Non-integrability of the fourth Painlevé equation in the Liouville–Arnold sense

Tsvetana Stoyanova

doi:10.1088/0951-7715/27/5/1029

Nonlinearity

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Non-integrability of the fourth Painlevé equation in the Liouville–Arnold sense

Tsvetana Stoyanova¹

Published 25 April 2014 • © 2014 IOP Publishing Ltd & London Mathematical Society
Nonlinearity, Volume 27, Number 5 Citation Tsvetana Stoyanova 2014 Nonlinearity 27 1029 DOI 10.1088/0951-7715/27/5/1029

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cveti@fmi.uni-sofia.bg

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¹ Department of Mathematics and Informatics, Sofia University, 5 J Bourchier Blvd, Sofia 1164, Bulgaria

Dates

Received 1 May 2013
Accepted 19 March 2014
Published 25 April 2014

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Abstract

In this paper we are concerned with the integrability of the fourth Painlevé equation (P_IV) from the point of view of the Hamiltonian dynamics. We prove that the fourth Painlevé equation

$\begin{equation}\ddot{y}=\frac{1}{2 y}\,(\dot{y})^2 + \frac{3}{2}\,y^3 + 4 t y^2 + 2 (t^2 - a)\,y + \frac{b}{y}, \end{equation} \tag{ 0.1 }$

with parameters a = m, b = −2(1 + 2n + m) where $m, n \in {\mathbb Z}$ $m, n \in {\mathbb Z}$ , is not integrable in the Liouville–Arnold sense by means of meromorphic first integrals. We explicitly compute formal and analytic invariants of the second variational equations which generate topologically the differential Galois group. In this way our calculations and the Ziglin–Ramis–Morales-Ruiz–Simó method yield the non-integrability results.

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Non-integrability of the fourth Painlevé equation in the Liouville–Arnold sense

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