On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations

TL;DR

This paper explores the Lagrangian structures of 3D consistent asymmetric quad-equations, extending discrete integrable systems and establishing flip-invariance of their action functionals, including novel asymmetric cases.

Contribution

It introduces Lagrangian structures and flip-invariance for a new class of 3D consistent asymmetric quad-equations, expanding understanding of discrete integrable systems.

Findings

Established Lagrangian structures for asymmetric systems

Proved flip-invariance of the action functional

Included novel asymmetric systems in the analysis

Abstract

Recently, the first-named author gave a classification of 3D consistent 6-tuples of quad-equations with the tetrahedron property; several novel asymmetric 6-tuples have been found. Due to 3D consistency, these 6-tuples can be extended to discrete integrable systems on Z^m. We establish Lagrangian structures and flip-invariance of the action functional for the class of discrete integrable systems involving equations for which some of the biquadratics are non-degenerate and some are degenerate. This class covers, among others, some of the above mentioned novel systems.