Inverse Spectral Problem for the Periodic Camassa-Holm Equation

Korotyaev, Evgeni

doi:10.2991/jnmp.2004.11.4.6

Cited by 7 publications

(6 citation statements)

References 34 publications

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“…When mO0 is sufficiently regular, the Liouville transformation can be used to convert the CH spectral operator This transformation has been employed by Constantin (2001), Lenells (2002Lenells ( , 2004, Constantin & Lenells (2003a,b), Johnson (2003) and McKean (2003). The Liouville transformation also works in the case of the periodic Camassa-Holm equation (see Constantin 1998;Constantin & McKean 1999;Korotyaev 2004). McKean considers potentials mO0 for which a G ZGN, whereas Constantin considers a modified Camassa-Holm equation, namely…”

Section: Factorization and Inverse Problemsmentioning

confidence: 99%

The string density problem and the Camassa–Holm equation

Beals

Sattinger

Szmigielski

2007

Phil. Trans. R. Soc. A.

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Recently, the string density problem, considered in the pioneering work of M. G. Kreĭn, has arisen naturally in connection with the Camassa-Holm equation for shallow water waves. In this paper we review the forward and inverse string density problems, with some numerical examples, and relate it to the Camassa-Holm equation, with special reference to multi-peakon/anti-peakon solutions. Under stronger assumptions, the Camassa-Holm spectral problem and the string density problem can be transformed to the Schrödinger spectral problem and its inverse problem. Recent results exploiting this transformation are reviewed briefly.

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Section: Factorization and Inverse Problemsmentioning

confidence: 99%

The string density problem and the Camassa–Holm equation

Beals

Sattinger

Szmigielski

2007

Phil. Trans. R. Soc. A.

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“…More precisely, a fairly complete spectral picture of the problem −f ′′ + 1 4 f = z ωf (1.5) has been given in [9,10,13] when ω is at least a continuous function. Under the additional assumption that ω is sufficiently smooth and positive, the inverse spectral problem for (1.5) has been considered in [3,11,30]. Somewhat related in this context, smooth finite gap solutions of the Camassa-Holm equation, respectively its two-component generalization, have been studied in [13,19,25].…”

Section: Introductionmentioning

confidence: 99%

Trace formulas and continuous dependence of spectra for the periodic conservative Camassa–Holm flow

Eckhardt

Kostenko

Nicolussi

2020

Journal of Differential Equations

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This article is concerned with the isospectral problemfor the periodic conservative Camassa-Holm flow, where ω is a periodic real distribution in H −1 loc (R) and υ is a periodic non-negative Borel measure on R. We develop basic Floquet theory for this problem, derive trace formulas for the associated spectra and establish continuous dependence of these spectra on the coefficients with respect to a weak * topology.

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“…It is remarkable that most of the essential properties of the Camassa-Holm equation are already observable for this class of solutions. Previous literature on periodic inverse spectral problems for (1.2) is very scarce and always restricted to the case when ω is a strictly positive continuous function; see [3,17,42]. Somewhat related, smooth finite gap solutions of the Camassa-Holm equation have been studied in [20,26,34].…”

Section: Introductionmentioning

confidence: 99%