TY - JOUR

T1 - Dark equations and their light integrability

AU - Blackmore, Denis

AU - Prykarpatski, Anatolij K.

N1 - Funding Information:
D.B. thanks the NSF (Grant CMMI-1029809) for support, and A.P. cordially thanks Prof. J. Cies´lińskiemu (Białystok University, Poland) and Prof. I. Mykytyuk (Pedagogical University of Krakow, Poland) for useful discussions. Thanks are due also to the reviewers for their constructive criticism.

PY - 2014/7/3

Y1 - 2014/7/3

N2 - 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.

AB - 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.

KW - Burgers type system

KW - Lax integrability

KW - asymptotic analysis

KW - commuting infinite hierarchies of dynamical systems

KW - conserved quantities

KW - differential-algebraic approach

KW - symmetry recursion operator

UR - http://www.scopus.com/inward/record.url?scp=84903526078&partnerID=8YFLogxK

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U2 - 10.1080/14029251.2014.936760

DO - 10.1080/14029251.2014.936760

M3 - Article

AN - SCOPUS:84903526078

VL - 21

SP - 407

EP - 428

JO - Journal of Nonlinear Mathematical Physics

JF - Journal of Nonlinear Mathematical Physics

SN - 1402-9251

IS - 3

ER -