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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2411.13132 |
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| _version_ | 1866915027287539712 |
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| author | Shen, Jia Wang, Yanni Zheng, Haohao |
| author_facet | Shen, Jia Wang, Yanni Zheng, Haohao |
| contents | This paper presents an investigation into the high-order asymptotic expansion for 2D and 3D cubic nonlinear Klein-Gordon equations in the non-relativistic limit regime. There are extensive numerical and analytic results concerning that the solution of NLKG can be approximated by first-order modulated Schrödinger profiles in terms of $e^{i\frac t {\varepsilon^2}}v + c.c. $, where $v$ is the solution of related NLS and ``$c.c.$" denotes the complex conjugate. Particularly, the best analytic result up to now is given in \cite{lei}, which proves that the $L_x^2$ norm of the error can be controlled by $\varepsilon^2 +(\varepsilon^2t)^{\frac α4}$ for $H^α_x$-data, $α\in [1, 4]$. As for the high-order expansion, to our best knowledge, there are only numerical results, while the theoretical one is lacking.
In this paper, we extend this study further and give the first high-order analytic result. We introduce the high-order expansion inspired by the numerical experiments in \cite{schratz2020, faou2014a}: \[
e^{i\frac t {\varepsilon^2}}v
+\varepsilon^2
\Big( \frac 18 e^{3i\frac t {\varepsilon^2} }v^3
+e^{i\frac t {\varepsilon^2}} w
\Big)
+c.c.,
\] where $w$ is the solution to some specific Schrödinger-type equation. We show that the $L_x^2$ estimate of the error is of higher order $\varepsilon^4+\left(\varepsilon^2t\right)^\fracα{4}$ for $H^α_x$-data, $α\in [4, 8]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13132 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | High-order asymptotic expansion for the nonlinear Klein-Gordon equation in the non-relativistic limit regime Shen, Jia Wang, Yanni Zheng, Haohao Analysis of PDEs 35B40, 35B65 This paper presents an investigation into the high-order asymptotic expansion for 2D and 3D cubic nonlinear Klein-Gordon equations in the non-relativistic limit regime. There are extensive numerical and analytic results concerning that the solution of NLKG can be approximated by first-order modulated Schrödinger profiles in terms of $e^{i\frac t {\varepsilon^2}}v + c.c. $, where $v$ is the solution of related NLS and ``$c.c.$" denotes the complex conjugate. Particularly, the best analytic result up to now is given in \cite{lei}, which proves that the $L_x^2$ norm of the error can be controlled by $\varepsilon^2 +(\varepsilon^2t)^{\frac α4}$ for $H^α_x$-data, $α\in [1, 4]$. As for the high-order expansion, to our best knowledge, there are only numerical results, while the theoretical one is lacking. In this paper, we extend this study further and give the first high-order analytic result. We introduce the high-order expansion inspired by the numerical experiments in \cite{schratz2020, faou2014a}: \[ e^{i\frac t {\varepsilon^2}}v +\varepsilon^2 \Big( \frac 18 e^{3i\frac t {\varepsilon^2} }v^3 +e^{i\frac t {\varepsilon^2}} w \Big) +c.c., \] where $w$ is the solution to some specific Schrödinger-type equation. We show that the $L_x^2$ estimate of the error is of higher order $\varepsilon^4+\left(\varepsilon^2t\right)^\fracα{4}$ for $H^α_x$-data, $α\in [4, 8]$. |
| title | High-order asymptotic expansion for the nonlinear Klein-Gordon equation in the non-relativistic limit regime |
| topic | Analysis of PDEs 35B40, 35B65 |
| url | https://arxiv.org/abs/2411.13132 |