Painlevé Property and Generating Functions for Asymptotics
A. V. Kitaev
TL;DR
The paper develops a two-variable generating-function framework for the asymptotic analysis of Painlevé equations, focusing on the degenerate third Painlevé equation. By representing solutions as $u(\tau)=\tau^r A(x(\tau),y(\tau))$ with $A(x,y)=\sum_{k\ge0} y^k A_k(x)$ and its conjugate $B_k(y)$, the authors derive PDEs that govern the generating functions and reveal a rich root-structure underpinning the asymptotics. They construct explicit, rational generating functions for the regular-singularity case and the irregular, trig- and elliptic-type regimes, linking their results to known Boutroux-type asymptotics and to truncated trans-series. The work provides a unified, generating-function-based route to uniform and cross-validated asymptotics across different singularity regimes and establishes connections to isomonodromy and monodromy data in DP3. Overall, it offers new analytic tools for obtaining and matching asymptotic expansions in DP3, with potential applicability to broader Painlevé systems.
Abstract
This paper proposes a new approach to the asymptotic analysis of Painlevé equations. The approach is based on representing solutions of the Painlevé equations using formal series in two variables, $\sum_{k=0}^{\infty}y^kA_k(x)$, with rational functions $A_k(x)$. The approach is applied to the asymptotic analysis of the third degenerate Painlevé equation.