On the theory of higher-order Painlevé equations

Abstract

One important property of Painlev´e 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 Painlev´e equations from Hamiltonian systems [10], and isomonodromic deformation of linear systems described by Painlev´e equations [11, 12]. In the present paper, we construct equivalent Hamiltonian systems for the first few equations in the series of higher-order Painlev´e equations obtained by reduction from higher-order Korteweg– de Vries equations [2], construct analogs of τ -functions, and study polynomial Hamiltonians.