Variational Poisson-Nijenhuis structures for partial differential equations

TL;DR

This paper develops a framework for variational Poisson-Nijenhuis structures on nonlinear PDEs, linking algebraic brackets with symmetry Lie brackets to advance the understanding of nonlocal structures.

Contribution

It introduces a novel approach connecting Schouten and Nijenhuis brackets with symmetry Lie brackets on PDE coverings, enabling the study of nonlocal structures.

Findings

Established relations between brackets on PDEs and their coverings

Constructed a framework for nonlocal Poisson-Nijenhuis structures

Enhanced understanding of symmetry and nonlocality in PDEs

Abstract

We explore variational Poisson-Nijenhuis structures on nonlinear PDEs and establish relations between Schouten and Nijenhuis brackets on the initial equation with the Lie bracket of symmetries on its natural extensions (coverings). This approach allows to construct a framework for the theory of nonlocal structures.