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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.06920 |
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| _version_ | 1866910560577126400 |
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| author | Bao, Weizhu Lu, Yong Zhang, Zhifei |
| author_facet | Bao, Weizhu Lu, Yong Zhang, Zhifei |
| contents | In this paper, we study the nonrelativistic limit of the cubic nonlinear Klein-Gordon equation in $\mathbb{R}^{3}$ with a small parameter $0<\varepsilon \ll 1$, which is inversely proportional to the speed of light. We show that the cubic nonlinear Klein-Gordon equation converges to the cubic nonlinear Schrödinger equation with a convergence rate of order $\varepsilon^{2}$. In particular, for the defocusing case and smooth initial data, we prove error estimates of the form $(1+t)\varepsilon^{2}$ at time $t$ which is valid up to long time of order $\varepsilon^{-1}$; while for nonsmooth initial data, we prove error estimates of the form $(1+t)\varepsilon$ at time $t$ which is valid up to long time of order $\varepsilon^{-\frac{1}{2}}$. These specific forms of error estimates coincide with the numerical results obtained in \cite{BZ19,SZ20}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_06920 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convergence rates in the nonrelativistic limit of the cubic Klein-Gordon equation Bao, Weizhu Lu, Yong Zhang, Zhifei Analysis of PDEs In this paper, we study the nonrelativistic limit of the cubic nonlinear Klein-Gordon equation in $\mathbb{R}^{3}$ with a small parameter $0<\varepsilon \ll 1$, which is inversely proportional to the speed of light. We show that the cubic nonlinear Klein-Gordon equation converges to the cubic nonlinear Schrödinger equation with a convergence rate of order $\varepsilon^{2}$. In particular, for the defocusing case and smooth initial data, we prove error estimates of the form $(1+t)\varepsilon^{2}$ at time $t$ which is valid up to long time of order $\varepsilon^{-1}$; while for nonsmooth initial data, we prove error estimates of the form $(1+t)\varepsilon$ at time $t$ which is valid up to long time of order $\varepsilon^{-\frac{1}{2}}$. These specific forms of error estimates coincide with the numerical results obtained in \cite{BZ19,SZ20}. |
| title | Convergence rates in the nonrelativistic limit of the cubic Klein-Gordon equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2304.06920 |