760 research outputs found
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
Algebro-Geometric Constraints on Solitons with Respect to Quasi-Periodic Backgrounds
We investigate the algebraic conditions the scattering data of short-range
perturbations of quasi-periodic finite-gap Jacobi operators have to satisfy. As
our main result we provide the Poisson-Jensen-type formula for the transmission
coefficient in terms of Abelian integrals on the underlying hyperelliptic
Riemann surface and give an explicit condition for its single-valuedness. In
addition, we establish trace formulas which relate the scattering data to the
conserved quantities in this case.Comment: 9 pages. Bull. London Math. Soc. (to appear
Soliton Solutions of the Toda Hierarchy on Quasi-Periodic Backgrounds Revisited
We investigate soliton solutions of the Toda hierarchy on a quasi-periodic
finite-gap background by means of the double commutation method and the inverse
scattering transform. In particular, we compute the phase shift caused by a
soliton on a quasi-periodic finite-gap background. Furthermore, we consider
short range perturbations via scattering theory. We give a full description of
the effect of the double commutation method on the scattering data and
establish the inverse scattering transform in this setting.Comment: 16 page
Trace Formulas in Connection with Scattering Theory for Quasi-Periodic Background
We investigate trace formulas for Jacobi operators which are trace class
perturbations of quasi-periodic finite-gap operators using Krein's spectral
shift theory. In particular we establish the conserved quantities for the
solutions of the Toda hierarchy in this class.Comment: 7 page
Relative Oscillation Theory for Sturm-Liouville Operators Extended
We extend relative oscillation theory to the case of Sturm--Liouville
operators with different 's. We show that the
weighted number of zeros of Wronskians of certain solutions equals the value of
Krein's spectral shift function inside essential spectral gaps.Comment: 16 page
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
Reconstruction of the Transmission Coefficient for Steplike Finite-Gap Backgrounds
We consider scattering theory for one-dimensional Jacobi operators with
respect to steplike quasi-periodic finite-gap backgrounds and show how the
transmission coefficient can be reconstructed from minimal scattering data.
This generalizes the Poisson-Jensen formula for the classical constant
background case.Comment: 9 page
Stability of the Periodic Toda Lattice in the Soliton Region
We apply the method of nonlinear steepest descent to compute the long-time
asymptotics of the periodic (and slightly more generally of the quasi-periodic
finite-gap) Toda lattice for decaying initial data in the soliton region. In
addition, we show how to reduce the problem in the remaining region to the
known case without solitons.Comment: 28 page
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