Liouvillian integrability of three dimensional vector fields

Autor:	Aziz, Waleed, Christopher, Colin, Pantazi, Chara, Walcher, Sebastian
Rok vydání:	2023
Předmět:	Nonlinear Sciences - Exactly Solvable and Integrable Systems Mathematics - Dynamical Systems 34A99 12H05 34M15
Druh dokumentu:	Working Paper
Popis:	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. Comment: 20 pages. Changed from first submitted version: Rewritten Abstract and Introduction; simplified and clarified proofs (Lemma 3, Lemmma 5, Theorem 4) Second change: Theorem 4 holds in any dimension $n\geq 3$ (with the same proof)
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2310.20451 Zobrazit plný text záznamu View this record from Arxiv