Journal of Nonlinear Mathematical Physics

Volume 21, Issue 3, June 2014, Pages 407 - 428

Dark Equations and Their Light Integrability

Authors
Denis Blackmore
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102-1982, United States of Americadeblac@m.njit.edu
Anatolij K. Prykarpatski
Department of Applied Mathematics, AGH University of Science and Technology, Krakow, Polandpryk.anat@ua.fm,prykanat@cybergal.com
Received 27 January 2014, Accepted 8 May 2014, Available Online 6 January 2021.
DOI
10.1080/14029251.2014.936760How to use a DOI?
Keywords
Burgers type system; differential-algebraic approach; asymptotic analysis; conserved quantities; Lax integrability; symmetry recursion operator; commuting infinite hierarchies of dynamical systems
Abstract

A relatively new approach to analyzing integrability, based upon differential-algebraic and symplectic techniques, is applied to some “dark equations ”of the type introduced by Boris Kupershmidt. These dark equations have unusual properties and are not particularly well-understood. In particular, dark three-component polynomial Burgers type systems are studied in detail. Their matrix Lax representations are constructed, and the related symmetry recursion operators and infinite hierarchies of integrable nonlinear dynamical systems along with their Lax representations are derived. New linear Lax spectral problems for dark integrable countable hierarchies of dynamical systems are proposed and some special cases are considered as a means of indicating that the approach presented is applicable to a far wider class of dark equations than analyzed here.

Copyright
© 2014 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
21 - 3
Pages
407 - 428
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2014.936760How to use a DOI?
Copyright
© 2014 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Denis Blackmore
AU  - Anatolij K. Prykarpatski
PY  - 2021
DA  - 2021/01/06
TI  - Dark Equations and Their Light Integrability
JO  - Journal of Nonlinear Mathematical Physics
SP  - 407
EP  - 428
VL  - 21
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2014.936760
DO  - 10.1080/14029251.2014.936760
ID  - Blackmore2021
ER  -