On a class of third-order nonlocal Hamiltonian operators

Abstract

Based on the theory of Poisson vertex algebras we calculate skew-symmetry conditions and Jacobi identities for a class of third-order nonlocal operators of differential-geometric type. Hamiltonian operators within this class are defined by a Monge metric and a skew-symmetric two-form satisfying a number of differentialgeometric constraints. Complete classification results in the 2-component and 3component cases are obtained. MSC: 37K05, 37K10, 37K20, 37K25.

WDVV equations and invariant bi-Hamiltonian formalism

“…A completely new result holds for Dubrovin's second canonical form (1.15). We stress that we could obtain this result only for recent developments in the calculations of Schouten brackets for weakly nonlocal operators [13,14]. Theorem 6.…”

WDVV equations: Hamiltonian operators and symbolic computations

“…They share many properties with WDVV equations and Dubrovin-Frobenius manifolds including the existence of an associated integrable dispersive hierarchy (see [1] for details). We observe, in particular, that the oriented associativity equation has an infinite hierarchy of nonlocal symmetries [22], a first-order local Hamiltonian operator of the same type as A 1 [19] and a third-order nonlocal Hamiltonian operator which is the straightforward generalization of A 2 [3,21].…”

“…Next for the third-order operator we need its metric h i j as well as constants c i j k . We input the metric, denoted by g3, as before and similarly to Christoffel symbols for the previous metric the constants c i j k are handled as a three-dimensional filed: c_hi := Array(1 .. N, 1 .. N, 1 .. N); c_hi [2,3,1]…”

Computing with Hamiltonian operators

“…When forming WDVV equations an essential parameter is the number N of independent variables. It is an opinion of the author that WDVV equations might have a bi-Hamiltonian formalism for an arbitrary value of N. It is a recent discovery that the same holds for another system of PDEs that is of fundamental importance in the geometric theory of integrable systems, the oriented associativity equation, in the case N = 3 [15]. The Readers might wish to try to find a bi-Hamiltonian formalism for the WDVV system in the easiest unknown case N = 5, to corroborate (or negate!)…”