A completeness study on a class of discrete, 'two by two' Lax pairs

TL;DR

This paper introduces a method to analyze all possible 'two by two' discrete Lax pairs with separable entries, leading to new higher-order lattice equations and establishing their uniqueness within this class.

Contribution

It provides a systematic approach to classify partial difference Lax pairs of a specific form and derives new integrable lattice equations, proving their exclusivity for this class.

Findings

Derived new higher-order lattice sine-Gordon and modified KdV equations

Established the non-existence of other equations for this Lax pair class

Provided a comprehensive classification method for these Lax pairs

Abstract

We propose a method by which to examine all possible partial difference Lax pairs that consist of 'two by two' discrete linear problems, where the matrices contain one separable term in each entry. We thereby derive new, higher-order versions of the lattice sine-Gordon and lattice modified KdV equations, while showing that there can be no other partial difference equations associated with this type of Lax pair.