Letters in Mathematical Physics
, Volume 32, Issue 2, pp 137151
First online:
The geometry of peaked solitons and billiard solutions of a class of integrable PDE's
 Mark S. AlberAffiliated withSchool of Mathematics, Institute for Advanced Study, Princeton and Department of Mathematics, University of Notre Dame
 , Roberto CamassaAffiliated withCenter for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory
 , Darryl D. HolmAffiliated withCenter for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory
 , Jerrold E. MarsdenAffiliated withDepartment of Mathematics, University of California
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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 [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,nsolitons, solitons with quasiperiodic background, billiard, andnpeakon 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 onNdimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.
Mathematics Subject Classifications (1991)
58F07 70H99 76B15 Title
 The geometry of peaked solitons and billiard solutions of a class of integrable PDE's
 Journal

Letters in Mathematical Physics
Volume 32, Issue 2 , pp 137151
 Cover Date
 199410
 DOI
 10.1007/BF00739423
 Print ISSN
 03779017
 Online ISSN
 15730530
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 58F07
 70H99
 76B15
 Authors

 Mark S. Alber ^{(1)}
 Roberto Camassa ^{(2)}
 Darryl D. Holm ^{(2)}
 Jerrold E. Marsden ^{(3)}
 Author Affiliations

 1. School of Mathematics, Institute for Advanced Study, Princeton and Department of Mathematics, University of Notre Dame, 46556, Notre Dame, IN, USA
 2. Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, 87545, Los Alamos, NM, USA
 3. Department of Mathematics, University of California, 94720, Berkeley, CA, USA