Two error bounds of the elliptic asymptotics for the fifth Painlev\'e transcendents

TL;DR

This paper derives explicit error bounds for the elliptic asymptotics of fifth Painlevé transcendents, confirming conjectured estimates and enhancing understanding of their asymptotic behavior near infinity.

Contribution

It provides the first explicit error bounds for the elliptic asymptotics of fifth Painlevé transcendents, advancing the precision of asymptotic analysis in this area.

Findings

Explicit error bounds for elliptic asymptotics derived

Confirmed conjectured magnitude estimates of error terms

Extended analysis to correction functions related to the Lagrangian

Abstract

For the fifth Painlev\'e equation it is known that a general solution is represented asymptotically by an elliptic function in cheese-like strips near the point at infinity. We present an explicit asymptotic formula for the error term of this expression, which leads to an estimate for its magnitude as was conjectured. Analogous formula is obtained for the error term of the correction function associated with the Lagrangian.