The Kupershmidt hydrodynamic chains and lattices

TL;DR

This paper explores the Kupershmidt hydrodynamic chains, detailing their Hamiltonian structures, reductions, and transformations, thereby advancing the understanding of integrable hydrodynamic systems.

Contribution

It introduces new local Hamiltonian structures, hydrodynamic reductions, and reciprocal transformations for Kupershmidt chains, expanding their mathematical framework.

Findings

Identification of an infinite set of local Hamiltonian structures

Development of hydrodynamic reductions parameterized by hypergeometric functions

Description of reciprocal transformations for the chains

Abstract

This paper is devoted to the very important class of hydrodynamic chains first derived by B. Kupershmidt and later re-discovered by M. Blaszak. An infinite set of local Hamiltonian structures, hydrodynamic reductions parameterized by the hypergeometric function and reciprocal transformations for the Kupershmidt hydrodynamic chains are described.