Letters in Mathematical Physics

, Volume 32, Issue 2, pp 137–151

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


  • Mark S. Alber
    • School of Mathematics, Institute for Advanced Study, Princeton and Department of MathematicsUniversity of Notre Dame
  • Roberto Camassa
    • Center for Nonlinear Studies and Theoretical DivisionLos Alamos National Laboratory
  • Darryl D. Holm
    • Center for Nonlinear Studies and Theoretical DivisionLos Alamos National Laboratory
  • Jerrold E. Marsden
    • Department of MathematicsUniversity of California

DOI: 10.1007/BF00739423

Cite this article as:
Alber, M.S., Camassa, R., Holm, D.D. et al. Lett Math Phys (1994) 32: 137. doi:10.1007/BF00739423


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)

Download to read the full article text

Copyright information

© Kluwer Academic Publishers 1994