Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2/5+}(\mathbb{T})$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916527328985088 |
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| author | Bernier, Joackim Grébert, Benoît Robert, Tristan |
| author_facet | Bernier, Joackim Grébert, Benoît Robert, Tristan |
| contents | In this paper, we succeed in integrating Strichartz estimates (encoding the dispersive effects of the equations) in Birkhoff normal form techniques. As a consequence, we deduce a result on the long time behavior of quintic NLS solutions on the circle for small but very irregular initial data (in $H^s$ for $s > 2/5$). Note that since $2/5 < 1$, we cannot claim conservation of energy and, more importantly, since $2/5 < 1/2$, we must dispense with the algebra property of $H^s$. This is the first dynamical result where we use the dispersive properties of NLS in a context of Birkhoff normal form. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_05236 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2/5+}(\mathbb{T})$ Bernier, Joackim Grébert, Benoît Robert, Tristan Analysis of PDEs In this paper, we succeed in integrating Strichartz estimates (encoding the dispersive effects of the equations) in Birkhoff normal form techniques. As a consequence, we deduce a result on the long time behavior of quintic NLS solutions on the circle for small but very irregular initial data (in $H^s$ for $s > 2/5$). Note that since $2/5 < 1$, we cannot claim conservation of energy and, more importantly, since $2/5 < 1/2$, we must dispense with the algebra property of $H^s$. This is the first dynamical result where we use the dispersive properties of NLS in a context of Birkhoff normal form. |
| title | Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2/5+}(\mathbb{T})$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2305.05236 |