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

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 $2\times 2\times 2$, yielding a multi-component system of coupled 3-leg quad-equations.

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Cited by in corpus (5)

• Field analogue of the Ruijsenaars-Schneider model
• Lax equations for relativistic $\mathrm{GL}(NM,\mathbb{C})$ Gaudin models on elliptic curve
• Classical integrable spin chains of Landau-Lifshitz type from R-matrix identities
• Elliptic Ruijsenaars-Toda and elliptic Toda chains: classical r-matrix structure and relation to XYZ chain
• Elliptic solutions of Boussinesq type lattice equations and the elliptic Nth root of unity