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arXiv · 1405.3927

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

N. Delice·F. W. Nijhoff·S. Yoo-Kong
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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.

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N. Delice, F. W. Nijhoff, S. Yoo-Kong. 2014-05-15. On Elliptic Lax Systems on the Lattice and a Compound Theorem for Hyperdeterminants. https://doi.org/10.1088/1751-8113%2F48%2F3%2F035206

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