Bi-Hamiltonian structure of the oriented associativity equation

Abstract

The Oriented Associativity equation plays a fundamental role in the theory of Integrable Systems. In this paper we prove that the equation, besides being Hamiltonian with respect to a first-order Hamiltonian operator, has a third-order non-local homogeneous Hamiltonian operator belonging to a class which has been recently studied, thus providing a highly non-trivial example in that class and showing intriguing connections with algebraic geometry.MSC2010: 37K05.

WDVV equations and invariant bi-Hamiltonian formalism

“…The generalization to the Oriented Associativity equations for Fmanifolds is an active research topic (see e.g. [6]) and quadratic line complexes occur also in that case [56] (see also [53]). A future role of such objects besides the bi-Hamiltonian structure that they provide is foreseeable.…”

Second-order integrable Lagrangians and WDVV equations

“…• They are non-diagonalisable (their Haantjes tensor does not vanish). This suggests that integrable Lagrangian densities (1) are related to the associativity (WDVV) equations where analogous systems were obtained in [22], see also [33,34] for related results. Such a link indeed exists and is discussed in Sect.…”

“…where is the modular discriminant. These three solutions (which correspond to rational, trigonometric and elliptic cases of the Weierstrass ℘function equation in (34)) are considered separately below. Note that both the rational and trigonometric cases lead to degenerate Lagrangians, so only the elliptic case is of interest.…”

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].…”

“…Until now, the results on Hamiltonian operators are known only for one of the simplest cases of oriented associativity equation; more calculations are needed in order to support a conjecture on the bi-Hamiltonianity of the F-manifold equation. Indeed, the compatibility of A 1 and A 2 is still an open question even in the simplest case, see [21] for a discussion.…”