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

SIGMA 9 (2013), 045, 12 pages      arXiv:1306.3628      https://doi.org/10.3842/SIGMA.2013.045

Euler Equations Related to the Generalized Neveu-Schwarz Algebra

Dafeng Zuo a, b
a) School of Mathematical Science, University of Science and Technology of China, Hefei 230026, P.R. China
b) Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, P.R. China

Received March 11, 2013, in final form June 12, 2013; Published online June 16, 2013

Abstract
In this paper, we study supersymmetric or bi-superhamiltonian Euler equations related to the generalized Neveu-Schwarz algebra. As an application, we obtain several supersymmetric or bi-superhamiltonian generalizations of some well-known integrable systems including the coupled KdV equation, the 2-component Camassa-Holm equation and the 2-component Hunter-Saxton equation. To our knowledge, most of them are new.

Key words: supersymmetric; bi-superhamiltonian; Euler equations; generalized Neveu-Schwarz algebra.

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References

  1. Antonowicz M., Fordy A.P., Super-extensions of energy dependent Schrödinger operators, Comm. Math. Phys. 124 (1989), 487-500.
  2. Aratyn H., Gomes J.F., Zimerman A.H., Deformations of N=2 superconformal algebra and supersymmetric two-component Camassa-Holm equation, J. Phys. A: Math. Theor. 40 (2007), 4511-4527, hep-th/0611192.
  3. Arnold V.I., Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966), 319-361.
  4. Arnold V.I., Khesin B.A., Topological methods in hydrodynamics, Applied Mathematical Sciences, Vol. 125, Springer-Verlag, New York, 1998.
  5. Brunelli J.C., Das A., Popowicz Z., Supersymmetric extensions of the Harry Dym hierarchy, J. Math. Phys. 44 (2003), 4756-4767, nlin.SI/0304047.
  6. Chen M., Liu S.-Q., Zhang Y., A two-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys. 75 (2006), 1-15, nlin.SI/0501028.
  7. Constantin A., Kappeler T., Kolev B., Topalov P., On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom. 31 (2007), 155-180.
  8. Constantin A., Kolev B., Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv. 78 (2003), 787-804, math-ph/0305013.
  9. Constantin A., Kolev B., Integrability of invariant metrics on the diffeomorphism group of the circle, J. Nonlinear Sci. 16 (2006), 109-122, arXiv:0911.5058.
  10. Devchand C., A Kuper-CH system, Unpublished note, 2005, Private communications, 2010.
  11. Devchand C., Schiff J., The supersymmetric Camassa-Holm equation and geodesic flow on the superconformal group, J. Math. Phys. 42 (2001), 260-273, solv-int/9811016.
  12. Ebin D.G., Marsden J., Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92 (1970), 102-163.
  13. Falqui G., On a Camassa-Holm type equation with two dependent variables, J. Phys. A: Math. Gen. 39 (2006), 327-342, nlin.SI/0505059.
  14. Guha P., Geodesic flow on extended Bott-Virasoro group and generalized two-component peakon type dual systems, Rev. Math. Phys. 20 (2008), 1191-1208.
  15. Guha P., Integrable geodesic flows on the (super)extension of the Bott-Virasoro group, Lett. Math. Phys. 52 (2000), 311-328.
  16. Guha P., Olver P.J., Geodesic flow and two (super) component analog of the Camassa-Holm equation, SIGMA 2 (2006), 054, 9 pages, nlin.SI/0605041.
  17. Ito M., Symmetries and conservation laws of a coupled nonlinear wave equation, Phys. Lett. A 91 (1982), 335-338.
  18. Khesin B., Topological fluid dynamics, Notices Amer. Math. Soc. 52 (2005), 9-19.
  19. Khesin B., Lenells J., Misioek G., Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann. 342 (2008), 617-656, arXiv:0803.3078.
  20. Khesin B., Misioek G., Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math. 176 (2003), 116-144.
  21. Khesin B., Wendt R., The geometry of infinite-dimensional groups, A Series of Modern Surveys in Mathematics, Vol. 51, Springer-Verlag, Berlin, 2009.
  22. Kolev B., Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365 (2007), 2333-2357, math-ph/0610001.
  23. Kupershmidt B.A., A super Korteweg-de Vries equation: an integrable system, Phys. Lett. A 102 (1984), 213-215.
  24. Lenells J., A bi-Hamiltonian supersymmetric geodesic equation, Lett. Math. Phys. 85 (2008), 55-63, arXiv:0806.4793.
  25. Lenells J., Lechtenfeld O., On the N=2 supersymmetric Camassa-Holm and Hunter-Saxton equations, J. Math. Phys. 50 (2009), 012704, 17 pages, arXiv:0809.0077.
  26. Liu Q.P., Popowicz Z., Tian K., Supersymmetric reciprocal transformation and its applications, J. Math. Phys. 51 (2010), 093511, 24 pages, arXiv:1002.1769.
  27. Liu Q.P., Popowicz Z., Tian K., The even and odd supersymmetric Hunter-Saxton and Liouville equations, Phys. Lett. A 375 (2010), 29-35, arXiv:1005.3109.
  28. Manakov S.V., A remark on the integration of the Eulerian equations of the dynamics of an n-dimensional rigid body, Funct. Anal. Appl. 10 (1976), 328-329.
  29. Marcel P., Ovsienko V., Roger C., Extension of the Virasoro and Neveu-Schwarz algebras and generalized Sturm-Liouville operators, Lett. Math. Phys. 40 (1997), 31-39, hep-th/9602170.
  30. Marsden J.E., Ratiu T.S., Introduction to mechanics and symmetry. A basic exposition of classical mechanical systems, Texts in Applied Mathematics, Vol. 17, 2nd ed., Springer-Verlag, New York, 1999.
  31. Mathieu P., Supersymmetric extension of the Korteweg-de Vries equation, J. Math. Phys. 29 (1988), 2499-2506.
  32. Mishchenko A.S., Integrals of geodesic flows on Lie groups, Funct. Anal. Appl. 4 (1970), 232-235.
  33. Misioek G., A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys. 24 (1998), 203-208.
  34. Ovsienko V.Yu., Coadjoint representation of Virasoro-type Lie algebras and differential operators on tensor-densities, in Infinite Dimensional Kähler Manifolds (Oberwolfach, 1995), DMV Sem., Vol. 31, Birkhäuser, Basel, 2001, 231-255, math-ph/0602009.
  35. Ovsienko V.Yu., Khesin B.A., Korteweg-de Vries superequation as an Euler equation, Funct. Anal. Appl. 21 (1987), 329-331.
  36. Popowicz Z., A 2-component or N=2 supersymmetric Camassa-Holm equation, Phys. Lett. A 354 (2006), 110-114, nlin.SI/0509050.
  37. Ratiu T., Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body, Proc. Nat. Acad. Sci. USA 78 (1981), 1327-1328.
  38. Zhang L., Zuo D., Integrable hierarchies related to the Kuper-CH spectral problem, J. Math. Phys. 52 (2011), 073503, 11 pages.
  39. Zuo D., A two-component μ-Hunter-Saxton equation, Inverse Problems 26 (2010), 085003, 9 pages, arXiv:1010.4454.

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