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

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 $E_{\mathrm{J}}$ of each equation ${\mathrm{P}}_{\mathrm{J}}$, $\mathrm{J}=\mathrm{I},\dots ,\mathrm{V}$, 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.