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Studies on the Painlev\'e Equations, V, Third Painlev\'e Equations of Special Type PIII(D7) and PIII(D8)

J. Math. Sci. Univ. Tokyo
Vol. 13 (2006), No. 2, Page 145--204.

OHYAMA, Yousuke; KAWAMUKO, Hiroyuki; SAKAI, Hidetaka; OKAMOTO, Kazuo
Studies on the Painlev\'e Equations, V, Third Painlev\'e Equations of Special Type PIII(D7) and PIII(D8)
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Abstract:
By means of geometrical classification (\cite{S}) of space of initial conditions, it is natural to consider the three types, PIII(D6), PIII(D7) and PIII(D8), for the third Painlev\'e equation. The fourth article of the series of papers \cite{O2} on the Painlev\'e equations is concerned with PIII(D6), generic type of the equation. The other two types, PIII(D7) and PIII(D8) are obtained as degeneration from PIII(D6); the present paper is devoted to investigating them in detail. Each of PIII(D7) and PIII(D8) is characterized through holonomic deformation of a linear differential equation and written as a Hamiltonian system. PIII(D7) contains a parameter and admits birational canonical transformations as symmetry, isomorphic to the affine Weyl group of type A(1)1. Sequence of τ-functions are defined for PIII(D7) by means of successive application of the translation of the symmetry of the equation; they satisfy the Toda equation. The τ-functions related to algebraic solutions of PIII(D7) are determined explicitly. The irreducibility of PIII(D7), as well as that of PIII(D8), is established, and there is no transcendental classical solution of these equations. A space of initial conditions is constructed for each of PIII(D7) and PIII(D8) by the use of successive blowing-up's of the projective plane P2.

Keywords: Painlev\'e equations, Isomonodromic deformation, B\"acklund transformation, Differential Galois theory, Space of initial conditions.

Mathematics Subject Classification (2000): 34M55, 34M15, 34M45.
Mathematical Reviews Number: MR2277519

Received: 2005-11-07