The geometry of peaked solitons and billiard solutions of a class of integrable PDE's
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  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.
© 1994 Springer Verlag. Received: 17 February 1994. Research supported in part by DOE CHAMMP and HPCC programs. Research partially supported by the Department of Energy, the Office of Naval Research and the Fields Institute for Research in the Mathematical Sciences. We thank Craig Evans for some useful comments about weak solutions. Mark Alber also thanks the Institute for Advanced Study in Princeton and the Center for Nonlinear Studies at Los Alamos National Laboratory for their hospitality during the Fall of 1993 and during two visits in August 1993 and January 1994.