Search for integrable two-component versions of the lattice equations in the ABS-list

TL;DR

This paper classifies two-component integrable lattice equations related to the ABS list, identifying specific pairs with symmetry properties and confirming their integrability via the CAC test, revealing a complete classification for certain cases.

Contribution

It provides a systematic classification of two-component integrable lattice equations related to the ABS list under specific symmetry and reduction conditions.

Findings

Complete classification for H1, H3, Q1, Q3 equations.

Only equations with an even number of y↔z replacements are possible.

Identified equations produce genuine Bäcklund transformations and Lax pairs.

Abstract

We search and classify two-component versions of the quad equations in the ABS list, under certain assumptions. The independent variables will be called $y,z$ and in addition to multilinearity and irreducibility the equation pair is required to have the following specific properties: (1) The two equations forming the pair are related by $y\leftrightarrow z$ exchange. (2) When $z=y$ both equations reduce to one of the equations in the ABS list. (3) Evolution in any corner direction is by a multilinear equation pair. One straightforward way to construct such two-component pairs is by taking some particular equation in the ABS list (in terms of $y$), using replacement $y \leftrightarrow z$ for some particular shifts, after which the other equation of the pair is obtained by property (1). This way we can get 8 pairs for each starting equation. One of our main results is that due to condition (3) this is in fact complete for H1, H3, Q1, Q3. (For H2 we have a further case, Q2, Q4 we did not check.) As for the CAC integrability test, for each choice of the bottom equations we could in principle have $8^2$ possible side-equations. However, we find that only equations constructed with an even number of $y \leftrightarrow z$ replacements are possible, and for each such equation there are two sets of "side" equation pairs that produce (the same) genuine B\"acklund transformation and Lax pair.