Generalised (bi-)Hamiltonian structures of hydrodynamic type and (bi-)flat F-manifolds

TL;DR

This paper introduces generalized Hamiltonian structures for hydrodynamic PDEs, linking them to geometric data and associating them with bi-flat F-manifolds, enhancing understanding of integrable systems.

Contribution

It extends the concept of Hamiltonian structures to a broader class of hydrodynamic systems and establishes a geometric correspondence with bi-flat F-manifolds.

Findings

Generalized Hamiltonian structures are characterized by geometric data.

A correspondence between these structures and bi-flat F-manifolds is established.

Compatibility with principal hierarchies is demonstrated.

Abstract

We introduce the notions of generalised (bi-)Hamiltonian structures which generalise naturally the (bi-)Hamiltonian structures of evolutionary partial differential equations. In the hydrodynamic case, these structures are characterised in terms of geometric data. Furthermore, we show that a generalised (bi)-Hamiltonian structure of hydrodynamic type can be associated with any (bi-)flat F-manifold, and it is compatible with the corresponding principal hierarchy.