Correlation between pole location and asymptotic behavior for Painlev\'e I solutions

Abstract

We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show that the extension provides a method of finding singularities of solutions of nonlinear differential equations, using asymptotic information. This transasymptotic matching method is applied to Painlev\'e's first equation. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions, obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole $x_p(C)$ on $\RR^+$ of a solution is monotonic in a parameter $C$ describing its asymptotics on antistokes lines$^1$, and obtain rigorous bounds for $x_p(C)$. We also derive the behavior of $x_p(C)$ for large $C\in\CC$. The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles.