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An augmentation of the phase space of the system of type \(A_4^{(1)}\). (English) Zbl 1074.34084

The author studies nonlinear systems with the Weyl group symmetry of type \(A_l^{(1)}\), \(l=2,4\), introduced by Noumi and Yamada. It is known that the system with \(l=2\) is equivalent to the classical fourth Painlevé equation PIV, the case \(l=3\) corresponds to PV, and \(l\geq4\) leads to higher-order generalizations of the Painlevé equations. A generic solution to a classical Painlevé equation is determined by the Cauchy data, while the parametrization of the total solution set is found by Okamoto using successive blowing-ups of the spaces of initial conditions at accessible singular points. For the higher Painlevé systems, however, the Okamoto’s method leads to comparably cumbersome computations. The author presents an elementary but effective approach to this problem.
Namely, it is known that the Laurent expansion to a Painlevé function is determined by two arbitrary constants which are the pole location and one of the minor term coefficients. The author observes that a similar assertion holds true for the system of type \(A_4^{(1)}\). Furthermore, he shows that some Bäcklund transformations turn the Laurent expansion into Taylor series which solve nonlinear systems of the first-order ODEs with a polynomial r.h.s. This proves the existence of a genuine meromorphic solution with the above formal Laurent expansion at a given point and provides its parametrization by the Cauchy data for the transformed system. The latter amounts to a construction of the desired parametrizing set as a disjoint union of subsets corresponding to families of holomorphic and various meromorphic solutions at the given point.

MSC:

34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
12H20 Abstract differential equations
34A26 Geometric methods in ordinary differential equations
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
33E17 Painlevé-type functions
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