${\mathbb{Z}}_N$ graded discrete Lax pairs and Yang-Baxter maps

Abstract

We recently introduced a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In this paper we introduce the corresponding Yang-Baxter maps. Many well known examples belong to this scheme for $N=2$, so, for $N\geq 3$, our systems may be regarded as generalisations of these. In particular, for each $N$ we introduce a generalisation of the map $H_{III}^B$ in the classification of scalar Yang-Baxter maps. For $N=3$ this is equivalent to the Yang-Baxter map associated with the discrete modified Boussinesq equation. For $N\geq 5$ (and odd) we introduce a new family of Yang-Baxter maps, which have no lower dimensional analogue. We also present multi-component versions of the Yang-Baxter maps $F_{IV}$ and $F_V$ (given in the ABS classification).