Three computational approaches to weakly nonlocal Poisson brackets

Poisson brackets for conserved quantities are quite common in the theory of integrable PDEs. A vast amount of them is defined through nonlocal (pseudodifferential) operators. Such operators make the task of checking the Jacobi identity particularly difficult. In this seminar we will propose and 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 will show that the three approaches lead to the same computations and results.