On the prolongation of a local hydrodynamic-type Hamiltonian operator to a nonlocal one

TL;DR

This paper introduces a new method to extend local hydrodynamic Hamiltonian operators to nonlocal ones, enriching the understanding of their structure and applications in integrable systems.

Contribution

It presents a novel prolongation construction for local Hamiltonian operators to their nonlocal counterparts, expanding the toolkit for analyzing hydrodynamic-type systems.

Findings

Constructed a prolongation method for Hamiltonian operators.

Applied the method to an isothermal no-slip drift flux system.

Demonstrated the method's effectiveness through examples.

Abstract

Nonlocal Hamiltonian operators of Ferapontov type are well-known objects that naturally arise local from Hamiltonian operators of Dubrovin-Novikov type with the help of three constructions, Dirac reduction, recursion scheme and reciprocal transformation. We provide an additional construction, namely the prolongation of a local hydrodynamic-type Hamiltonian operator of a subsystem to its nonlocal counterpart for the entire system. We exemplify this construction by a system governing an isothermal no-slip drift flux.