From pencils of Novikov algebras of St\"ackel type to soliton hierarchies

Abstract

In this article we construct evolutionary soliton hierarchies from pencils of Novikov algebras of St\"ackel type. We start by defining a special class of associative Novikov algebras, which we call Novikov algebras of St\"ackel type, as they are associated with classical St\"ackel metrics in Vi\`ete coordinates. We obtain sufficient conditions for pencils of these algebras so that the corresponding Dubrovin-Novikov Hamiltonian operators can be centrally extended, producing sets of pairwise compatible Poisson operators. These operators lead to coupled Korteweg-de~Vries (cKdV) and coupled Harry Dym (cHD) hierarchies, as well as to a triangular cKdV hierarchy and a triangular cHD hierarchy.