147 research outputs found
The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data
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
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
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
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
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
on an interval , where is a
real-valued distribution in , is a
non-negative Borel measure on and 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
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
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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