On the Theory of Higher-Order Painlevé Equations

Abstract

One important property of Painleve equations is their representability in the form of equivalent Hamiltonian systems with polynomial Hamiltonians. This property, originally discovered in [1] and later used in a number of papers [2–8], is especially important for the analysis of τ -functions [9], direct construction of analogs of Painleve equations from Hamiltonian systems [10], and isomonodromic deformation of linear systems described by Painleve equations [11, 12]. In the present paper, we construct equivalent Hamiltonian systems for the first few equations in the series of higher-order Painleve equations obtained by reduction from higher-order Korteweg– de Vries equations [2], construct analogs of τ -functions, and study polynomial Hamiltonians. The solutions of the first Painleve equation q′′ = 6q + t (P1) are meromorphic functions on C with local representation