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PERIODIC PEAKONS AND CALOGERO–FRANÇOISE FLOWS

Published online by Cambridge University Press:  10 February 2005

R. Beals ,
D. H. Sattinger  and
J. Szmigielski
R. Beals
Affiliation:
Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06520-8283, USA
D. H. Sattinger
Affiliation:
Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06520-8283, USA
J. Szmigielski
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan S7N 5E6, Canada
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Abstract

It has long been known that a number of periodic completely integrable systems are associated to hyperelliptic curves, for which the Abel map linearizes the flow (at least in part). We show that this is true for a relatively recent such system: the periodic discrete reduction of the shallow water equation derived by Camassa and Holm. The associated spectral problem has the same form and evolves in the same way as the spectral problem for a family of finite-dimensional non-periodic Hamiltonian flows introduced by Calogero and Françoise. We adapt the Weyl function method used earlier by us to solve the peakon problem to give an explicit solution to both the periodic discrete Camassa–Holm system and the (non-periodic) Calogero–Françoise system in terms of theta functions.

AMS 2000 Mathematics subject classification: Primary 35Q51; 37J35; 35Q53

Keywords

spectral problemsWeyl functionAbel map
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 4 , Issue 1 , January 2005 , pp. 1 - 27
Copyright
2005 Cambridge University Press

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Footnotes

Please note that the DOIs in the printed edition of volume 4 issue 1 contained errors. Correct DOIs for each article in this issue are published here online. We apologise for any inconvenience caused