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Hyperasymptotics for nonlinear ODEs. II: The first Painlevé equation and a second-order Riccati equation. (English) Zbl 1206.34077

Summary: This paper is a sequel to the author’s first part [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, No. 2060, 2503–2520 (2005; Zbl 1186.34076)] 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.

MSC:
34E05Asymptotic expansions (ODE)
34M40Stokes phenomena and connection problems (ODE in the complex domain)
34M55Painlevé and other special equations; classification, hierarchies
34M60Singular perturbation problems in the complex domain