An Asymptotic Reduction of a Painleve' VI equation to a Painleve' III (January 2011)

TL;DR

This paper demonstrates how, near a critical point, Painleve' VI equations can be asymptotically simplified to Painleve' III equations, enabling easier analysis of their critical behaviors.

Contribution

It introduces an asymptotic reduction method from PVI to PIII equations near critical points, facilitating the study of PVI transcendents.

Findings

PVI can be asymptotically reduced to PIII near critical points

Leading terms of PVI critical behaviors can be derived from PIII behaviors

Provides a new approach to analyze Painleve equations asymptotically

Abstract

When the independent variable is close to a critical point, it is shown that PVI can be asymptotically reduced to PIII. In this way, it is possible to compute the leading term of the critical behaviors of PVI transcendents starting from the behaviors of PIII transcendents.