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On Billiard Solutions of Nonlinear PDE's

Alber, Mark S. and Camassa, Roberto and Fedorov, Yuri N. and Holm, Darryl D. and Marsden, Jerrold E. (1999) On Billiard Solutions of Nonlinear PDE's. Physics Letters A, 264 (2-3). pp. 171-178. ISSN 0375-9601. http://resolver.caltech.edu/CaltechAUTHORS:20100701-140021943

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Abstract

This Letter presents some special features of a class of integrable PDEs admitting billiard-type solutions, which set them apart from equations whose solutions are smooth, such as the KdV equation. These billiard solutions are weak solutions that are piecewise smooth and have first derivative discontinuities at peaks in their profiles. A connection is established between the peak locations and finite dimensional billiard systems moving inside n-dimensional quadrics under the field of Hooke potentials. Points of reflection are described in terms of theta-functions and are shown to correspond to the location of peak discontinuities in the PDEs weak solutions. The dynamics of the peaks is described in the context of the algebraic-geometric approach to integrable systems.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/S0375-9601(99)00784-7DOIUNSPECIFIED
Additional Information:© 1999 Published by Elsevier Science B.V. Available online 7 January 2000. Communicated by A.R. Bishop. Research partially supported by NSF grant DMS 9626672 and NATO grant CRG 950897. Research supported in part by US DOE CHAMMP and HPCC programs and NATO grant CRG 950897. Research supported in part by the Center for Applied Mathematics, University of Notre Dame. Research supported in part by US DOE CHAMMP and HPCC programs. Research partially supported by Caltech and NSF grant DMS 9802106.
Funders:
Funding AgencyGrant Number
NSFDMS 9626672
NATOCRG 950897
Department of Energy/Computer Hardware, Advanced Mathematics, and Model Physics (CHAMMP) and HPCCUNSPECIFIED
NATOCRG 950897
Center for Applied Mathematics, University of Notre DameUNSPECIFIED
Department of Energy/Computer Hardware, Advanced Mathematics, and Model Physics (CHAMMP) and HPCCUNSPECIFIED
CaltechUNSPECIFIED
NSFDMS 9802106
Classification Code:PACS: 05.45.Yv; 03.40.Gc; 11.10.Ef; 68.10.-m; AMS Subj. Class: 58F07, 70H99, 76B15
Record Number:CaltechAUTHORS:20100701-140021943
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100701-140021943
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:18900
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:06 Jul 2010 15:24
Last Modified:26 Dec 2012 12:11

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