Elliptic Painlevé equations from next-nearest-neighbor translations on the $E_8^{(1)}$ lattice

Abstract

The well known elliptic discrete Painlevé equation of Sakai is constructed by a standard translation on the $E_8^{(1)}$ lattice, given by nearest neighbor vectors. In this paper, we give a new elliptic discrete Painlevé equation obtained by translations along next-nearest-neighbor vectors. This equation is a generic (8-parameter) version of a 2-parameter elliptic difference equation found by reduction from Adler's partial difference equation, the so-called Q4 equation. We also provide a projective reduction of the well known equation of Sakai.

14 pages, 1 figure

References in corpus (2)

• Geometric Aspects of Painlevé Equations
• Geometry of an elliptic-difference equation related to Q4

Cited by in corpus (2)

• Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations
• Full-parameter discrete Painlevé systems from non-translational Cremona isometries