On Elliptic Lax Systems on the Lattice and a Compound Theorem for Hyperdeterminants

TL;DR

This paper introduces a general elliptic Lax scheme for lattice systems, focusing on higher-rank analogues of known equations, and presents novel representations and hyperdeterminant-based analyses for specific cases.

Contribution

It develops a comprehensive elliptic Lax framework for higher-rank lattice systems and provides new Lax representations and hyperdeterminant applications for particular models.

Findings

New Lax representation of Adler's elliptic lattice equation in 3-leg form

Analysis of rank 3 case using Cayley's hyperdeterminant

Identification of higher-rank analogues of Landau-Lifschitz and Krichever-Novikov equations

Abstract

A general elliptic $N\times N$ 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 $2\times2\times2$, yielding a multi-component system of coupled 3-leg quad-equations.