DMCA
The Geometry of Peaked Solitons and Billiard Solutions of a Class of Integrable PDE’s ∗
BibTeX
@MISC{Holm_thegeometry,
author = {Darryl D. Holm and Jerrold E. Marsden},
title = {The Geometry of Peaked Solitons and Billiard Solutions of a Class of Integrable PDE’s ∗},
year = {}
}
Share
 |
 |
 |
 |
||
OpenURL
Abstract
The purpose of this letter is to investigate the geometry of new classes of solitonlike 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. 1
Keyphrases
peaked soliton integrable pde billiard solution shallow water equation quasiperiodic solution explicit formula n-dimensional quadric riemann surface dym hierarchy complex integrable hamiltonian system corresponding angle representation new solution new class asymptotic reduction soliton solution quasiperiodic background geodesic flow special limiting procedure solitonlike solution integrable nonlinear equation phase shift n-peakon solution complex angle representation forthcoming paper
