Jacobi Structures of Evolutionary Partial Differential Equations

TL;DR

This paper introduces infinite dimensional Jacobi structures to analyze nonlocal Hamiltonian systems in evolutionary PDEs, demonstrating their invariance under reciprocal transformations and exploring their cohomologies and integrability.

Contribution

It develops a new framework of infinite dimensional Jacobi structures for nonlocal PDEs, extending classical geometric tools and invariance properties.

Findings

Invariance of Jacobi structures under reciprocal transformations

Computation of Lichnerowicz-Jacobi cohomologies for these structures

Introduction of bi-Jacobi structures and their role in PDE integrability

Abstract

In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian evolutionary PDEs. We prove that our class of infinite dimensional Jacobi structures is invariant under reciprocal transformations. The main technical tool is in a suitable generalization of the classical Schouten-Nijenhuis bracket to the space of the so called quasi-local multi-vectors, and a simple realization of this structure in the framework of supermanifolds. These constructions are used to the computation of the Lichnerowicz-Jacobi cohomologies of Jacobi structures. We also introduce the notion of bi-Jacobi structures and consider the integrability of a system of evolutionary PDEs that possesses a bi-Jacobi structure.