Asymmetric integrable quad-graph equations

TL;DR

This paper introduces a method to identify integrable quad-graph equations by analyzing conservation laws and algebraic entropy, discovering new integrable systems and classifying their properties.

Contribution

It presents a novel empirical approach to detect integrability in difference equations and identifies new integrable equations, including one unrelated to known systems.

Findings

Identified several types of integrable equations.

Discovered a new integrable system unrelated to known ones.

Provided a classification based on conservation laws and algebraic entropy.

Abstract

Integrable difference equations commonly have more low-order conservation laws than occur for nonintegrable difference equations of similar complexity. We use this empirical observation to sift a large class of difference equations, in order to find candidates for integrability. It turns out that all such candidates have an equivalent affine form. These are tested by calculating their algebraic entropy. In this way, we have found several types of integrable equations, one of which seems to be entirely unrelated to any known discrete integrable system. We also list all single-tile conservation laws for the integrable equations in the above class.