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

Alber, Mark S. and Camassa, Roberto and Holm, Darryl D. and Marsden, Jerrold E. (1994) The geometry of peaked solitons and billiard solutions of a class of integrable PDE's. Letters in Mathematical Physics, 32 (2). pp. 137-151. ISSN 0377-9017.

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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 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.

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Additional Information:© 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.
Funding AgencyGrant Number
Department of Energy (DOE)/Computer Hardware Advanced Mathematics and Model Physics (CHAMMP)UNSPECIFIED
High Performance Computing and Communications (HPCC)UNSPECIFIED
Department of EnergyUNSPECIFIED
Office of Naval ResearchUNSPECIFIED
Fields Institute for Research in the Mathematical SciencesUNSPECIFIED
Issue or Number:2
Classification Code:PACS: 03.40.Gc, 11.10.Ef, 68.10.-m; AMS Subject Classification: 58F07, 70H99, 76B15
Record Number:CaltechAUTHORS:20100706-142301021
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:18913
Deposited By: Ruth Sustaita
Deposited On:07 Jul 2010 15:31
Last Modified:03 Oct 2019 01:49

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