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

## TL;DR

This paper explores the connection between pluri-Lagrangian structures and variational symmetries in integrable PDE hierarchies, extending recent results from ODEs to PDEs, and establishing conditions for the existence of multi-time Lagrangian forms.

## Contribution

It generalizes the concept of variational symmetries and pluri-Lagrangian structures from ODEs to 2D PDE hierarchies, providing a framework for their coexistence.

## Key findings

- Existence of pluri-Lagrangian 2-forms under symmetry conditions
- Multi-time Euler-Lagrange equations match original PDE systems
- Hierarchies with commuting evolutionary flows induced by symmetries

## Abstract

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 differential equations the recent findings in [Petrera, Suris. J. Nonlinear Math. Phys. 24:sup1, 121--145 (2017)] for ordinary differential equations. We consider hierarchies of $2$-dimensional Lagrangian PDEs (many of which have a natural $(1+1)$-dimensional space-time interpretation) and show that if the flow of each PDE is a variational symmetry of all others, then there exists a pluri-Lagrangian 2-form for the hierarchy. The corresponding multi-time Euler-Lagrange equations coincide with the original system supplied with commuting evolutionary flows induced by the variational symmetries.