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Symmetry, Integrability and Geometry: Methods and Applications, 2015, Volume 11, 045, 20 pp.
DOI: https://doi.org/10.3842/SIGMA.2015.045
(Mi sigma1026)
 

This article is cited in 5 scientific papers (total in 5 papers)

Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville

Ognyan Christova, Georgi Georgievb

a Faculty of Mathematics and Informatics, Sofia University, 1164 Sofia, Bulgaria
b Department of Mathematics and Informatics, University of Transport, 1574, Sofia, Bulgaria
Full-text PDF (430 kB) Citations (5)
References:
Abstract: In this paper we study the equation
$$ w^{(4)} = 5 w'' (w^2 - w') + 5 w (w')^2 - w^5 + (\lambda z + \alpha)w + \gamma, $$
which is one of the higher-order Painlevé equations (i.e., equations in the polynomial class having the Painlevé property). Like the classical Painlevé equations, this equation admits a Hamiltonian formulation, Bäcklund transformations and families of rational and special functions. We prove that this equation considered as a Hamiltonian system with parameters $\gamma/\lambda = 3 k$, $\gamma/\lambda = 3 k - 1$, $k \in \mathbb{Z}$, is not integrable in Liouville sense by means of rational first integrals. To do that we use the Ziglin–Morales-Ruiz–Ramis approach. Then we study the integrability of the second and third members of the $\mathrm{P}_{\mathrm{II}}$-hierarchy. Again as in the previous case it turns out that the normal variational equations are particular cases of the generalized confluent hypergeometric equations whose differential Galois groups are non-commutative and hence, they are obstructions to integrability.
Keywords: Painlevé type equations; Hamiltonian systems; differential Galois groups; generalized confluent hypergeometric equations.
Received: December 10, 2014; in final form June 10, 2015; Published online June 17, 2015
Bibliographic databases:
Document Type: Article
Language: English
Citation: Ognyan Christov, Georgi Georgiev, “Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville”, SIGMA, 11 (2015), 045, 20 pp.
Citation in format AMSBIB
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\by Ognyan~Christov, Georgi~Georgiev
\paper Non-Integrability of Some Higher-Order Painlev\'e Equations in~the~Sense of~Liouville
\jour SIGMA
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\vol 11
\papernumber 045
\totalpages 20
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  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Symmetry, Integrability and Geometry: Methods and Applications
     
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