A variational perspective on continuum limits of ABS and lattice GD equations

A pluri-Lagrangian structure is an attribute of integrability for lattice equations and for hierarchies of differential equations. It combines the notion of multi-dimensional consistency (in the discrete case) or commutativity of the flows (in the continuous case) with a variational principle. Recently we developed a continuum limit procedure for pluri-Lagrangian systems, which we now apply to most of the ABS list and some members of the lattice Gelfand-Dickey hierarchy. We obtain pluri-Lagrangian structures for many hierarchies of integrable PDEs for which such structures where previously unknown. This includes the Krichever-Novikov hierarchy, the double hierarchy of sine-Gordon and modified KdV equations, and a first example of a continuous multi-component pluri-Lagrangian system.

💡 Research Summary

The paper develops a systematic continuum‑limit procedure for pluri‑Lagrangian (also called Lagrangian multiform) systems and applies it to a large portion of the Adler‑Bobenko‑Suris (ABS) list of integrable quad equations as well as selected members of the lattice Gelfand–Dickey (GD) hierarchy. A pluri‑Lagrangian structure combines multidimensional consistency (in the discrete setting) or commutativity of flows (in the continuous setting) with a variational principle: a field is a critical point of an action defined on any discrete (or continuous) surface simultaneously.

The authors begin by recalling the discrete pluri‑Lagrangian principle: a discrete 2‑form (L) assigns a Lagrangian value to each elementary square (quad) of a lattice, and the field satisfies a set of corner‑wise Euler–Lagrange equations that guarantee consistency around a cube. They then introduce the continuous analogue, where a 2‑form ( \mathcal L = \sum_{i<j} \mathcal L_{ij}