Construction of solutions of nonlinear irregular singular differential equations by Borel summable functions and an application to Painlev\'{e} equations

TL;DR

This paper develops a method using Borel summability to construct solutions for nonlinear irregular singular differential equations and applies it to Painlevé equations, enhancing the understanding of their asymptotic behavior.

Contribution

It introduces a refined approach to asymptotic expansions of solutions using Borel summability, specifically for nonlinear irregular singular differential equations, with applications to Painlevé equations.

Findings

Established Borel summability for solutions of nonlinear irregular singular equations

Provided a new framework for asymptotic analysis of Painlevé equations

Enhanced understanding of solution behavior near singularities

Abstract

A system of nonlinear differential equations $x^{1+\gamma}\frac{dY}{dx}= F_0(x)+A(x)Y+F(x,Y)$ is considered. We study more precisely the meaning of asymptotic expansion of transformations and solutions than preceding pioneering works, by using the theory of Borel summable functions in asymptotic analysis, and apply results to Painlev\'{e} equations.