Real solutions of the first Painlev\'e equation with large initial data

TL;DR

This paper analyzes the behavior of solutions to the first Painlevé equation with large initial data, using uniform asymptotics to classify solution types and establish connection formulas.

Contribution

It provides a rigorous proof of solution properties on the negative axis and refines the relation between large initial data and solution behavior.

Findings

Proof of properties of PI solutions on negative real axis

Relation between large initial data and solution types

Derivation of limiting connection formulas

Abstract

We consider three special cases of the initial value problem of the first Painlev\'e equation (PI). Our approach is based on the method of uniform asymptotics introduced by Bassom, Clarkson, Law and McLeod. A rigorous proof of a property of the PI solutions on the negative real axis, recently revealed by Bender and Komijani, is given by approximating the Stokes multipliers. Moreover, we build more precise relation between the large initial data of the PI solutions and their three different types of behavior as the independent variable tends to negative infinity. In addition, some limiting form connection formulas are obtained.