A direct method to find Stokes multipliers in closed form for P1 and more general integrable systems

TL;DR

This paper presents a new rigorous method based on Borel summability to analyze nonlinear ODEs near infinity, providing explicit Stokes multipliers for Painlevé I and related integrable systems without relying on linearization techniques.

Contribution

The authors develop a novel approach to compute Stokes multipliers directly from the Painlevé-Kowalevski property, applicable to a broad class of integrable and near-integrable second order ODEs.

Findings

Closed-form Stokes multipliers for Painlevé I

Global asymptotics for solutions with power-like behavior

Method applicable to a wider class of second order ODEs

Abstract

We introduce a new rigorous method, based on Borel summability and asymptotic constants of motion generalizing \cite{invent} and \cite{ode1}, to analyze singular behavior of nonlinear ODEs in a neighborhood of infinity and provide global information about their solutions. In equations with the Painlev\'e-Kowalevski (P-K) property (stating that movable singularities are not branched) it allows for solving connection problems. The analysis in carried in detail for P$_1$, $y"=6y^2+z$, for which we find the Stokes multipliers in closed form and global asymptotics for solutions having power-like behavior in some direction in $\CC$, in particular for the tritronqu\'ees. Calculating the Stokes multipliers solely relies on the P-K property and does not use linearization techniques such as Riemann-Hilbert or isomonodromic reformulations. We discuss how the approach would work to calculate connection constants for a larger class of P-K integrable equations. We develop methods for finding asymptotic expansions in sectors where solutions have infinitely many singularities. These techniques do not rely on integrability and apply to more general second order ODEs which, after normalization, are asymptotically close to autonomous Hamiltonian systems.