Asymptotic stability near the soliton for quartic Klein-Gordon in 1D
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| Format: | Preprint |
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2022
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| _version_ | 1866929529612664832 |
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| author | Kairzhan, Adilbek Pusateri, Fabio |
| author_facet | Kairzhan, Adilbek Pusateri, Fabio |
| contents | We consider the nonlinear focusing Klein-Gordon equation in $1 + 1$ dimensions and the global space-time dynamics of solutions near the unstable soliton. Our main result is a proof of optimal decay, and local decay, for even perturbations of the static soliton originating from well-prepared initial data belonging to a subset of the stable manifold constructed in Bates-Jones (Dynamics reported, 1989) and Kowalczyk-Martel-Muñoz (J. Eur. Math. Soc., 2021). Our results complement those of Kowalczyk-Martel-Muñoz (J. Eur. Math. Soc., 2021) and confirm numerical results of Bizon-Chmaj-Szpak (J. Math. Phys., 2011) when considering nonlinearities $u^p$ with $p \geq 4$. In particular, we provide new information both local and global in space about asymptotically stable perturbations of the soliton under localization assumptions on the data. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_15008 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Asymptotic stability near the soliton for quartic Klein-Gordon in 1D Kairzhan, Adilbek Pusateri, Fabio Analysis of PDEs 43A32, 42B37, 35P25, 35Q55 We consider the nonlinear focusing Klein-Gordon equation in $1 + 1$ dimensions and the global space-time dynamics of solutions near the unstable soliton. Our main result is a proof of optimal decay, and local decay, for even perturbations of the static soliton originating from well-prepared initial data belonging to a subset of the stable manifold constructed in Bates-Jones (Dynamics reported, 1989) and Kowalczyk-Martel-Muñoz (J. Eur. Math. Soc., 2021). Our results complement those of Kowalczyk-Martel-Muñoz (J. Eur. Math. Soc., 2021) and confirm numerical results of Bizon-Chmaj-Szpak (J. Math. Phys., 2011) when considering nonlinearities $u^p$ with $p \geq 4$. In particular, we provide new information both local and global in space about asymptotically stable perturbations of the soliton under localization assumptions on the data. |
| title | Asymptotic stability near the soliton for quartic Klein-Gordon in 1D |
| topic | Analysis of PDEs 43A32, 42B37, 35P25, 35Q55 |
| url | https://arxiv.org/abs/2206.15008 |