Miura transformations for discrete Painlevé equations coming from the affine E${}_8$ Weyl group

Abstract

We derive integrable equations starting from autonomous mappings with a general form inspired by the additive systems associated to the affine Weyl group E${}_8^{(1)}$. By deautonomisation we obtain two hitherto unknown systems, one of which turns out to be a linearisable one, and we show that both these systems arise from the deautonomisation of a non-QRT mapping. In order to unambiguously prove the integrability of these nonautonomous systems, we introduce a series of Miura transformations which allows us to prove that one of these systems is indeed a discrete Painlevé equation, related to the affine Weyl group E${}_7^{(1)}$, and to cast it in canonical form. A similar sequence of Miura transformations allows us to effectively linearise the second system we obtain. An interesting off-shoot of our calculations is that the series of Miura transformations, when applied at the autonomous limit, allows one to transform a non-QRT invariant into a QRT one.

10 pages, no figures

References in corpus (1)

• Geometric Aspects of Painlevé Equations

Cited by in corpus (2)

• Studies on spaces of initial conditions for nonautonomous mappings of the plane
• Multiplicative equations related to the affine Weyl group E${}_8$