Lagrangian multiforms and multidimensional consistency

TL;DR

This paper introduces a Lagrangian multiform framework for 2D integrable lattice equations, demonstrating their multidimensional consistency and establishing a variational principle that extends to higher dimensions.

Contribution

It formulates a novel Lagrangian multiform approach for integrable lattice systems, linking multidimensional consistency with a variational principle.

Findings

Lagrangians obey a closure relation in higher-dimensional lattices

Lagrangian multiforms provide a variational description of integrable systems

Connection established between multidimensional consistency and the variational principle

Abstract

We show that well-chosen Lagrangians for a class of two-dimensional integrable lattice equations obey a closure relation when embedded in a higher dimensional lattice. On the basis of this property we formulate a Lagrangian description for such systems in terms of Lagrangian multiforms. We discuss the connection of this formalism with the notion of multidimensional consistency, and the role of the lattice from the point of view of the relevant variational principle.