Letters in Mathematical Physics

, Volume 59, Issue 2, pp 117–131

Geometric Integrability of the Camassa–Holm Equation

  • Enrique G. Reyes
Article

Abstract

It is observed that the Camassa–Holm equation describes pseudo-spherical surfaces and that therefore, its integrability properties can be studied by geometrical means. An sl(2, R)-valued linear problem whose integrability condition is the Camassa–Holm equation is presented, a ‘Miura transform’ and a ‘modified Camassa–Holm equation’ are introduced, and conservation laws for the Camassa–Holm equation are then directly constructed. Finally, it is pointed out that this equation possesses a nonlocal symmetry, and its flow is explicitly computed.

Camassa–Holm equation pseudo-spherical surfaces geometric integrability Miura transformation conservation law nonlocal symmetry 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, I. M.: The Vessiot Handbook. Formal geometry and mathematical physics technical report, Utah State University, 2000. Website: http://www.math.usu.edu/~fg_mp/Google Scholar
  2. 2.
    Beals, R., Sattinger, D. H. and Szmigielski, J: Acoustic scattering and the extended Korteweg-de Vries hierarchy, Adv. Math. 140 (1998), 190–206.Google Scholar
  3. 3.
    Bluman, G. W. and Kumei, S.: Symmetry-based algorithms to relate partial differential equations. II. Linearization by nonlocal symmetries, European J. Appl. Math. 1(1) (1990), 217–223.Google Scholar
  4. 4.
    Camassa, R. and Holm, D. D.: An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71(11) (1993), 1661–1664.Google Scholar
  5. 5.
    Camassa, R., Holm, D. D. and Hyman, J. M.: A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994), 1–33.Google Scholar
  6. 6.
    Camassa, R. and Zenchuk, A. I.: On the initial value problem for a completely integrable shallow water wave equation, Phys. Lett. A 281 (2001), 26–33.Google Scholar
  7. 7.
    Chern, S. S. and Tenenblat, K.: Pseudo-spherical surfaces and evolution equations, Stud. Appl. Math. 74 (1986), 55–83.Google Scholar
  8. 8.
    Constantin, A.: The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math. 15(1) (1997), 53–85.Google Scholar
  9. 9.
    Fisher, M. and Schiff, J.: The Camassa Holm equation: conserved quantities and the initial value problem, Phys. Lett. A 259(5) (1999), 371–376.Google Scholar
  10. 10.
    Fokas, A. S.: On a class of physically important integrable equations, Physica D 87 (1995), 145–150.Google Scholar
  11. 11.
    Fokas, A. S., Olver, P. J. and Rosenau, P.: A plethora of integrable bi-Hamiltonian equations, In: A. S. Fokas and I. M. Gel'fand (eds), Algebraic Aspects of Integrable Systems: In memory of Irene Dorfman, Prog. Nonlinear Differential Equations 26, Birkhauser, Boston, 1996, pp. 93–101.Google Scholar
  12. 12.
    Fuchssteiner, B. and Fokas, A. S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D 4 (1981), 47–66.Google Scholar
  13. 13.
    Fuchssteiner, B.: The Lie algebra structure of degenerate hamiltonian and bi-hamiltonian systems, Progr. Theoret. Phys. 68 (1982), 1082–1104.Google Scholar
  14. 14.
    Fuchssteiner, B.: Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Physica D 95 (1996), 229–243.Google Scholar
  15. 15.
    Galas, F.: New nonlocal symmetries with pseudopotentials, J. Phys. A 25(15) (1992), L981–L986.Google Scholar
  16. 16.
    Hunter, J. K. and Saxton, R.: Dynamics of director fields, SIAM J. Appl. Math. 51(6) (1991), 1498–1521.Google Scholar
  17. 17.
    Kraenkel, R. A. and Zenchuk, A.: Camassa-Holm equation: transformation to deformed sinh-Gordon equations, cuspon and soliton solutions, J. Phys. A: Math. Gen. 32 (1999), 4733–4747.Google Scholar
  18. 18.
    Krasil'shchik, I. S. and Vinogradov, A. M. ( eds): Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Transl. Math. Monogr. 182, Amer. Math. Soc., Providence, 1999.Google Scholar
  19. 19.
    Leo, M., Leo, R., Soliani, G. and Tempesta, P.: On the relation between Lie symmetries and prolongation structures of nonlinear field equations—non-local symmetries, Progr. Theoret. Phys. 105(1) (2001), 77–97.Google Scholar
  20. 20.
    Olver, P. J. and Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E 53 (1996), 1900–1906.Google Scholar
  21. 21.
    Reyes, E. G.: Some geometric aspects of integrability of differential equations in two independent variables, Acta Appl. Math. 64(2/3) (2000), 75–109.Google Scholar
  22. 22.
    Reyes, E. G.: On geometrically integrable equations and hierarchies of pseudo-spherical type. In: J. A. Leslie and T. Robart (eds), Proc. NSF-CBMS Conference on the Geometrical Study of Differential Equations, Contemp. Math., Amer. Math. Soc., 2001.Google Scholar
  23. 23.
    Reyes, E. G.: The soliton content of the Camassa-Holm and Hunter-Saxton equations. In: Proc. Fourth International Conference on Symmetry in Nonlinear Mathematical Physics, Proc. Inst. Math. NAS Ukraine, 2001.Google Scholar
  24. 24.
    Schiff, J.: Zero curvature formulations of dual hierarchies, J. Math. Phys. 37(4) (1996), 1928–1938.Google Scholar
  25. 25.
    Schiff, J.: The Camassa-Holm equation: a loop group approach, Phys. D 121(1-2) (1998), 24–43.Google Scholar
  26. 26.
    Schiff, J.: Symmetries of KdV and loop groups, Preprint, solv-int/9606004 Los Alamos arXiv.org e-Print archive.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Enrique G. Reyes
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenU.S.A.

Personalised recommendations