Non-symmetric discrete Toda systems from quad-graphs

TL;DR

This paper links non-symmetric discrete relativistic Toda equations to 3D consistent quad-equation systems, providing a method for deriving zero curvature representations and extending results to continuous time cases.

Contribution

It introduces a novel relation between non-symmetric Toda systems and 3D consistent quad-equations, expanding understanding beyond symmetric cases.

Findings

Established relation between non-symmetric Toda systems and 3D consistent quad-equations

Developed an algorithmic method for zero curvature representations

Extended results to continuous time cases

Abstract

For all non-symmetric discrete relativistic Toda type equations we establish a relation to 3D consistent systems of quad-equations. Unlike the more simple and better understood symmetric case, here the three coordinate planes of $\mathbb Z^3$ carry different equations. Our construction allows for an algorithmic derivation of the zero curvature representations and yields analogous results also for the continuous time case.