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


SIGMA 6 (2010), 018, 10 pages      arXiv:0910.2389      http://dx.doi.org/10.3842/SIGMA.2010.018
Contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries

The Integrability of New Two-Component KdV Equation

Ziemowit Popowicz
Institute for Theoretical Physics, University of Wroclaw, Wroclaw 50204, Poland

Received October 19, 2009, in final form February 04, 2010; Published online February 12, 2010

Abstract
We consider the bi-Hamiltonian representation of the two-component coupled KdV equations discovered by Drinfel'd and Sokolov and rediscovered by Sakovich and Foursov. Connection of this equation with the supersymmetric Kadomtsev-Petviashvilli-Radul-Manin hierarchy is presented. For this new supersymmetric equation the Lax representation and odd Hamiltonian structure is given.

Key words: KdV equation; Lax representation; integrability; supersymmetry.

pdf (241 kb)   ps (152 kb)   tex (15 kb)

References

  1. Svinopulov S., Jordan algebras and integrable systems, Funct. Anal. Appl. 27 (1993), 257-265.
    Svinopulov S., Jordan algebras and generalized Korteweg-de Vries equations, Theoret. and Math. Phys. 87 (1991), 611-620.
  2. Gürses M., Karasu A., Integrable coupled KdV systems, J. Math. Phys. 39 (1998), 2103-2111, solv-int/9711015.
    Gürses M., Karasu A., Integrable KdV systems: recursion operators of degree four, Phys. Lett. A 251 (1999), 247-249, solv-int/9811013.
  3. Antonowicz M., Fordy A.P., Coupled KdV equations with multi-Hamiltonian structures, Phys. D 28 (1987), 345-357.
  4. Ma W.-X., A class of coupled KdV systems and their bi-Hamiltonian formulation, J. Phys. A: Math. Gen. 31 (1998), 7585-7591, solv-int/9803009.
  5. Foursov M.V., Towards the complete classification of homogeneous two-component integrable equations, J. Math. Phys. 44 (2003), 3088-3096.
  6. Drinfel'd V.G., Sokolov V.V., New evolutionary equations possessing an (L,A)-pair, Trudy Sem. S.L. Soboleva (1981), no. 2, 5-9 (in Russian).
  7. Kupershmidt B.A., Elements of superintegrable systems. Basic techniques and results, Mathematics and its Applications, Vol. 34, D. Reidel Publishing Co., Dordrecht, 1987.
  8. Gürses M., Oguz O., A super AKNS scheme, Phys. Lett. A 108 (1985), 437-440.
  9. Kulish P.P., Quantum osp-invariant nonlinear Schrödinger equation, Lett. Math. Phys. 10 (1985), 87-93.
  10. Manin Yu.I., Radul A.O., A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy, Comm. Math. Phys. 98 (1985), 65-77.
  11. Mathieu P., Supersymmetric extension of the Korteweg-de Vries equation, J. Math. Phys. 29 (1988), 2499-2506.
  12. Laberge C.A., Mathieu P., N=2 superconformal algebra and integrable O(2) fermionic extensions of the Korteweg-de Vries equation, Phys. Lett. B 215 (1988), 718-722.
  13. Labelle P., Mathieu P., A new N=2 supersymmetric Korteweg-de Vries equation, J. Math. Phys. 32 (1991), 923-927.
  14. Chaichian M., Lukierski J., N=1 super-WZW and N=1,2,3,4 super-KdV models as D=2 current superfield theories, Phys. Lett. B 212 (1988), 451-466.
  15. Oevel W., Popowicz Z., The bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems, Comm. Math. Phys. 139 (1991), 441-460.
  16. Hirota R., Satsuma J., Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A 85 (1981), 407-408.
  17. Ito M., Symmetries and conservation laws of a coupled nonlinear wave equation, Phys. Lett. A 91 (1982), 335-338.
  18. Sakovich S.Yu., Coupled KdV equations of Hirota-Satsuma type, J. Nonlinear Math. Phys. 6 (1999), 255-262, solv-int/9901005.
    Sakovich S.Yu., Addendum to: "Coupled KdV equations of Hirota-Satsuma type", J. Nonlinear Math. Phys. 8 (2001), 311-312, nlin.SI/0104072.
  19. Dodd R., Fordy A., On the integrability of a system of coupled KdV equations, Phys. Lett. A 89 (1982), 168-170.
  20. Popowicz Z., The extended supersymmetrization of the multicomponent Kadomtsev-Petviashvilli hierarchy, J. Phys. A: Math. Gen. 19 (1996), 1281-1291.
  21. Blaszak M., Multi-Hamiltonian theory of dynamical systems, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1998.
  22. Gürses M., Karasu A., Sokolov V.V., On construction of recursion operators from Lax representation, J. Math. Phys. 40 (1999), 6473-6490, solv-int/9909003.
  23. Popowicz Z., Odd Hamiltonian structure for supersymmetric Sawada-Kotera equation, Phys. Lett. A 373 (2009), 3315-3323, arXiv:0902.2861.
  24. Tian K., Liu Q.P., A supersymmetric Sawada-Kotera equation, Phys. Lett. A 373 (2009), 1807-1810, arXiv:0802.4011.
  25. Aratyn H., Nissimov E., Pacheva S., Supersymmetric Kadomtsev-Petviashvili hierarchy: "ghost" symmetry structure, reductions, and Darboux-Bäcklund solutions, J. Math. Phys. 40 (1999), 2922-2932, solv-int/9801021.
    Aratyn H., Nissimov E., Pacheva S., Berezinian construction of super-solitons in supersymmetric constrained KP hierarchy, solv-int/9808004.

Previous article   Next article   Contents of Volume 6 (2010)