On non-commutative leapfrog map

TL;DR

This paper explores the integrability of the non-commutative leapfrog map by deriving explicit formulas, analyzing its continuous limit, and establishing its Poisson structure using non-commutative algebraic tools.

Contribution

It introduces a new explicit formula for the non-commutative leapfrog map and its zero-curvature equation, and formulates its Poisson structure via non-commutative networks.

Findings

Derived explicit formula for the non-commutative leapfrog map

Established the discrete zero-curvature equation

Formulated the Poisson structure for the map

Abstract

We investigate the integrability of the non-commutative leapfrog map in this paper. Firstly, we derive the explicit formula for the non-commutative leapfrog map and corresponding discrete zero-curvature equation by employing the concept of non-commutative cross-ratio. Then we revisit this discrete map, as well as its continuous limit, from the perspective of non-commutative Laurent bi-orthogonal polynomials. Finally, the Poisson structure for this discrete non-commutative map is formulated with the help of a non-commutative network. We aim to enhance our understanding of the integrability properties of the non-commutative leapfrog map and its related mathematical structures through these analysis and constructions.