Algebraic entropy for semi-discrete equations

TL;DR

This paper extends algebraic entropy to semi-discrete equations, demonstrating that zero entropy characterizes integrability in these systems through calculations on various examples.

Contribution

It introduces a new approach to measure integrability in semi-discrete equations by extending algebraic entropy, previously used for purely discrete or continuous systems.

Findings

Vanishing entropy indicates integrability in semi-discrete equations.

Entropy calculation distinguishes integrable from non-integrable systems.

The method applies to both integrable and non-integrable semi-discrete equations.

Abstract

We extend the definition of algebraic entropy to semi-discrete (difference-differential) equations. Calculating the entropy for a number of integrable and non integrable systems, we show that its vanishing is a characteristic feature of integrability for this type of equations.