Discrete Painlevé equations from singularity patterns: the asymmetric trihomographic case

Abstract

We derive the discrete Painlevé equations associated to the affine Weyl group E${}_8^{(1)}$ that can be represented by an (in the QRT sense) "asymmetric" trihomographic system. The method used in this paper is based on singularity confinement. We start by obtaining all possible singularity patterns for a general asymmetric trihomographic system and discard those patterns which cannot lead to confined singularities. Working with the remaining ones we implement the confinement conditions and derive the corresponding discrete Painlevé equations, which involve two variables. By eliminating either of these variables we obtain a "symmetric" equation. Examining all these equations of a single variable, we find that they coincide exactly with those derived in previous works of ours, thereby establishing the completeness of our results.

20 pages, 1 figure

References in corpus (5)

• Geometric Aspects of Painlevé Equations
• Deautonomisation by singularity confinement: an algebro-geometric justification
• The redemption of singularity confinement
• A systematic method for constructing discrete Painlevé equations in the degeneration cascade of the E${}_8$ group
• Restoring discrete Painlevé equations from an E${}_8^{(1)}$-associated one

Cited by in corpus (1)

• Singularity confinement in delay-differential Painlevé equations