Linear quadrilateral lattice equations and multidimensional consistency

TL;DR

This paper demonstrates that all scalar linear quadrilateral lattice equations belong to two compatible families, each parametrized, ensuring multidimensional consistency across higher-dimensional lattices.

Contribution

It classifies scalar linear quadrilateral lattice equations into two compatible families with explicit parametrizations, establishing their multidimensional consistency.

Findings

Two distinct families of compatible scalar linear quadrilateral lattice equations identified.

Explicit parametrizations provided for each family.

All such equations are shown to be compatible in higher dimensions.

Abstract

It is shown that every scalar linear quadrilateral lattice equation lies within a family of similar equations, members of which are compatible between one another on a higher dimensional lattice. There turn out to be two such families, a natural parametrisation is given for each.