Lagrangians and integrability for additive fourth-order difference equations

TL;DR

This paper characterizes all invertible fourth-order difference equations linear in extremal values using discrete Lagrangians, explores their integrability, relates them to known classifications, and discusses their continuum limits.

Contribution

It provides a complete characterization of such difference equations via a discrete Lagrangian framework and investigates their integrability and continuum limits.

Findings

All invertible fourth-order difference equations linear in extremal values are characterized.

Some integrability properties of the classified equations are established.

Connections with existing classifications and continuum limits are discussed.

Abstract

We use a recently found method to characterise all the invertible fourth-order difference equations linear in the extremal values based on the existence of a discrete Lagrangian. We also give some result on the integrability properties of the obtained family and we put it in relation with known classifications. Finally, we discuss the continuum limits of the integrable cases.