PERIODIC PEAKONS AND CALOGERO–FRANÇOISE FLOWS

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 eq...

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Bibliographic Details
Published inJournal of the Institute of Mathematics of Jussieu Vol. 4; no. 1; pp. 1 - 27
Main Authors Beals, R., Sattinger, D. H., Szmigielski, J.
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.01.2005
Subjects
Weyl function
spectral problems
Abel map
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Summary: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
ISSN:1474-7480
1475-3030
DOI:10.1017/S1474748005000010