Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations

TL;DR

This paper develops alternative discrete Lagrangian multiforms for ABS equations, ensuring their Euler-Lagrange equations match the original equations and addressing branch cut issues to preserve key properties.

Contribution

It introduces new Lagrangian multiforms with equations equivalent to ABS equations and incorporates branch cut considerations through integer-valued fields.

Findings

New Lagrangian multiforms match ABS equations' Euler-Lagrange equations.

Branch cut choices affect the existence of three-leg forms and closure properties.

Including integer-valued fields restores key properties affected by branch choices.

Abstract

Discrete Lagrangian multiform theory is a variational perspective on lattice equations that are integrable in the sense of multidimensional consistency. The Lagrangian multiforms for the equations of the ABS classification formed the start of this theory, but the Lagrangian multiforms that are usually considered in this context produce equations that are slightly weaker than the ABS equations. In this work, we present alternative Lagrangian multiforms that have Euler-Lagrange equations equivalent to the ABS equations. In addition, the treatment of the ABS Lagrangian multiforms in the existing literature fails to acknowledge that the complex functions in their definitions have branch cuts. The choice of branch affects both the existence of an additive three-leg form for the ABS equations and the closure property of the Lagrangian multiforms. We give counterexamples for both these properties, but we recover them by including integer-valued fields, related to the branch choices, in the action sums.