Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds

TL;DR

This paper introduces a computational method to verify weakly nonlocal Poisson brackets in integrable systems by linking differential operators to superfunctions on supermanifolds, facilitating the analysis of their Poisson properties.

Contribution

It presents a novel approach that uses supermanifold theory to identify and analyze weakly nonlocal Poisson brackets, simplifying their verification process.

Findings

Provides a new computational framework for weakly nonlocal Poisson brackets

Establishes a connection between differential operators and superfunctions on supermanifolds

Facilitates the proof of Poisson bracket properties in integrable systems

Abstract

Poisson brackets between conserved quantities are a fundamental tool in the theory of integrable systems. The subclass of weakly nonlocal Poisson brackets occurs in many significant integrable systems. Proving that a weakly nonlocal differential operator defines a Poisson bracket can be challenging. We propose a computational approach to this problem through the identification of such operators with superfunctions on supermanifolds.