# Connection Formulae for Asymptotics of the Fifth Painlev\'e Transcendent on the Imaginary Axis: I

## TL;DR

This paper derives asymptotic expansions and connection formulas for solutions of the fifth Painlevé equation on the imaginary axis, using monodromy data and numerical verification to enhance understanding of its complex behavior.

## Contribution

It introduces new connection formulas for the fifth Painlevé transcendent's asymptotics on the imaginary axis, based on monodromy data and numerical validation.

## Key findings

- Asymptotic expansions for solutions as t→i∞ are obtained.
- Connection formulas linking asymptotics are derived.
- Numerical verification confirms the theoretical results.

## Abstract

Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlev\'e equation as $t\to\imath\infty$ are found. These asymptotics are parameterized by monodromy data of the associated linear ODE. $$ \frac{d}{d\lambda}Y= \left(\frac t2\sigma_3 + \frac{A_0}\lambda+\frac{A_1}{\lambda-1}\right)Y. $$ The parametrization allows one to derive connection formulas for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulas are also considered.