Liouville canonical form for compatible nonlocal Poisson brackets of hydrodynamic type, and integrable hierarchies

TL;DR

This paper develops a canonical form for compatible nonlocal Poisson brackets of hydrodynamic type, facilitating the construction of integrable hierarchies associated with these brackets.

Contribution

It introduces a method to reduce compatible nonlocal Poisson brackets to a canonical form, aiding the development of integrable hierarchies.

Findings

Canonical form for compatible nonlocal Poisson brackets derived

Effective construction method for integrable hierarchies established

Applicable to brackets generated by metrics of constant Riemannian curvature

Abstract

We solve the problem of reducing to the simplest and convenient for our purposes, canonical form for an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature in order to get an effective construction of the integrable hierarchies related to all these compatible Poisson brackets.