Algebraic entropy, symmetries and linearization of quad equations consistent on the cube

TL;DR

This paper investigates the algebraic entropy and symmetries of certain nonlinear difference equations, demonstrating their linearizability and providing methods to construct associated lattice equations, Bäcklund transformations, and Lax pairs.

Contribution

It introduces a systematic approach to analyze and linearize specific quad equations consistent around the cube, expanding understanding of their integrability properties.

Findings

The $H^4$ trapezoidal and $H^6$ families are linearizable.

Explicit linearization methods are provided for selected examples.

Algebraic entropy calculations confirm the linearizability of these equations.

Abstract

We discuss the non autonomous nonlinear partial difference equations belonging to Boll classification of quad graph equations consistent around the cube. We show how starting from the compatible equations on a cell we can construct the lattice equations, its B\"acklund transformations and Lax pairs. By carrying out the algebraic entropy calculations we show that the $H^4$ trapezoidal and the $H^6$ families are linearizable and in a few examples we show how we can effectively linearize them.