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From pencils of Novikov algebras of Stäckel type to soliton hierarchies

Maciej Błaszak, Krzysztof Marciniak, Błażej M. Szablikowski

Abstract

In this article we construct evolutionary soliton hierarchies from pencils of Novikov algebras of Stäckel type. We start by defining a special class of associative Novikov algebras, which we call Novikov algebras of Stäckel type, as they are associated with classical Stäckel metrics in Viète coordinates. We obtain sufficient conditions for pencils of these algebras so that the corresponding Dubrovin-Novikov Hamiltonian operators can be centrally extended, producing sets of pairwise compatible Poisson operators. These operators lead to coupled Korteweg-de~Vries (cKdV) and coupled Harry Dym (cHD) hierarchies, as well as to a triangular cKdV hierarchy and a triangular cHD hierarchy.

From pencils of Novikov algebras of Stäckel type to soliton hierarchies

Abstract

In this article we construct evolutionary soliton hierarchies from pencils of Novikov algebras of Stäckel type. We start by defining a special class of associative Novikov algebras, which we call Novikov algebras of Stäckel type, as they are associated with classical Stäckel metrics in Viète coordinates. We obtain sufficient conditions for pencils of these algebras so that the corresponding Dubrovin-Novikov Hamiltonian operators can be centrally extended, producing sets of pairwise compatible Poisson operators. These operators lead to coupled Korteweg-de~Vries (cKdV) and coupled Harry Dym (cHD) hierarchies, as well as to a triangular cKdV hierarchy and a triangular cHD hierarchy.

Paper Structure

This paper contains 10 sections, 6 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Novikov algebras, the associated Poisson operators and their central extensions
  3. Novikov algebras of Stäckel type
  4. Novikov pencils of Stäckel type
  5. Multi-Hamiltonian hierarchies of evolutionary type
  6. Coupled Harry Dym hierarchy
  7. Coupled Korteweg-de Vries hierarchy
  8. Triangular coupled Harry Dym hierarchy
  9. Triangular coupled Korteweg-de Vries hierarchy
  10. Appendix A

Key Result

Lemma 1

All $\mathcal{A}^{m}$ are associative.

Theorems & Definitions (12)

  • Definition 1
  • Remark 1
  • Lemma 1
  • Example 1
  • Remark 2
  • Theorem 1
  • Lemma 2
  • Example 2
  • Lemma 3
  • Theorem 2
  • ...and 2 more