Non-integrability of Painlevé V equations in the Liouville sense and Stokes phenomenon
The six Painlevé equations were introduced by P. Painlevé and B. Gambier who classified rational differential equations of the second order with no movable critical points. Later, these equations appeared in numerous physical applications (see introduction to [\textit{A. S. Fokas} et al., Painlevé transcendents. The Riemann-Hilbert approach. Mathematical Surveys and Monographs 128. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1111.34001)]).NEWLINENEWLINE The solutions of these equations turn out to be in general a new class of special functions, that is they cannot be expressed as a finite algebraic combination of elementary and elliptic functions and a finite number of contour integrals and quadratures of these functions [\textit{H. Umemura}, ``Birational automorphism groups and differential equations'', Nagoya Math. J. 119, 1--80 (1990; Zbl 0714.12009)].NEWLINENEWLINE The main object of the investigation in the paper under review is the fifth Painlevé equation NEWLINE\[NEWLINEy''=\biggl(\frac{1}{2y}+\frac{1}{y-1}\biggr)(y')^2 -\frac{1}{t}y'+\frac{(y-1)^2}{2t^2}(\kappa_{\infty}y-\frac{\kappa_0^2}{y})-(\theta+1) \frac{y}{t}+\frac{\delta y(y+1)}{(y-1)}.NEWLINE\]NEWLINE All Painlevé equations are equivalent to time-dependent Hamiltonian systems [\textit{K. Okamoto}, ``Studies on the Painlevé equations. II: Fifth Painlevé equation \(P_ V\)'', Jap. J. Math., New Ser. 13, No. 1, 47--76 (1987; Zbl 0694.34005)], [\textit{J. Malmquist}, ``Sur les équations différentielles du second ordre dont l'intégrale générale a ses points critiques fixes'', Ark. för Mat., Astron. och Fys. 17, No. 8, 89 p. (1923; JFM 49.0305.02)]. In the case of the Painlevé II equation the corresponding Hamiltonian system is non-integrable [\textit{J. J. Moralez-Ruiz}, ``A remark about the Painlevé transcendents'', Séminaires et Congrès 14, 229--235 (2006; Zbl 1140.37016)]. Also the corresponding Hamiltonian system is nonintegrable for Painlevé \(VI\) for some values of parameters [\textit{E. Horozov} and \textit{T. Stoyanova}, ``Non-integrability of some Painlevé VI-equations and dilogarithms'', Regul. Chaotic Dyn. 12, No. 6, 622--629 (2007; Zbl 1229.37057)], [\textit{T. Stoyanova}, ``Non-integrability of Painlevé VI equations in the Liouville sense'', Nonlinearity 22, No. 9, 2201--2230 (2009; Zbl 1179.34103)].NEWLINENEWLINE The main result of the paper under review is the proof of non-integrability of the Hamiltonian system corresponding to the fifth Painlevé equation with \(\kappa_{\infty}=0\), \(\kappa_0=-\theta\), \(\theta\notin \mathbb{N}\), and for \(\kappa_{\infty}=0\), \(\kappa_0=-\theta=1\).
scientific article; zbMATH DE number 6050659