Turbulent Threshold for Continuum Calogero-Moser Models
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909348672831488 |
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| author | Hogan, James Kowalski, Matthew |
| author_facet | Hogan, James Kowalski, Matthew |
| contents | We determine the sharp mass threshold for Sobolev norm growth for the focusing continuum Calogero--Moser model. It is known that below the mass of $2π$, solutions to this completely integrable model enjoy uniform-in-time $H^s$ bounds for all $s \geq 0$. In contrast, we show that for arbitrarily small $\varepsilon > 0$ there exists initial data $u_0 \in H^\infty_+$ of mass $2π+ \varepsilon$ such that the corresponding maximal lifespan solution $u : (T_-, T_+) \times \mathbb{R} \to \mathbb{C}$ satisfies $\lim_{t \to T_\pm} \|u(t)\|_{H^s} = \infty$ for all $s > 0$. As part of our proof, we demonstrate an orbital stability statement for the soliton and a dispersive decay bound for solutions with suitable initial data. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_16609 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Turbulent Threshold for Continuum Calogero-Moser Models Hogan, James Kowalski, Matthew Analysis of PDEs Mathematical Physics 37K10, 35Q55 We determine the sharp mass threshold for Sobolev norm growth for the focusing continuum Calogero--Moser model. It is known that below the mass of $2π$, solutions to this completely integrable model enjoy uniform-in-time $H^s$ bounds for all $s \geq 0$. In contrast, we show that for arbitrarily small $\varepsilon > 0$ there exists initial data $u_0 \in H^\infty_+$ of mass $2π+ \varepsilon$ such that the corresponding maximal lifespan solution $u : (T_-, T_+) \times \mathbb{R} \to \mathbb{C}$ satisfies $\lim_{t \to T_\pm} \|u(t)\|_{H^s} = \infty$ for all $s > 0$. As part of our proof, we demonstrate an orbital stability statement for the soliton and a dispersive decay bound for solutions with suitable initial data. |
| title | Turbulent Threshold for Continuum Calogero-Moser Models |
| topic | Analysis of PDEs Mathematical Physics 37K10, 35Q55 |
| url | https://arxiv.org/abs/2401.16609 |