Letters in Mathematical Physics

, Volume 32, Issue 2, pp 137–151 | Cite as

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

  • Mark S. Alber
  • Roberto Camassa
  • Darryl D. Holm
  • Jerrold E. Marsden


The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [10] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.

Mathematics Subject Classifications (1991)

58F07 70H99 76B15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ablowitz, M. J. and Segur, H.,Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.Google Scholar
  2. 2.
    Alber, S. J., Investigation of equations of Koteweg-de Vries type by the method of recurrence relations,J. London Math. Soc. 19, 467–480 (1979).Google Scholar
  3. 3.
    Alber, M. S., On integrable systems and semiclassical solutions of the stationary Schrödinger equations,Inverse Problems 5, 131–148 (1989).Google Scholar
  4. 4.
    Alber, M. S. and Alber, S. J., Hamiltonian formalism for finite-zone solutions of integrable equations,C.R. Acad. Sci. Paris 301, 777–781 (1985).Google Scholar
  5. 5.
    Alber, M. S. and Alber, S. J., Hamiltonian formalism for nonlinear Schrödinger equations and sine-Gordon equations,J. London Math. Soc. 36, 176–192 (1987).Google Scholar
  6. 6.
    Alber, M. S., Camassa, R., Holm, D. D., and Marsden, J. E., in preparation (1994).Google Scholar
  7. 7.
    Alber, M. S. and Marsden, J. E., On geometric phases for soliton equations,Comm. Math. Phys. 149, 217–240 (1992).Google Scholar
  8. 8.
    Alber, M. S. and Marsden, J. E.,Geometric Phases and Monodromy at Singularities, NATO Advanced Study Institute, Series C 1994, to appear.Google Scholar
  9. 9.
    Alber, M. S. and Marsden, J. E., Resonant geometric phases for soliton equations,Fields Institute Commun. 1994, to appear.Google Scholar
  10. 10.
    Camassa, R. and Holm, D. D., An integrable shallow water equation with peaked solitons,Phys. Rev. Lett,71, 1661–1664 (1993).Google Scholar
  11. 11.
    Camassa, R., Holm, D. D., and Hyman, J. M., A new integrable shallow water equation,Adv. Appl. Mech. (1993), to appear.Google Scholar
  12. 12.
    Ercolani, N. and McKean, H. P., Geometry of KdV(4): Abel sums, Jacobi variety, and theta function in the scattering case,Invent. Math. 99, 483 (1990).Google Scholar
  13. 13.
    Flaschka, H. and McLaughlin, D. W., Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions,Prog. Theoret. Phys. 55, 438–456 (1976).Google Scholar
  14. 14.
    Ge, Z., Kruse, H. P., Marsden, J. E., and Scovel, C., Poisson brackets in the shallow water approximation, preprint (1993).Google Scholar
  15. 15.
    Green, A. E. and Naghdi, P. M., A derivation of equations for wave propagation in water of variable depth,J. Fluid Mech. 78, 237–246 (1976).Google Scholar
  16. 16.
    Kruskal, M. D., Nonlinear wave equations, in J. Moser (ed),Dynamical Systems, Theory and Applications, Lecture Notes in Physics 38, Springer, New York, 1975.Google Scholar
  17. 17.
    Marsden, J. E., Montgomery, R., and Ratiu, T., Cartan-Hannay-Berry phases and symmetry,Contemp. Math. 97, 279 (1989); see alsoMem. Amer. Math. Soc. 436.Google Scholar
  18. 18.
    McKean, H. P.,Integrable Systems and Algebraic Curves, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1979.Google Scholar
  19. 19.
    McKean, H. P., Theta functions, solitons, and singular curves, in C. I. Byrnes (ed),PDE and Geometry, Proc. of Park City Conference, 1977.Google Scholar
  20. 20.
    Morse, P. M. and Feshbach, H.,Methods of Theoretical Physics, McGraw-Hill, New York, 1953.Google Scholar
  21. 21.
    Whitham, G. B.,Linear and Nonlinear Waves, Wiley, New York, 1974, p. 585.Google Scholar
  22. 22.
    Whitham, G. B., Notes from the course ‘Special Topics in Nonlinear Wave Propagation’, California Institute of Technology, Pasadena CA, 1988.Google Scholar
  23. 23.
    Weinstein, A., Connections of Berry and Hannay type for moving Lagrangian submanifolds,Adv. in Math. 82, 133–159 (1990).Google Scholar
  24. 24.
    Wadati, M., Ichikawa, Y. H., and Shimizu, T., Cusp soliton of a new integrable nonlinear evolution equation,Prog. Theoret. Phys. 64, 1959–1967 (1980).Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Mark S. Alber
    • 1
  • Roberto Camassa
    • 2
  • Darryl D. Holm
    • 2
  • Jerrold E. Marsden
    • 3
  1. 1.School of Mathematics, Institute for Advanced Study, Princeton and Department of MathematicsUniversity of Notre DameNotre DameUSA
  2. 2.Center for Nonlinear Studies and Theoretical DivisionLos Alamos National LaboratoryLos AlamosUSA
  3. 3.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations