Skip to main content
SpringerLink
Log in

A Bi-Hamiltonian Supersymmetric Geodesic Equation

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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. Camassa R., Holm D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  5. Constantin A., Escher J.: Global existence and blow-up for a shallow water equation. Annali Sc. Norm. Sup. Pisa 26, 303–328 (1998)

    MATH  MathSciNet  Google Scholar 

  6. Constantin A., Gerdjikov V.S., Ivanov R.I.: Inverse scattering transform for the Camassa–Holm equation. Inverse Probl. 22, 2197–2207 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Constantin A., Kolev B.: On the geometric approach to the motion of inertial mechanical systems. J. Phys. A 35, R51–R79 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Depireux D.A., Mathieu P.: Integrable supersymmetry breaking perturbations of N = 1, 2 superconformal minimal models. Phys. Lett. B 308, 272–278 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  9. Devchand C., Schiff J.: The supersymmetric Camassa–Holm equation and geodesic flow on the superconformal group. J. Math. Phys. 42, 260–273 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. Fokas A.S.: Symmetries and integrability. Stud. Appl. Math. 77, 253–299 (1987)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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)

    Article  MATH  ADS  Google Scholar 

  13. Hunter J.K., Saxton R.: Dynamics of director fields. SIAM J. Appl. Math. 51, 1498–1521 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hunter J.K., Zheng Y.: On a completely integrable nonlinear hyperbolic variational equation. Phys. D 79, 361–386 (1994)

    MATH  MathSciNet  Google Scholar 

  15. Khesin B., Misiołek G.: Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176, 116–144 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kulish P.P.: Quantum OSP-invariant nonlinear Schrödinger equation. Lett. Math. Phys. 10, 87–93 (1985)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Kupershmidt B.A.: A super Korteweg-de Vries equation: an integrable system. Phys. Lett. A 102, 213–215 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  18. Lenells J.: Traveling wave solutions of the Camassa–Holm equation. J. Differ. Equ. 217, 393–430 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lenells J.: The Hunter–Saxton equation describes the geodesic flow on a sphere. J. Geom. Phys. 57, 2049–2064 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Manin Y.I., Radul A.O.: A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy. Commun. Math. Phys. 98, 65–77 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  21. Mathieu P.: Supersymmetric extension of the Korteweg–de Vries equation. J. Math. Phys. 29, 2499–2506 (1988)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. Misiołek G.: A shallow water equation as a geodesic flow on the Bott–Virasoro group. J. Geom. Phys. 24, 203–208 (1998)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. Morosi C., Pizzocchero L.: osp(3,2) and gl(3,3) supersymmetric KdV hierarchies. Phys. Lett. A 185, 241–252 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. Oevel W., Popowicz Z.: The bi-Hamiltonian structure of fully supersymmetric Korteweg–de Vries systems. Commun. Math. Phys. 139, 441–460 (1991)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. Ovsienko V., Khesin B.: The (super) KdV equation as an Euler equation. Funct. Anal. Appl. 21(4), 81–82 (1987)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Shkoller S.: Geometry and curvature of diffeomorphism groups with H 1 metric and mean hydrodynamics. J. Funct. Anal. 160, 337–365 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK

    Jonatan Lenells

Authors
  1. Jonatan Lenells
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonatan Lenells.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lenells, J. A Bi-Hamiltonian Supersymmetric Geodesic Equation. Lett Math Phys 85, 55–63 (2008). https://doi.org/10.1007/s11005-008-0257-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-008-0257-4

Mathematics Subject Classification (2000)

Keywords

Search

Navigation