Letters in Mathematical Physics

, Volume 85, Issue 1, pp 55–63 | Cite as

A Bi-Hamiltonian Supersymmetric Geodesic Equation



A supersymmetric extension of the Hunter–Saxton equation is constructed. We present its bi-Hamiltonian structure and show that it arises geometrically as a geodesic equation on the space of superdiffeomorphisms of the circle that leave a point fixed endowed with a right-invariant metric.

Mathematics Subject Classification (2000)

37K10 17A70 


supersymmetry integrable systems Hunter–Saxton equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alber M., Camassa R., Holm D.D., Marsden J.E.: The geometry of peaked solitons and billiard solutions of a class of integrable PDE’s. Lett. Math. Phys. 32, 137–151 (1994)MATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Arnold V.: Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses application à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16, 319–361 (1966)Google Scholar
  3. 3.
    Camassa R., Holm D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Chaichian M., Kulish P.P.: On the method of inverse scattering problem and Bäcklund transformations for supersymmetric equations. Phys. Lett. B 78, 413–417 (1978)CrossRefADSGoogle Scholar
  5. 5.
    Constantin A., Escher J.: Global existence and blow-up for a shallow water equation. Annali Sc. Norm. Sup. Pisa 26, 303–328 (1998)MATHMathSciNetGoogle Scholar
  6. 6.
    Constantin A., Gerdjikov V.S., Ivanov R.I.: Inverse scattering transform for the Camassa–Holm equation. Inverse Probl. 22, 2197–2207 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Constantin A., Kolev B.: On the geometric approach to the motion of inertial mechanical systems. J. Phys. A 35, R51–R79 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Depireux D.A., Mathieu P.: Integrable supersymmetry breaking perturbations of N = 1, 2 superconformal minimal models. Phys. Lett. B 308, 272–278 (1993)CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Devchand C., Schiff J.: The supersymmetric Camassa–Holm equation and geodesic flow on the superconformal group. J. Math. Phys. 42, 260–273 (2001)MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Fokas A.S.: Symmetries and integrability. Stud. Appl. Math. 77, 253–299 (1987)MATHMathSciNetGoogle Scholar
  11. 11.
    Fokas A.S., Anderson R.L.: On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems. J. Math. Phys. 23, 1066–1073 (1982)MATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M.: Method for solving the Korteweg–de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)MATHCrossRefADSGoogle Scholar
  13. 13.
    Hunter J.K., Saxton R.: Dynamics of director fields. SIAM J. Appl. Math. 51, 1498–1521 (1991)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hunter J.K., Zheng Y.: On a completely integrable nonlinear hyperbolic variational equation. Phys. D 79, 361–386 (1994)MATHMathSciNetGoogle Scholar
  15. 15.
    Khesin B., Misiołek G.: Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176, 116–144 (2003)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kulish P.P.: Quantum OSP-invariant nonlinear Schrödinger equation. Lett. Math. Phys. 10, 87–93 (1985)MATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Kupershmidt B.A.: A super Korteweg-de Vries equation: an integrable system. Phys. Lett. A 102, 213–215 (1984)CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Lenells J.: Traveling wave solutions of the Camassa–Holm equation. J. Differ. Equ. 217, 393–430 (2005)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lenells J.: The Hunter–Saxton equation describes the geodesic flow on a sphere. J. Geom. Phys. 57, 2049–2064 (2007)MATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Manin Y.I., Radul A.O.: A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy. Commun. Math. Phys. 98, 65–77 (1988)CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Mathieu P.: Supersymmetric extension of the Korteweg–de Vries equation. J. Math. Phys. 29, 2499–2506 (1988)MATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Misiołek G.: A shallow water equation as a geodesic flow on the Bott–Virasoro group. J. Geom. Phys. 24, 203–208 (1998)MATHCrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Morosi C., Pizzocchero L.: osp(3,2) and gl(3,3) supersymmetric KdV hierarchies. Phys. Lett. A 185, 241–252 (1994)MATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Oevel W., Popowicz Z.: The bi-Hamiltonian structure of fully supersymmetric Korteweg–de Vries systems. Commun. Math. Phys. 139, 441–460 (1991)MATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Ovsienko V., Khesin B.: The (super) KdV equation as an Euler equation. Funct. Anal. Appl. 21(4), 81–82 (1987)MATHMathSciNetGoogle Scholar
  26. 26.
    Roelofs G.H.M., Kersten P.H.M.: Supersymmetric extensions of the nonlinear Schrödinger equation: symmetries and coverings. J. Math. Phys. 33, 2185–2206 (1992)MATHCrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Shkoller S.: Geometry and curvature of diffeomorphism groups with H 1 metric and mean hydrodynamics. J. Funct. Anal. 160, 337–365 (1998)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK

Personalised recommendations