Tri-Hamiltonian structure of the Ablowitz–Ladik hierarchy

Abstract

We construct a local tri-Hamiltonian structure of the Ablowitz–Ladik hierarchy, and compute the central invariants of the associated bihamiltonian structures. We show that the central invariants of one of the bihamiltonian structures are equal to 124, and the dispersionless limit of this bihamiltonian structure coincides with the one that is defined on the jet space of the Frobenius manifold associated with the Gromov–Witten invariants of local P1. This result provides support for the validity of Brini’s conjecture on the relation of these Gromov–Witten invariants with the Ablowitz–Ladik hierarchy.