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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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.
|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.|
|Classification Code:||PACS: 05.45.Yv; 03.40.Gc; 11.10.Ef; 68.10.-m; AMS Subj. Class: 58F07, 70H99, 76B15|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Ruth Sustaita|
|Deposited On:||06 Jul 2010 15:24|
|Last Modified:||26 Dec 2012 12:11|
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