The Geometry of Peaked Solitons and Billiard Solutions of a Class of Integrable PDE's
Alber, M. S., R. Camassa, D. D. Holm and J. E. Marsden
Lett. Math. Phys., 32, 137-151
Abstract:
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 [1993] for a shallow water equation. We put this equation in the
framework of complex integrable Hamiltonian systems on Riemann surfaces and using special limiting
procedures, draw some consequences from this setting. Among these consequences, one obtains new
solutions such as quasiperiodic solutions, n-solitons, solitons with quasiperiodic background,
billiard, and n-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 on N-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.