Searching for integrable lattice maps using factorization

TL;DR

This paper investigates lattice maps through factorization and algebraic entropy to identify integrable models, discovering new polynomial-growth models including a nonsymmetric generalization related to KdV equations.

Contribution

It introduces a novel factorization-based method to classify integrable lattice maps and identifies new models with polynomial growth.

Findings

Found new models with polynomial growth linked to integrability

Discovered a nonsymmetric generalization of quadratic maps related to KdV

Verified consistency around a cube for the new model

Abstract

We analyze the factorization process for lattice maps, searching for integrable cases. The maps were assumed to be at most quadratic in the dependent variables, and we required minimal factorization (one linear factor) after 2 steps of iteration. The results were then classified using algebraic entropy. Some new models with polynomial growth (strongly associated with integrability) were found. One of them is a nonsymmetric generalization of the homogeneous quadratic maps associated with KdV (modified and Schwarzian), for this new model we have also verified the "consistency around a cube".