Kapaev's global asymptotics of the fourth Painlev\'e transcendents. Elliptic asymptotics

TL;DR

This paper rigorously derives elliptic asymptotic representations for the fourth Painlevé transcendents, confirming and extending Kapaev's earlier announced results with detailed proofs and correction functions.

Contribution

It provides the first rigorous proofs of elliptic asymptotics for the fourth Painlevé transcendents, including correction functions, previously only announced without proof.

Findings

Elliptic asymptotic representations are established for the fourth Painlevé transcendents.

The paper confirms Kapaev's announced results with rigorous proofs.

New correction functions related to the asymptotics are obtained.

Abstract

For the fourth Painlev\'e transcendents we derive elliptic asymptotic representations, which were announced by late Professor Kapaev without proofs. Then we newly obtain related results including the correction function.