A Lax pair of the discrete Euler top in terms of quaternions

TL;DR

This paper derives a Lax pair for the discrete Euler top using a generalized eigenvalue problem and introduces a quaternion-based Lax pair, advancing the understanding of its integrability structure.

Contribution

It provides the first known Lax pair for the discrete Euler top and introduces a novel quaternion-based formulation.

Findings

Derived a Lax pair from a generalized eigenvalue problem

Introduced a quaternion-based Lax pair for the discrete Euler top

Enhanced the understanding of the integrability of the discrete Euler top

Abstract

We proposed the discrete Euler top in 2000. In that paper, exact solutions and conserved quantities are described. However, a Lax pair of our proposed discrete Euler top is not contained. Moreover, the Lax pair is still unknown. In this paper, from a generalized eigenvalue problem, we obtain the Lax pair of the discrete Euler top. In addition, we introduce another Lax pair of the discrete Euler top in terms of quaternions.