Geometry of an elliptic-difference equation related to Q4

Abstract

In this paper, we investigate a nonlinear non-autonomous elliptic difference equation, which was constructed by Ramani, Carstea and Grammaticos by integrable deautonomization of a periodic reduction of the discrete Krichever-Novikov equation, or Q4. We show how to construct it as a birational mapping on a rational surface blown up at eight points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$, and find its affine Weyl symmetry, placing it in the geometric framework of the Painlevé equations. The initial value space is ell-$A_0^{(1)}$ and its symmetry group is $W(F_4^{(1)})$. We show that the deautonomization is consistent with the lattice-geometry of Q4 by giving an alternative construction, which is a reduction from Q4 in the usual sense. A more symmetric reduction of the same kind provides another example of a second-order integrable elliptic difference equation.

arXiv comment22 pages, 4 figures

References in corpus (10)

• Classification of integrable equations on quad-graphs. The consistency approach
• Algebraic entropy
• Lax pair for the Adler (lattice Krichever-Novikov) System
• Bäcklund transformation for the Krichever-Novikov equation
• Algebraic entropy and the space of initial values for discrete dynamical systems
• Riccati Solutions of Discrete Painlevé Equations with Weyl Group Symmetry of Type $E_8^{(1)}$
• Cauchy problem for integrable discrete equations on quad-graphs
• Projective reduction of the discrete Painlevé system of type $(A_2+A_1{)}^{(1)}$
• Singularities of Type-Q ABS Equations
• Solutions of Adler's lattice equation associated with 2-cycles of the Backlund transformation

Cited by in corpus (6)

• Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations
• Lax pairs of discrete Painlevé equations: $(A_2+A_1{)}^{(1)}$ case
• Full-parameter discrete Painlevé systems from non-translational Cremona isometries
• Elliptic Painlevé equations from next-nearest-neighbor translations on the $E_8^{(1)}$ lattice
• Orthogonal Polynomials for the Gaussian Weight with a Jump and Discrete Painlevé Equations
• Two variations on $(A_3\times A_1\times A_1{)}^{(1)}$ type discrete Painlevé equations