On elliptic Lax systems on the lattice and a compound theorem for hyperdeterminants

Abstract

A general elliptic N × N matrix Lax scheme is presented, leading to two classes of elliptic lattice systems, one which we interpret as the higher-rank analogue of the Landau-Lifschitz equations, while the other class we characterize as the higher-rank analogue of the lattice Krichever-Novikov equation (or Adler’s lattice). We present the general scheme, but focus mainly of the latter type of models. In the case N = 2 we obtain a novel Lax representation of Adler’s elliptic lattice equation in its so-called 3-leg form. The case of rank N = 3 is analysed using Cayley’s hyperdeterminant of format 2×2×2, yielding a multi-component system of coupled 3-leg quad-equations.