Asymptotic Behaviours Given by Elliptic Functions in $P_I$--$P_V$

Abstract

Following the study of complex elliptic-function-type asymptotic behaviours of the Painlev\'e equations by Boutroux and Joshi and Kruskal for $P_I$ and $P_{II}$, we provide new results for elliptic-function-type behaviours admitted by $P_{III}$, $P_{IV}$, and $P_{V}$, in the limit as the independent variable $z$ approaches infinity. We show how the Hamiltonian $E_{\rm J}$ of each equation $\rm P_{\rm J}$, $\rm J=I, \ldots , V$, varies across a local period parallelogram of the leading-order behaviour, by applying the method of averaging in the complex $z$-plane. Surprisingly, our results show that all the equations $P_I-P_{V}$ share the same modulation of $E$ to the first two orders.