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

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.