A direct method to find Stokes multipliers in closed form for P$_1$ and more general integrable systems

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.