On the Lagrangian structure of integrable quad-equations

TL;DR

This paper proves a universal flip invariance property of action functionals in multidimensional lattices for all integrable quad-equations, revealing a fundamental Lagrangian relation within elementary quadrilaterals.

Contribution

It provides a simple, case-independent proof of flip invariance for all integrable quad-equations and uncovers a new fundamental Lagrangian relation.

Findings

Universal flip invariance proof for all integrable quad-equations

Discovery of a new fundamental Lagrangian relation within quadrilaterals

Simplification of understanding of Lagrangian structures in discrete integrable systems

Abstract

The new idea of flip invariance of action functionals in multidimensional lattices was recently highlighted as a key feature of discrete integrable systems. Flip invariance was proved for several particular cases of integrable quad-equations by Bazhanov, Mangazeev and Sergeev and by Lobb and Nijhoff. We provide a simple and case-independent proof for all integrable quad-equations. Moreover, we find a new relation for Lagrangians within one elementary quadrilateral which seems to be a fundamental building block of the various versions of flip invariance.