What is integrability of discrete variational systems?

TL;DR

This paper introduces the concept of pluri-Lagrangian problems as a variational analog of multi-dimensional consistency, providing a framework to analyze integrability of discrete systems through multi-time Euler-Lagrange equations and corner equations.

Contribution

It develops the theory of pluri-Lagrangian problems, derives multi-time Euler-Lagrange equations, and connects these to integrable quad-equations, extending the understanding of variational integrability.

Findings

Derived multi-time Euler-Lagrange equations for discrete pluri-Lagrangian systems.

Established the consistency of corner equations for integrable quad-equations.

Provided an example of a pluri-Lagrangian system not based on multidimensional consistency.

Abstract

We propose a notion of a pluri-Lagrangian problem, which should be understood as an analog of multi-dimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems initiated in 2009 by Lobb and Nijhoff, however having its more remote roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics and their quasiclassical limit, as well as in the theory of variational symmetries going back to Noether. A d-dimensional pluri-Lagrangian problem can be described as follows: given a d-form L on an m-dimensional space (called multi-time, m>d), whose coefficients depend on a sought-after function x of m independent variables (called field), find those fields x which deliver critical points to the action functionals $S_{\Sigma}=\int_{\Sigma} L$ for any d-dimensional manifold $\Sigma$ in the multi-time. We derive the main building blocks of the multi-time Euler-Lagrange equations for a discrete pluri-Lagrangian problem with d=2, the so called corner equations, and discuss the notion of consistency of the system of corner equations. We analyze the system of corner equations for a special class of three-point 2-forms, corresponding to integrable quad-equations of the ABS list. This allows us to close a conceptual gap of the work by Lobb and Nijhoff by showing that the corresponding 2-forms are closed not only on solutions of (non-variational) quad-equations, but also on general solutions of the corresponding corner equations. We also find an example of a pluri-Lagrangian system not coming from a multidimensionally consistent system of quad-equations.