Dark Equations and Their Light Integrability

Abstract

In memory of Boris Kupershmidt ( †2010), a mathematical light in the mysterious world of "dark"equations 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.

Symplectic Geometry Aspects of the Parametrically-Dependent Kardar–Parisi–Zhang Equation of Spin Glasses Theory, Its Integrability and Related Thermodynamic Stability

“…To solve this problem effectively, we made use of the gradient-holonomic algorithm, motivated by symplectic geometry techniques on functional manifolds [14,15], recently devised for studying the integrability properties of nonlinear dynamical systems with hidden symmetries and, in part, related with the optimal control theory [16,17] approach to parametrically dependent processes. Being applied to the parametrically-extended Kardar-Parisi-Zhang Equation ( 1) this algorithm allowed one to state that it belongs to a so called dark type class [18][19][20][21] of integrable Hamiltonian dynamical systems on functional manifolds with hidden symmetry. Namely, the parametrically-extended Kardar-Parisi-Zhang system of Equation ( 4) reduces to the evolution flow…”

On parametric generalizations of the Kardar-Parisi-Zhang equation and their integrability

“…To solve this problem effectively, we made use of the symplectic geometry based analytic gradient-holonomic algorithm [5,19], recently devised for studying integrability properties of nonlinear dynamical systems with hidden symmetries and partially motivated by the classical optimal control theory [3,17] approach. Being applied to the parametrically extended Kardar-Parisi-Zhang equation (2), this algorithm allowed us to solve the posed above problem and to to state that it belongs to the class of completely integrable dark type [5,6,4,12,15,16] Hamiltonian systems on functional manifolds with hidden symmetry. Namely, there was stated the following proposition.…”

On the Lie-Algebraic Integrability of the Calogero-Degasperis Dynamical System and Its Generalizations

“…Subject to the dynamical systems of the C-type in many cases there was proved that they could be linearized by means of some, in general, nonlocal transformations of functional manifolds. In particular, the differential-algebraic aspects of the nonlinear C-type dynamical system on a smooth 2π-periodic functional manifold M ⊂ C(R /{2πZ}; R) and the related linearization mapping were studied in detail in [14][15][16][17] by means of the gradientholonomic integrability scheme, devised previously in [3,4,18]. In addition, in [15] there was generalized the main result of [16] about the Hamiltonian structure of the Burg-ers evolution flow nonlinear dynamical system (1) was demonstrated to be a completely integrable biHamiltonian system on a suitably constructed infinite-dimensional functional manifold, possessing an infinite hierarchy of commuting to each other nonlocal conservation laws.…”