arXiv subjects
Publications and source records attributed to Elynor Liu.
We study the asymptotic behaviour of two multiplicative- ($q$-) discrete Painlevé equations as their respective independent variable goes to infinity. It is shown that the generic asymptotic behaviours are given by elliptic functions. We extend the method of averaging to these equations to show that the energies are slowly varying. The Picard-Fuchs equation is derived for a special case of $q$-P$_{\rm III}$
Following the study of complex elliptic-function-type asymptotic behaviours of the Painlevé 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.