Extended symmetry of higher Painlevé equations of even periodicity and their rational solutions
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929605087068160 |
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| author | Aratyn, Henrik Gomes, José Francisco Lobo, Gabriel Vieira Zimerman, Abraham Hirsz |
| author_facet | Aratyn, Henrik Gomes, José Francisco Lobo, Gabriel Vieira Zimerman, Abraham Hirsz |
| contents | The structure of extended affine Weyl symmetry group of higher Painlevé equations of $N$ periodicity depends on whether $N$ is even or odd. We find that for even $N$, the symmetry group ${\widehat A}^{(1)}_{N-1}$ contains the conventional Bäcklund transformations $s_j, j=1,{\ldots},N$, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of $N$ points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to existence of degenerated solutions and for $N=4$ we explicitly show how the reflection automorphisms around even points cause degeneracy of a class of rational solutions obtained on the orbit of translation operators of ${\widehat A}^{(1)}_{3}$. We obtain the closed expressions for solutions and their degenerated counterparts in terms of determinants of Kummer polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03534 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extended symmetry of higher Painlevé equations of even periodicity and their rational solutions Aratyn, Henrik Gomes, José Francisco Lobo, Gabriel Vieira Zimerman, Abraham Hirsz Exactly Solvable and Integrable Systems The structure of extended affine Weyl symmetry group of higher Painlevé equations of $N$ periodicity depends on whether $N$ is even or odd. We find that for even $N$, the symmetry group ${\widehat A}^{(1)}_{N-1}$ contains the conventional Bäcklund transformations $s_j, j=1,{\ldots},N$, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of $N$ points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to existence of degenerated solutions and for $N=4$ we explicitly show how the reflection automorphisms around even points cause degeneracy of a class of rational solutions obtained on the orbit of translation operators of ${\widehat A}^{(1)}_{3}$. We obtain the closed expressions for solutions and their degenerated counterparts in terms of determinants of Kummer polynomials. |
| title | Extended symmetry of higher Painlevé equations of even periodicity and their rational solutions |
| topic | Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2409.03534 |