Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 001, 11 pages     nlin.SI/0601007      http://dx.doi.org/10.3842/SIGMA.2006.001

Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces

Oksana Ye. Hentosh
Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, 3B Naukova Str., Lviv, 79060 Ukraine

Received October 27, 2005, in final form December 21, 2005; Published online January 04, 2006

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.

Key words: Lax type flows; "ghost" symmetries; the Davey-Stewartson system.

pdf (229 kb)   ps (174 kb)   tex (14 kb)

References

  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, V.50, N 2, 219-248.
  2. Lax P.D., Periodic solutions of the KdV equation, Comm. Pure Appl. Math., 1975, V.28, 141-188.
  3. Novikov S.P. (Editor), Soliton theory: method of the inverse problem, Moscow, Nauka, 1980 (in Russian).
  4. Prykarpatsky A.K., Mykytiuk I.V., Algebraic integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects, Dordrecht-Boston-London, Kluwer Academic Publishers, 1998.
  5. Blaszak M., Multi-Hamiltonian theory of dynamical systems, Verlag-Berlin-Heidelberg, Springer, 1998.
  6. Manin Yu.I., Radul A.O., A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy, Comm. Math. Phys., 1985, V.98, 65-77.
  7. Oevel W., Popowicz Z., The bi-Hamiltonian structure of fully supersymmetrin Korteweg-de Vries systems, Comm. Math. Phys., 1991, V.139, 441-460.
  8. Semenov-Tian-Shansky M.A., What is the A-matrix, Funct. Anal. Appl., 1983, V.17, N 4, 17-33 (in Russian).
  9. Oevel W., A-structures, Yang-Baxter equations and related involution theorems, J. Math. Phys., 1989, V.30, N 5, 1140-1149.
  10. Oevel W., Strampp W., Constrained KP hierarchy and bi-Hamiltonian structures, Comm. Math. Phys., 1993, V.157, 51-81.
  11. Prykarpatsky Ya.A., Structure of integrable Lax flows on nonlocal manifolds: dynamical systems with sources, Math. Methods and Phys.-Mech. Fields, 1997, V.40, N 4, 106-115 (in Ukrainian).
  12. Prykarpatsky A.K., Hentosh O.Ye., The Lie-algebraic structure of (2+1)-dimensional Lax type integrable nonlinear dynamical systems, Ukrainian Math. J., 2004, V.56, N 7, 939-946.
  13. Nissimov E., Pacheva S., Symmetries of supersymmetric integrable hierarchies of KP type, J. Math. Phys., 2002, V.43, N 5, 2547-2586, nlin.SI/0102010.
  14. Berezin F.A., Introduction to algebra and analysis with anticommuting variables, Moscow Univ., 1983 (in Russian).
  15. Shander V.N., Analogues of the Frobenius and Darboux theorems, Reports of Bulgarian Academy of Sciences, 1983, V.36, N 3, 309-311.
  16. Ablowitz M.J., Segur H., Solitons and the inverse scattering transform, Philadelphia, SIAM, 1981.
  17. Kulish P.P., Lipovsky V.D., On Hamiltonian interpretation of the inverse problem method for the Davey-Stewartson equation, LOMI Proceedings, Leningrad, Nauka, 1987, V.161, 54-71 (in Russian).
  18. Prykarpatsky A.K., Hentosh O.E., Blackmore D.L., The finite-dimensional Moser type reductions of modified Boussinesq and super-Korteweg-de Vries Hamiltonian systems via the gradient-holonomic algorithm and the dual moment maps. I, J. Nonlinear Math. Phys., 1997, V.4, N 3-4, 455-469.
  19. Prykarpatsky A.K., Blackmore D., Strampp W., Sydorenko Yu., Samuliak R., Some remarks on Lagrangian and Hamiltonian formalism, related to infinite-dimensional dynamical systems with symmetries, Condensed Matter Phys., 1995, N 6, 79-104.
  20. Sato M., Soliton equations as dynamical systems on infinite Grassmann manifolds, RIMS Kokyuroku, Kyoto Univ., 1981, V.439, 30-40.
  21. Reiman A.G., Semenov-Tian-Shansky M.A., The Hamiltonian structure of Kadomtsev-Petviashvili type equations, LOMI Proceedings, Leningrad, Nauka, 1987, V.164, 212-227 (in Russian).
  22. Prykarpatsky A.K., Samoilenko V.Hr., Andrushkiw R.I., Mitropolsky Yu.O., Prytula M.M., Algebraic structure of the gradient-holonomic algorithm for Lax integrable nonlinear systems. I, J. Math. Phys., 1994, V.35, N 4, 1763-1777.

Next article   Contents of Volume 2 (2006)