Three computational approaches to weakly nonlocal Poisson brackets

Abstract

We compare three different ways of checking the Jacobi identity for weakly nonlocal Poisson brackets using the theory of distributions, pseudo-differential operators, and Poisson vertex algebras, respectively. We show that the three approaches lead to similar computations and same results. K E Y W O R D Smathematical physics, partial differential equations, solitons and integrable systems 412

Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds

“…In [2] we developed an algorithm to compute Schouten brackets of such operators using three different formalisms: distributions, pseudodifferential operators, Poisson vertex algebras. In this paper we propose an alternative approach based on the identification of weakly non local hamiltonian operators with superfunctions on supermanifolds.…”

“…A rigorous approach to the computation of Schouten bracket for a wide class of nonlocal Hamiltonian operators has been proposed in [3,4], and it is based on the notion of nonlocal Poisson Vertex Algebra. In the case of weakly nonlocal Hamiltonian operators a computational solution to the problem has been recently proposed through the parallel development of an algorithm in the three different languages of distributions, operators, and Poisson Vertex Algebras [2].…”

“…This implies that in order to check that the expression vanishes one should calculate its Euler-Lagrange operator and see if the result is zero. When writing [2] we considered the possibility to extend the results of the paper to the formalism of superfunctions. Nonlocal terms in operators can be represented by introducing new odd nonlocal variables.…”

“…where r α are new nonlocal odd variables defined by ∂ x r α = w j α p j . Unfortunately, a naive rephrasing of the arguments of [2] in terms of superfunctions does not work, as superfunctions do not allow to keep the role of the arguments ψ 1 , ψ 2 , ψ 3 distinct in the three-vectors.…”

Computing with Hamiltonian operators

“…In the framework of the algebraic approach to the Hamiltonian formalism for PDEs [6] the Mathematica packages MasterPVA and WAlg [17] provide procedures for computing the Schouten bracket between local differential operators. The recent paper [16] shows that the algebraic approach to Schouten brackets leads to the same computations and results as more traditional approaches, also for nonlocal operators.…”