Algebraic structure of the gradient-holonomic algorithm for Lax integrable nonlinear dynamical systems. I

Abstract

The algebraic structure of the gradient-holonomic algorithm for Lax integrable dynamical systems is discussed. A generalization of the ℛ-structure approach for the case of operator-valued affine Lie algebras is used to prove the bi-Hamiltonian formulation of nonlinear integrable dynamical systems in multidimensions. The monodromy transfer matrix is constructed to describe the operator manifold in relation to canonical Lie–Poisson bracket on initial affine Lie algebra with gauge central extension. As an illustration, the two-dimensional operator Benney–Kaup integrable hierarchy is considered and their bi-Hamiltonicity is proved.

Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces

“…By now several regular Lie-algebraic approaches existed to constructing Lax integrable (2 + 1)-dimensional nonlinear dynamical systems on functional manifolds, which were presented in [12,20,21,22]. In this paper a new Lie-algebraic method is devised for introducing one more commuting variable into (1|1 + 1)-dimensional dynamical systems with preserving their integrability by Lax.…”

“…If N > 2 in the representation (22), the hierarchies of additional symmetries can be used for constructing Lax integrable ((1 + K)|1 + 1)-dimensional supersymmetric systems, where K = 1, N − 1.…”

Lie-algebraic structure of integrable nonlinear dynamical systems on extended functional manifolds

“…In this case, forall a,b~ G, Rb], where R:G R >~ is a standard R-structure on G (see [5] and [3]), and the scalar product (., . )p is given by the relation…”

“…This method was efficiently applied to the investigation of various nonlinear systems and partial differential equations in mathematical and theoretical physics. The algebraic structure of the gradient-holonomic method was studied in [3] and [4]. The present paper is a continuation of [3] and deals with the general Liealgebraic scheme of construction of integrable nonlinear dynamical systems on extended functional manifolds of eigenfunctions of the operator of associated Lax-type representation.…”

ClassicalR-matrix theory of dispersionless systems: II. (2 1) dimension theory

“…The central extension was considered in early works by Reyman and Semenov-Tian-Shansky [9,10] and also by Prykarpatsky [11,12]. The central extension approach to integrable field and lattice-field systems was presented as well in [13,14].…”