Non-integrability of the rational Sasano system of type $A^{(2)}_4$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910436987764736 |
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| author | Stoyanova, Tsvetana |
| author_facet | Stoyanova, Tsvetana |
| contents | In this paper we study the integrability of the Sasano system of type $A^{(2)}_4$ from the point of view of the Hamiltonian dynamics. We prove rigorously that for these values of the parameters for which the Sasano system of type $A^{(2)}_4$ has a particular rational solution it is not integrable by rational first integrals. By an explicit computation we show that the connected component $G^0$ of the unit element of the differential Galois group of the normal variational equations along a simple particular rational solution is a direct product of two groups $SL_2(\mathbb{C})$. This the Morales-Ramis theory leads to a non-integrable result. Moreover using Bäcklund transformations we extend the obtained particular non-integrable result to the entire orbit of the parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14351 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-integrability of the rational Sasano system of type $A^{(2)}_4$ Stoyanova, Tsvetana Mathematical Physics Classical Analysis and ODEs 34M55, 37M30, 33C15, 37J65 In this paper we study the integrability of the Sasano system of type $A^{(2)}_4$ from the point of view of the Hamiltonian dynamics. We prove rigorously that for these values of the parameters for which the Sasano system of type $A^{(2)}_4$ has a particular rational solution it is not integrable by rational first integrals. By an explicit computation we show that the connected component $G^0$ of the unit element of the differential Galois group of the normal variational equations along a simple particular rational solution is a direct product of two groups $SL_2(\mathbb{C})$. This the Morales-Ramis theory leads to a non-integrable result. Moreover using Bäcklund transformations we extend the obtained particular non-integrable result to the entire orbit of the parameters. |
| title | Non-integrability of the rational Sasano system of type $A^{(2)}_4$ |
| topic | Mathematical Physics Classical Analysis and ODEs 34M55, 37M30, 33C15, 37J65 |
| url | https://arxiv.org/abs/2402.14351 |