Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces

Abstract

Abstract. We obtain via Bäcklund transformation the Hamiltonian representation for a Lax type nonlinear dynamical system hierarchy on a dual space to the Lie algebra of super-integral-differential operators of one anticommuting variable, extended by evolutions of the corresponding spectral problem eigenfunctions and adjoint eigenfunctions, as well as for the hierarchies of their additional symmetries. The relation of these hierarchies with the integrable by Lax (2|1 + 1)-dimensional supersymmetric Davey-Stewartson system is investigated.

The Lax Integrable Differential-Difference Dynamical Systems on Extended Phase Spaces

“…They considered non-periodic lattices of a Toda type [10,33,37,41] related to coadjoint orbits of solvable matrix Lie algebras.The R-matrix approach [5,14,29,35,37,40] being useful for the Lie-algebraic description of Lax integrable nonlinear dynamical systems on functional manifolds [1, 23] turned out to be suitable for the Lie-algebraic description of Lax integrable (1 + 1)-dimensional lattice and nonlocal differential-difference systems by means of the Lie algebra of shift operators [4,11,22,27,31].The Lax integrable (2 + 1)-dimensional differential-difference systems were obtained via the Sato procedure [39] in [32,43,44] whereas in papers [6,7,8,17,20, 34] such differential-difference systems were considered as Hamiltonian flows on the dual spaces to the central extensions by the Maurer-Cartan 2-cocycle of shift operator Lie algebras.Taking into account that every flow from the Lax-type hierarchy on the dual space to the shift operator Lie algebra or its central extension can be written as a compatibility condition of the spectral relationship for the corresponding operator and the suitable eigenfunction evolution an important problem of finding the Hamiltonian representation for the hierarchy of Lax-type flows coupled with the evolutions of eigenfunctions and appropriate adjoint eigenfunctions naturally arises. In the case when the spectral relationship admits a finite set of eigenvalues it was partly solved in the papers [16,18,19,30,36] for the Lie algebra of integral-differential operators [30,36] ⋆ This paper is a contribution to the…”

“…and its supergeneralizations [16,19] as well as for the corresponding central extension [18] by means of the variational property of Casimir functionals under some Bäcklund transformation. Section 2 deals with a general Lie-algebraic scheme for constructing the hierarchy of Lax-type flows as Hamiltonian ones on a dual space to the Lie algebra of shift operators [4,8].…”

“…In Section 3 the Hamiltonian structure for the related coupled Lax-type hierarchy is obtained by means of the Bäcklund transformation technique developed in [16,18,19,36].…”

“…In Section 4 the corresponding hierarchies of squared eigenfunction symmetries [3,12,13,28,44,45] for the coupled Lax-type flows are established to be Hamiltonian also. It is proved that the additional hierarchy of Hamiltonian flows is generated by the Poisson structure being obtained from the tensor product of the R-deformed canonical Lie-Poisson bracket with the standard Poisson bracket on the related eigenfunctions and adjoint eigenfunctions superspace [16,18,19,36] and the corresponding natural powers of a suitable eigenvalue are their Hamiltonians. These hierarchies are applied to constructing Lax integrable (2 + 1)-dimensional differential-difference systems and their triple Lax-type linearizations.…”

“…Earlier the squared eigenfunction symmetry hierarchies associated with the Lie algebras of integral-differential and super-integral-differential operators in commutator forms as well as their relations to some (2+1)-, (2|1+1)-and (2|2+1)-dimensional Lax integrable nonlinear dynamical systems on functional manifolds and supermanifolds were investigated in [3,12,13,16,18,19,28]. In paper [44] the commutator-type squared eigenfunction symmetries were considered for the shift operator Lie algebra.…”