From Kp/Uc Hierarchies to Painlevé Equations

Abstract

We study the underlying relationship between Painlevé equations and infinite-dimensional integrable systems, such as the KP and UC hierarchies. We show that a certain reduction of these hierarchies by requiring homogeneity and periodicity yields Painlevé equations, including their higher order generalization. This result allows us to clearly understand various aspects of the equations, e.g., Lax formalism, Hirota bilinear relations for τ-functions, Weyl group symmetry, and algebraic solutions in terms of the character polynomials, i.e., the Schur function and the universal character.

Six-dimensional Painlevé systems and their particular solutions in terms of rigid systems

“…In this section, we recall the Hamiltonian of the six-dimensional Painlevé systems which have been already derived in [7,23,13,2].…”

On Solutions of the Fuji-Suzuki-Tsuda System

“…The purpose of this note is to initiate a systematic study of rank N Fuji-Suzuki-Tsuda system, abbreviated below as FST N . This Hamiltonian system of nonlinear non-autonomous ODEs first appeared as a particular reduction of the Drinfeld-Sokolov hierarchy [12,28], and independently in [29] as a reduction of the universal character hierarchy. Its fundamental significance comes from the isomonodromic theory [11], where it describes deformations of rank N Fuchsian systems with 4 regular singular points, 2 of which have special spectral type (N − 1, 1) [29,31].…”

A quantum isomonodromy equation and its application to {\cal N}=2SU(N) gauge theories

“…The first problem is to look for the suitable isomonodromy system with higher rank symmetries. Fortunately, a nice candidate appeared in recent work [15] [17]. The system, which we call Fuji-Suzuki-Tsuda (FST) equation 2 , can be described as an isomonodromy deforsimilarity reduction of his 'UC-hierarchy' (certain generalization of KP hierarchy).…”