Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations

Abstract

It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated. In this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. %By choosing a particular Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painlevé equations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group and some additional geometric data. This technique allows us to obtain factorized expressions of discrete Painlevé equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.

48 pages

References in corpus (3)

• Geometric Aspects of Painlevé Equations
• Geometry of an elliptic-difference equation related to Q4
• Elliptic Painlevé equations from next-nearest-neighbor translations on the $E_8^{(1)}$ lattice

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• Orthogonal Polynomials for the Gaussian Weight with a Jump and Discrete Painlevé Equations
• Solving the Quispel-Roberts-Thompson maps using Kajiwara-Noumi-Yamada's representation of elliptic curves
• Singularity confinement and proliferation of tau functions for a general differential-difference Sawada-Kotera equation