Hyperasymptotics for nonlinear ODEs II. The first Painlevé equation and a second-order Riccati equation

Abstract

This paper is a sequel to Olde Daalhuis (Olde Daalhuis 2005 Proc. R. Soc. A 461 , 2503–2520) in which we constructed hyperasymptotic expansions for a simple first-order Riccati equation. In this paper we illustrate that the method also works for more complicated nonlinear ordinary differential equations, and that in those cases the Riemann sheet structure of the so-called Borel transform is much more interesting. The two examples are the first Painlevé equation and a second-order Riccati equation. The main tools that we need are transseries expansions and Stokes multipliers. Hyperasymptotic expansions determine the solutions uniquely. Some details are given about solutions that are real-valued on the positive real axis.