138 research outputs found

    The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data

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    We establish the inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data. In particular, it is employed to prove existence of weak solutions for the corresponding Cauchy problem.Comment: 27 page

    On the connection between the Hilger and Radon--Nikodym derivatives

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    We show that the Hilger derivative on time scales is a special case of the Radon--Nikodym derivative with respect to the natural measure associated with every time scale. Moreover, we show that the concept of delta absolute continuity agrees with the one from measure theory in this context.Comment: 7 page

    An isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation

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    We introduce a generalized isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation. Utilizing the solution of the indefinite moment problem given by M. G. Krein and H. Langer, we show that the conservative Camassa-Holm equation is integrable by the inverse spectral transform in the multi-peakon case.Comment: 25 page

    The Classical Moment Problem and Generalized Indefinite Strings

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    We show that the classical Hamburger moment problem can be included in the spectral theory of generalized indefinite strings. Namely, we introduce the class of Krein-Langer strings and show that there is a bijective correspondence between moment sequences and this class of generalized indefinite strings. This result can be viewed as a complement to the classical results of M. G. Krein on the connection between the Stieltjes moment problem and Krein-Stieltjes strings and I. S. Kac on the connection between the Hamburger moment problem and 2x2 canonical systems with Hamburger Hamiltonians.Comment: 25 page

    The inverse spectral problem for indefinite strings

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    Motivated by the study of certain nonlinear wave equations (in particular, the Camassa-Holm equation), we introduce a new class of generalized indefinite strings associated with differential equations of the form βˆ’u"=z u ω+z2u υ-u"=z\,u\,\omega+z^2u\,\upsilon on an interval [0,L)[0,L), where Ο‰\omega is a real-valued distribution in Hlocβˆ’1[0,L)H^{-1}_{\mathrm{loc}}[0,L), Ο…\upsilon is a non-negative Borel measure on [0,L)[0,L) and zz is a complex spectral parameter. Apart from developing basic spectral theory for these kinds of spectral problems, our main result is an indefinite analogue of M. G. Krein's celebrated solution of the inverse spectral problem for inhomogeneous vibrating strings.Comment: 27 page

    The inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation

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    We solve the inverse spectral problem associated with periodic conservative multi-peakon solutions of the Camassa-Holm equation. The corresponding isospectral sets can be identified with finite dimensional tori.Comment: 18 pages, 1 figur

    A coupling problem for entire functions and its application to the long-time asymptotics of integrable wave equations

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    We propose a novel technique for analyzing the long-time asymptotics of integrable wave equations in the case when the underlying isospectral problem has purely discrete spectrum. To this end, we introduce a natural coupling problem for entire functions, which serves as a replacement for the usual Riemann-Hilbert problem, which does not apply in these cases. As a prototypical example, we investigate the long-time asymptotics of the dispersionless Camassa-Holm equation.Comment: 11 page
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