Painlev\'{e} Property and Generating Functions for Asymptotics

TL;DR

This paper introduces a novel method for asymptotic analysis of Painlevé equations using formal series representations, demonstrated on the third degenerate Painlevé equation.

Contribution

It presents a new approach based on formal series in two variables for analyzing Painlevé equations asymptotically.

Findings

Effective asymptotic analysis of the third degenerate Painlevé equation

Representation of solutions using formal series with rational functions

Potential for broader application to Painlevé equations

Abstract

This paper proposes a new approach to the asymptotic analysis of Painlev\'e equations. The approach is based on representing solutions of the Painlev\'e 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 Painlev\'e equation.