Discrete integrable equations on face-centred cubics: consistency and Lax pairs of corner equations

Abstract

A new set of discrete integrable equations, called face-centred quad equations, was recently obtained using new types of interaction-round-a-face solutions of the classical Yang–Baxter equation. These equations satisfy a new formulation of multidimensional consistency, known as consistency-around-a-face-centred-cube (CAFCC), which requires consistency of an overdetermined system of 14 five-point equations on the face-centred cubic unit cell. In this paper, a new formulation of CAFCC is introduced where so-called type C equations are centred at faces of the face-centred cubic unit cell, whereas previously they were only centred at corners. This allows type C equations to be regarded as independent multidimensionally consistent integrable systems on higher dimensional lattices and is used to establish their Lax pairs.