Integrable lattice equations and their growth properties

TL;DR

This paper explores the growth properties of two-dimensional lattice equations, demonstrating that integrable equations exhibit polynomial degree growth, unlike nonintegrable ones which grow exponentially, aiding in identifying integrability.

Contribution

The paper introduces a degree growth criterion to distinguish integrable from nonintegrable lattice equations and identifies their deautonomisations and growth behaviors.

Findings

Integrable lattice equations have polynomial degree growth.

Nonintegrable equations exhibit exponential degree growth.

Linearisable equations grow slower than those integrable via Inverse Scattering.

Abstract

In this paper we investigate the integrability of two-dimensional partial difference equations using the newly developed techniques of study of the degree of the iterates. We show that while for generic, nonintegrable equations, the degree grows exponentially fast, for integrable lattice equations the degree growth is polynomial. The growth criterion is used in order to obtain the integrable deautonomisations of the equations examined. In the case of linearisable lattice equations we show that the degree growth is slower than in the case of equations integrable through Inverse Scattering Transform techniques.