Elliptic Asymptotic Behaviours of Continuous and Discrete Painlevé Equations

Abstract

This thesis investigates the asymptotic behaviours of both continuous and discrete Painlevé equations as their independent variables approach complex infinity. We focus on the third to the fifth Painlevé equations and three discrete Painlevé equations referred to as d-PI, q-PI and q-PIII in the literature. In each case, the generic asymptotic behaviours are found to be given by elliptic functions. We deduce the properties of the respective elliptic functions in terms of energy-like parameters which are Hamiltonians and invariants of the corresponding autonomous continuous and discrete Painlevé equations. By using the method of averaging, we show that the Hamiltonians and invariants vary slowly across a local period parallelogram of the leading-order behaviour. For the continuous Painlevé equations we show the surprising result that all the equations PI-PV share the same modulation to the first two orders. We also show that the Hamiltonians are bounded on a path to infinity at any fixed angle. The Picard-Fuchs equations are derived for the related elliptic integrals. We solve the Picard-Fuchs equation at its regular singular points to find expansions of the approximate-periods at their degenerate points. The method of averaging is extended to discrete Painlevé equations to show that the invariants are also slowly varying. We also find the singular points of the invariant curves. The Picard-Fuchs equation is derived for q-PIII for its periods. The expansion of the periods at their degenerate points are also given. The main new results of this thesis are summarised in Theorems 1 and 2.1-2.3.