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
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U2 - 10.1080/14029251.2014.936760
DO - 10.1080/14029251.2014.936760
M3 - Article
AN - SCOPUS:84903526078
SN - 1402-9251
VL - 21
SP - 407
EP - 428
JO - Journal of Nonlinear Mathematical Physics
JF - Journal of Nonlinear Mathematical Physics
IS - 3
ER -