Abstract
Following the study of complex elliptic-function-type asymptotic behaviours of the Painlevé equations by Boutroux, Joshi and Kruskal for and , we provide new results for elliptic-function-type behaviours admitted by , , and , in the limit as the independent variable z approaches infinity. We show how the Hamiltonian EJ of each equation , , 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 – share the same modulation of E to the first two orders.
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