On the isospectral problem of the dispersionless Camassa–Holm equation

We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa–Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral proble...

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Bibliographic Details
Published in:	Advances in mathematics (New York. 1965) Vol. 235; pp. 469 - 495
Main Authors:	Eckhardt, Jonathan, Teschl, Gerald
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 01-03-2013
Subjects:	Camassa–Holm equation Inverse spectral theory Isospectral problem secondary Inverse spectral theory Camassa–Holm equation Isospectral problem primary
Online Access:	Get full text
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Summary:	We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa–Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa–Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.
ISSN:	0001-8708 1090-2082
DOI:	10.1016/j.aim.2012.12.006