Department of Nonlinear Mathematical Analysis
Institute for Applied Problems of Mechanics & Mathematics
3b Naukova Str.,
Lviv 79060,
E-mail: dept25@iapmm.lviv.ua, hento@ua.fm

The Lax integrable supersymmetric hierarchies on extended phase spaces

Since the paper of M. Adler [1] there was understood that a wide class of Lax type integrable Korteweg-de Vries nonlinear systems and their supersymmetric analogs [2, 3] of one and two anticommuting variables could be considered as Hamiltonian flows, generated by the R-deformed canonical Lie-Poisson structure and Casimir functionals, on a dual space to the Lie algebra of integral-differential operators. The existence of Hamiltonian  representation for these flows, added by correponding evolutions of associated spectral problem eigenfunctions and adjoint eigenfunctions, in the case of super-integro-differential operators of one and two anticommuting variables is investigated by use of the invariant Casimir functionals' property under some Lie-Backlund transformation [4].

The following hierarchies of additional symmetries in a Lax type form are proven to be Hamiltonian ones. It is shown that the additional symmetry are generated by the Poisson structure, being a tensor product of the canonical Lie-Poisson and some canonical finite-dimensional super-Poisson ones, and a functional, being a sum of Casimir one and the corresponding power of a spectral eigenvalue. The connection of additional symmetry hierarchies with (2|1+1)- and (2|2+1)-dimensional supersymmetric Davey-Stewartson equations of one and two anticommuting variables accordingly and their triple linear representations are established.

  1. Adler M. On a trace functional for formal pseudo-differential operators and the symplectic structures of a Korteweg-de Vries equation, Invent. Math., 1979, 50, 219.
  2. Manin Yu.I. and Radul A.O. A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy, Comm. Math. Phys., 1985, 98, 65.
  3. Popowicz Z. The extended supersymmetrization of the nonlinear Schrodinger equation, Physics Letters A, 1994, 194, 375.
  4. Prykarpatsky A.K. and Hentosh O.Ye. The Lie-algebraic structure of (2+1)-dimensional Lax type integrable nonlinear dynamical systems, Ukr. Math. J., 2004, 56, N 7, 939.