# Three computational approaches to weakly nonlocal Poisson brackets

## TL;DR

This paper compares three computational methods—distributions, pseudodifferential operators, and Poisson vertex algebras—for verifying the Jacobi identity in weakly nonlocal Poisson brackets, demonstrating their equivalence.

## Contribution

It provides a unified comparison of three approaches to checking the Jacobi identity, highlighting their similarities and confirming their consistency.

## Key findings

- All three methods yield similar computations.
- The approaches are equivalent in verifying the Jacobi identity.
- The methods are consistent across different mathematical frameworks.

## Abstract

We compare three different ways of checking the Jacobi identity for weakly nonlocal Poisson brackets using the theory of distributions, of pseudodifferential operators and of Poisson vertex algebras, respectively. We show that the three approaches lead to similar computations and same results.