## 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 $P_{\rm III}(D_8)$

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