Liouvillian integrability of three dimensional vector fields
Fuente:
arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912769507328000 |
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| author | Aziz, Waleed Christopher, Colin Pantazi, Chara Walcher, Sebastian |
| author_facet | Aziz, Waleed Christopher, Colin Pantazi, Chara Walcher, Sebastian |
| contents | We consider a three dimensional complex polynomial, or rational, vector field (equivalently, a two-form in three variables) which admits a Liouvillian first integral. We prove that there exists a first integral whose differential is the product of a rational 1-form with a Darboux function, or there exists a Darboux Jacobi multiplier. Moreover, we prove that Liouvillian integrability {in any dimension $n\geq 3$} always implies the existence of a first integral that is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20451 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Liouvillian integrability of three dimensional vector fields Aziz, Waleed Christopher, Colin Pantazi, Chara Walcher, Sebastian Exactly Solvable and Integrable Systems Dynamical Systems 34A99, 12H05, 34M15 We consider a three dimensional complex polynomial, or rational, vector field (equivalently, a two-form in three variables) which admits a Liouvillian first integral. We prove that there exists a first integral whose differential is the product of a rational 1-form with a Darboux function, or there exists a Darboux Jacobi multiplier. Moreover, we prove that Liouvillian integrability {in any dimension $n\geq 3$} always implies the existence of a first integral that is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field. |
| title | Liouvillian integrability of three dimensional vector fields |
| topic | Exactly Solvable and Integrable Systems Dynamical Systems 34A99, 12H05, 34M15 |
| url | https://arxiv.org/abs/2310.20451 |