Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds

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.

Weakly nonlocal Poisson brackets: Tools, examples, computations

“…Recently, a canonical form of the Schouten bracket for weakly nonlocal operators has been achieved in three different formalisms (distributions, differential operators and Poisson Vertex Algebras) leading to three different (but equivalent) algorithms for the computation of the Schouten bracket [4] (see [20] for the supermanifold approach).…”

“…we define the velocity matrix of the system of PDEs; in particular, the Reduce matrix element av(i,j) corresponds to ∂ V i /∂u j x in the system of PDEs u i t = (V i ) x (20).…”

“…The second index of matrix w labels the tail vector. The vector V contains the components V i , needed to compute the velocity matrix ∂ V i /∂u j of the system of PDEs (20). We combine the local and nonlocal part of the λ-brackets (48) as follows P = BuildBracket[PLoc, c, w, Nw];…”

Weakly nonlocal Poisson brackets: tools, examples, computations

“…Let Q() denote the second Hamiltonian operator of the Heisenberg magnet equation defined in equation (19). The corresponding λ-brackets among generators are…”

“…Recently, a canonical form of the Schouten bracket for weakly nonlocal operators has been achieved in three different formalisms (distributions, differential operators and Poisson Vertex Algebras) leading to three different (but equivalent) algorithms for the computation of the Schouten bracket [4] (see [19] for the supermanifold approach).…”