4-Dimensional Power Geometry and its Application to $P_1$ -- $P_5$

TL;DR

This paper introduces 4-dimensional Power Geometry for second-order polynomial ODEs and applies it to analyze the first five Painlevé equations, providing a new geometric framework for their study.

Contribution

The paper develops a novel 4-dimensional Power Geometry approach and demonstrates its application to classical Painlevé equations, enhancing analytical tools for these nonlinear ODEs.

Findings

New geometric framework for second-order polynomial ODEs

Application of Power Geometry to Painlevé equations P1–P5

Potential for improved understanding of solution structures

Abstract

In the first section of this work we introduce 4-dimensional Power Geometry for second-order ODEs of a polynomial form. In the next five sections we apply this construction to the first five Painlev\'e equations.