_1}(x)ψ_{1}-μ_{1}|ψ_{1}|^{2}ψ_{1}-β|ψ_{2}|^{2}ψ_{1} & \\ i \hslash \partial_{t}ψ_{2}=-\frac{\hslash^2}{2m_{2}}Δψ_{2}+ {V_2}(x)ψ_{2}-μ_{2}|ψ_{2}|^{2}ψ_{2}-β|ψ_{1}|^{2}ψ_{2},& \end{cases} \] where $V_1$ and $V_2$ are radial potentials bounded from below. We address the case $m_{1}\sim \hslash^2\to0$, $m_2$ constant, and prove the existence of a standing wave solution with both nontrivial components satisfying a prescribed asymptotic profile. In particular, the second component of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature. |
| title | Partially concentrating standing waves for weakly coupled Schrödinger systems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2311.11648 |