Tri-Hamiltonian structure of an asymmetric generalized Ablowitz-Ladik hierarchy and a Frobenius manifold

Abstract

We construct a local tri-Hamiltonian structure of the asymmetric (3,1)-type generalized Ablowitz-Ladik (gAL) hierarchy at the full-dispersive level and rigorously prove its validity using the supervariable technique. All central invariants of the corresponding bi-Hamiltonian structures are computed. In addition, we construct a Frobenius manifold M arising from the dispersionless limit of this hierarchy and show that the dispersionless limits of the first flows of the (3,1)-type gAL hierarchy belong to the Principal Hierarchy of M.