# What is integrability of discrete variational systems?

@article{Boll2013WhatII, title={What is integrability of discrete variational systems?}, author={Raphael Boll and Matteo Petrera and Yuri B. Suris}, journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, year={2013}, volume={470} }

We propose a notion of a pluri-Lagrangian problem, which should be understood as an analogue 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…

## 30 Citations

### Discrete Pluriharmonic Functions as Solutions of Linear Pluri-Lagrangian Systems

- Mathematics
- 2014

Pluri-Lagrangian systems are variational systems with the multi-dimensional consistency property. This notion has its roots in the theory of pluriharmonic functions, in the Z-invariant models of…

### A Variational Principle for Discrete Integrable Systems

- Mathematics
- 2018

For multidimensionally consistent systems we can consider the Lagrangian as a form, closed on the multidimensional equations of motion. For 2-dimensional systems this allows us to define an action on…

### Quantum variational principle and quantum multiform structure: The case of quadratic Lagrangians

- MathematicsNuclear Physics B
- 2019

### Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

- MathematicsOpen Communications in Nonlinear Mathematical Physics
- 2021

Many integrable hierarchies of differential equations allow a variational
description, called a Lagrangian multiform or a pluri-Lagrangian structure. The
fundamental object in this theory is not a…

### Semi-discrete Lagrangian 2-forms and the Toda hierarchy

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2022

We present a variational theory of integrable differential-difference equations (semi-discrete integrable systems). This is an extension of the ideas known by the names ‘Lagrangian multiforms’ and…

### Continuum limits of pluri-Lagrangian systems

- Mathematics, Computer ScienceJournal of Integrable Systems
- 2019

A continuum limit procedure for pluri-Lagrangian systems, where the lattice parameters are interpreted as Miwa variables, describing a particular embedding in continuous multi-time of the mesh on which the discrete system lives.

### A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2019

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…

### Lagrangian multiforms on Lie groups and non-commuting flows

- Mathematics
- 2022

We describe a variational framework for non-commuting ﬂows, extending the theories of Lagrangian multiforms and pluri-Lagrangian systems, which have gained prominence in recent years as a variational…

### Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems

- Mathematics
- 2015

We establish the pluri-Lagrangian structure for families of Bäcklund transformations of relativistic Toda-type systems. The key idea is a novel embedding of these discrete-time (one-dimensional)…

### Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

- MathematicsEuropean Journal of Mathematics
- 2020

We investigate the relation between pluri-Lagrangian hierarchies of 2-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial…

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