Weak Lax pairs for lattice equations (Q2903932)

The authors discuss the strength of Lax pairs, or the ``zero curvature condition'' (ZCC), for various 2D lattice equations and show several cases where the ZCC does not uniquely determine the evolution. For example, in the flipped \(\mathrm{H1}_\epsilon\) model, the ZCC allows two possibilities. If one builds an infinite lattice by arbitrarily choosing for each cell one of the two allowed relations (``black and white patterns''), the result is sometimes integrable and sometimes shows nonzero entropy. If the ZCC is pushed to a \(2 \times 2\) sublattice, more examples are obtained where it is ambiguous and yields both regular solutions as well as an exotic one.NEWLINENEWLINEIn the course of their analysis, the authors discuss 3D consistency, Lax pairs, Bäcklund transformations, and exemplify these concepts for the lattice potential KdV, which describes the permutability property of continuous KdV. A number of black and white lattice models are introduced, as well as variants of the functional Yang-Baxter equation.