Affine linear and D4 symmetric lattice equations : symmetry analysis and reductions

TL;DR

This paper analyzes affine linear lattice equations with D4 symmetry, deriving their properties, symmetry structures, and reductions to discrete Painlevé equations, advancing understanding of their integrability and symmetry features.

Contribution

It systematically characterizes the symmetry generators of affine linear lattice equations with D4 symmetry and explores their reductions to discrete Painlevé equations.

Findings

Derived basic properties and conservation laws of the equations.

Established the generic form of symmetry generators.

Performed symmetry reductions to discrete Painlevé equations.

Abstract

We consider lattice equations on ${\mathds{Z}}^2$ which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and a conservation law. A systematic analysis of the Lie point and the generalized three- and five-point symmetries is presented. It leads to the generic form of the symmetry generators of all the equations in this class, which satisfy a certain non-degeneracy condition. Finally, symmetry reductions of certain lattice equations to discrete analogues of the Painlev\'e equations are considered.