Zero dispersion limit of the Calogero-Moser derivative NLS equation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909374870454272 |
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| author | Badreddine, Rana |
| author_facet | Badreddine, Rana |
| contents | We study the zero-dispersion limit of the Calogero-Moser derivative NLS equation $$i\partial_tu+\partial_x^2 u \pm\,2DΠ(|u|^2)u=0, \qquad x\in\mathbb{R},$$ starting from an initial data $u_0\in L^2_+(\mathbb{R})\cap L^\infty (\mathbb{R}),$ where $D=-i\partial_x,$ and $Π$ is the Szegő projector defined as $\widehat{Πu}(ξ)=1_{[0,+\infty)}(ξ)\widehat{u}(ξ).$ We characterize the zero-dispersion limit solution by an explicit formula. Moreover, we identify it, in terms of the branches of the multivalued solution of the inviscid Burgers-Hopf equation. Finally, we infer that it satisfies a maximum principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00119 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zero dispersion limit of the Calogero-Moser derivative NLS equation Badreddine, Rana Analysis of PDEs 37K10 primary, 30H10 secondary We study the zero-dispersion limit of the Calogero-Moser derivative NLS equation $$i\partial_tu+\partial_x^2 u \pm\,2DΠ(|u|^2)u=0, \qquad x\in\mathbb{R},$$ starting from an initial data $u_0\in L^2_+(\mathbb{R})\cap L^\infty (\mathbb{R}),$ where $D=-i\partial_x,$ and $Π$ is the Szegő projector defined as $\widehat{Πu}(ξ)=1_{[0,+\infty)}(ξ)\widehat{u}(ξ).$ We characterize the zero-dispersion limit solution by an explicit formula. Moreover, we identify it, in terms of the branches of the multivalued solution of the inviscid Burgers-Hopf equation. Finally, we infer that it satisfies a maximum principle. |
| title | Zero dispersion limit of the Calogero-Moser derivative NLS equation |
| topic | Analysis of PDEs 37K10 primary, 30H10 secondary |
| url | https://arxiv.org/abs/2403.00119 |