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


SIGMA 2 (2006), 030, 19 pages      math-ph/0511071      https://doi.org/10.3842/SIGMA.2006.030

Supersymmetric Representations and Integrable Fermionic Extensions of the Burgers and Boussinesq Equations

Arthemy V. Kiselev a, b and Thomas Wolf c
a) Department of Higher Mathematics, Ivanovo State Power University, 34 Rabfakovskaya Str., Ivanovo, 153003 Russia
b) Department of Physics, Middle East Technical University, 06531 Ankara, Turkey
c) Department of Mathematics, Brock University, 500 Glenridge Ave., St. Catharines, Ontario, Canada L2S 3A1

Received November 26, 2005, in final form February 25, 2006; Published online February 28, 2006

Abstract
We construct new integrable coupled systems of N = 1 supersymmetric equations and present integrable fermionic extensions of the Burgers and Boussinesq equations. Existence of infinitely many higher symmetries is demonstrated by the presence of recursion operators. Various algebraic methods are applied to the analysis of symmetries, conservation laws, recursion operators, and Hamiltonian structures. A fermionic extension of the Burgers equation is related with the Burgers flows on associative algebras. A Gardner's deformation is found for the bosonic super-field dispersionless Boussinesq equation, and unusual properties of a recursion operator for its Hamiltonian symmetries are described. Also, we construct a three-parametric supersymmetric system that incorporates the Boussinesq equation with dispersion and dissipation but never retracts to it for any values of the parameters.

Key words: integrable super-equations; fermionic extensions; Burgers equation; Boussinesq equation.

pdf (363 kb)   ps (209 kb)   tex (26 kb)

References

  1. Andrea S., Restuccia A., Sotomayor A., The Gardner category and nonlocal conservation laws for N=1 Super KdV, J. Math. Phys., 2005, V.46, 103513, 11 pages, hep-th/0504149.
  2. Bilge A.H., On the equivalence of linearization and formal symmetries as integrability tests for evolution equations, J. Phys. A: Math. Gen., 1993, V.26, 7511-7519.
  3. Hlavatý L., The Painlevé analysis of fermionic extensions of KdV and Burgers equations, Phys. Lett. A, 1989, V.137, N 4-5, 173-178.
  4. Kersten P., Krasil'shchik I., Verbovetsky A., Hamiltonian operators and l*-coverings, J. Geom. Phys., 2004, V.50, N 1-4, 273-302, math.DG/0304245.
  5. Kersten P., Krasil'shchik I., Verbovetsky A., (Non)local Hamiltonian and symplectic structures, recursions and hierarchies: a new approach and applications to the N=1 supersymmetric KdV equation, J. Phys. A: Math. Gen., 2004, V.37, 5003-5019, nlin.SI/0305026.
  6. Kiselev A.V., Karasu A., Hamiltonian deformations of the Boussinesq equations, Proc. Workshop `Quantization, Dualities, and Integrable Systems' (January 23-27, 2006, Denizli, Turkey), Preprint ISPUmath-1/2006, 12 pages.
  7. Kiselev A.V., Wolf T., On weakly non-local, nilpotent, and super-recursion operators for N=1 homogeneous super-equations, Proc. Int. Workshop `Supersymmetries and Quantum Symmetries - 2005' (July 26-31, 2005, Dubna, Russia), to appear, nlin.SI/0511056.
  8. Kiselev A.V., Wolf T., The SSTOOLS environment for classification of integrable super-equations, Comp. Phys. Commun., 2006, to appear.
  9. Krasil'shchik I.S., Kersten P.H.M., Symmetries and recursion operators for classical and supersymmetric differential equations, Dordrecht, Kluwer Acad. Publ., 2000.
  10. Krasil'shchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of jet spaces and nonlinear partial differential equations, New York, Gordon & Breach Sci. Publ., 1986.
  11. Kupershmidt B.A., Singular symmetries of integrable curves and surfaces, J. Math. Phys., 1982, V.23, 364-366.
  12. Kupershmidt B.A., Deformations of integrable systems, Proc. Roy. Irish Acad. A, 1983, V.83, 45-74.
  13. Laberge C.A., Mathieu P., N=2 superconformal algebra and integrable O(2) fermionic extensions of the Korteweg-de Vries equation, Phys. Lett. B, 1988, V.215, 718-722.
  14. Maltsev A.Ya., Novikov S.P., On the local systems Hamiltonian in the weakly non-local Poisson brackets, Phys. D, 2001, V.156, 53-80, nlin.SI/0006030.
  15. Manin Yu.I., Radul A.O., A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy, Comm. Math. Phys., 1985, V.98, 65-77.
  16. Mathieu P., Supersymmetric extension of the Korteweg-de Vries equation, J. Math. Phys., 1988, V.29, 2499-2506.
  17. Mathieu P., Open problems for the super KdV equations. Bäcklund and Darboux transformations. The geometry of solitons, CRM Proc. Lecture Notes, 2001, V.29, 325-334, math-ph/0005007.
  18. Miura R.M., Gardner C.S., Kruskal M.D., Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Math. Phys., 1968, V.9, 1204-1209.
  19. Olver P.J. , 2nd ed., New York, Springer-Verlag, 1993.
  20. Olver P.J., Sokolov V.V., Integrable evolution equations on associative algebras, Comm. Math. Phys., 1998, V.193, 245-268.
    Olver P.J., Sokolov V.V., Non-abelian integrable systems of the derivative nonlinear Schrödinger type, Inverse Problems, 1998, V.14, L5-L8.
  21. Rozdestvenski B.L., Janenko N.N., Systems of quasilinear equations and their applicatons to gas dynamics, Translations of Mathematical Monographs, Vol. 55, Providence, RI, AMS, 1983.
  22. Sergyeyev A., Locality of symmetries generated by nonhereditary, inhomogeneous, and time-dependent recursion operators: a new application for formal symmetries, Acta Appl. Math., 2004, V.83, 95-109, nlin.SI/0303033.
    Sergyeyev A., Why nonlocal recursion operators produce local symmetries: new results and applications, J. Phys. A: Math. Gen., 2005, V.38, 3397-3407, nlin.SI/0410049.
  23. Svinolupov S.I., On the analogues of the Burgers equation, Phys. Lett. A, 1989, V.135, 32-36.
  24. Tsuchida T., Wolf T., Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I, J. Phys. A: Math. Gen., 2005, V.38, 7691-7733, nlin.SI/0412003.
  25. Weiss J., Tabor M., Carnevale G., The Painlevé property for partial differential equations, J. Math. Phys., 1983, V.24, 522-526.
  26. Wolf T., Applications of CRACK in the classification of integrable systems, CRM Proc. Lecture Notes, 2004, V.37, 283-300, nlin.SI/0301032.
  27. Wolf T., Supersymmetric evolutionary equations with higher order symmetries, 2003,
    http://beowulf.ac.brocku.ca/~twolf/htdocs/susy/all.html (please contact T. W. for access).

Previous article   Next article   Contents of Volume 2 (2006)