Lax pairs for the discrete reduced Nahm systems

TL;DR

This paper develops a discretised Lax pair for reduced Nahm systems, demonstrating its equivalence with a known discretisation method and establishing integrability, while also highlighting limitations in characterising integrability for related discretisations.

Contribution

It introduces a discretised Lax pair for reduced Nahm systems and proves its equivalence with the Kahan-Hirota-Kimura discretisation, advancing understanding of integrability in discrete systems.

Findings

Discretised Lax pair guarantees integrability of reduced Nahm systems.

Proves equivalence with Kahan-Hirota-Kimura discretisation.

Shows limitations of Nahm systems in characterising integrability.

Abstract

We discretise the Lax pair for the reduced Nahm systems and prove its equivalence with the Kahan-Hirota-Kimura discretisation procedure. We show that these Lax pairs guarantee the integrability of the discrete reduced Nahm systems providing an invariant. Also, we show with an example that Nahm systems cannot solve the general problem of characterisation of the integrability for Kahan-Hirota-Kimura discretisations.