Studies on the Painlev\'e Equations, V, Third Painlev\'e Equations of Special Type $P_{\rm III}(D_7)$ and $P_{\rm III}(D_8)$

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Studies on the Painlev\'e Equations, V, Third Painlev\'e Equations of Special Type $P_{\rm III}(D_7)$ and $P_{\rm III}(D_8)$

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 $P_{\rm III}(D_7)$ and $P_{\rm III}(D_8)$
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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,

$P_{\rm III}(D_6)$ ,

$P_{\rm III}(D_7)$ and

$P_{\rm III}(D_8)$ , for the third Painlev\'e equation. The fourth article of the series of papers \cite{O2} on the Painlev\'e equations is concerned with

$P_{\rm III}(D_6)$ , generic type of the equation. The other two types,

$P_{\rm III}(D_7)$ and

$P_{\rm III}(D_8)$ are obtained as degeneration from

$P_{\rm III}(D_6)$ ; the present paper is devoted to investigating them in detail. Each of

$P_{\rm III}(D_7)$ and

$P_{\rm III}(D_8)$ is characterized through holonomic deformation of a linear differential equation and written as a Hamiltonian system.

$P_{\rm III}(D_7)$ contains a parameter and admits birational canonical transformations as symmetry, isomorphic to the affine Weyl group of type

$A_1^{(1)}$ . Sequence of

$\tau$ -functions are defined for

$P_{\rm III}(D_7)$ by means of successive application of the translation of the symmetry of the equation; they satisfy the Toda equation. The

$\tau$ -functions related to algebraic solutions of

$P_{\rm III}(D_7)$ are determined explicitly. The irreducibility of

$P_{\rm III}(D_7)$ , as well as that of

$P_{\rm III}(D_8)$ , is established, and there is no transcendental classical solution of these equations. A space of initial conditions is constructed for each of

$P_{\rm III}(D_7)$ and

$P_{\rm III}(D_8)$ by the use of successive blowing-up's of the projective plane

${\mathbb P}^2$ .

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

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Studies on the Painlev\'e Equations, V, Third Painlev\'e Equations of Special Type PIII(D7)P_{\rm III}(D_7) and PIII(D8)P_{\rm III}(D_8)

Studies on the Painlev\'e Equations, V, Third Painlev\'e Equations of Special Type $P_{\rm III}(D_7)$ and $P_{\rm III}(D_8)$