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

Stanislav Opanasenko, Roman O. Popovych

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.

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

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.

Paper Structure

This paper contains 2 sections, 3 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Example

Key Result

Theorem 1

An operator given by IDFM:eq:NonlocalHamOper is Hamiltonian if and only if the tensor $g=(g^{ij})$ defines a (pseudo-)Riemannian metric, the connection $\nabla$ is symmetric and compatible with the metric, $\nabla_k g^{ij}=0$, the tensor $g$ and the affinors $w_\alpha:=(w^j_{\alpha l})$ satisfy the and the affinors $w_\alpha$ commute, $[w_\alpha,w_\beta]=0$, $\alpha,\beta=1,\dots,N$. Here $R^{ij}

Theorems & Definitions (5)

  • Theorem 1: Ferapontov1991a
  • Theorem 2
  • Theorem 3
  • proof
  • Remark 4