89 research outputs found
B\"{a}cklund and Darboux transformations for the nonstationary Schr\"{o}dinger equation
Potentials of the nonstationary Schr\"{o}dinger operator constructed by means
of recursive B\"{a}cklund transformations are studied in detail.
Corresponding Darboux transformations of the Jost solutions are introduced. We
show that these solutions obey modified integral equations and present their
analyticity properties. Generated transformations of the spectral data are
derived.Comment: to be published in Proc. of the Steklov Inst. of Mathematics, Moscow,
Russi
Towards spectral theory of the Nonstationary Schr\"{o}dinger equation with a two-dimensionally perturbed one-dimensional potential
The Nonstationary Schr\"{o}dinger equation with potential being a
perturbation of a generic one-dimensional potential by means of a decaying
two-dimensional function is considered here in the framework of the extended
resolvent approach. The properties of the Jost solutions and spectral data are
investigated.Comment: 22 pages, no figure
A shape optimization problem on planar sets with prescribed topology
We consider shape optimization problems involving functionals depending on
perimeter, torsional rigidity and Lebesgue measure. The scaling free cost
functionals are of the form and the
class of admissible domains consists of two-dimensional open sets
satisfying the topological constraints of having a prescribed number of
bounded connected components of the complementary set. A relaxed procedure is
needed to have a well-posed problem and we show that when an optimal
relaxed domain exists. When the problem is ill-posed and for
the explicit value of the infimum is provided in the cases and
Integrable discretizations of the sine-Gordon equation
The inverse scattering theory for the sine-Gordon equation discretized in
space and both in space and time is considered.Comment: 18 pages, LaTeX2
Building extended resolvent of heat operator via twisting transformations
Twisting transformations for the heat operator are introduced. They are used,
at the same time, to superimpose a` la Darboux N solitons to a generic smooth,
decaying at infinity, potential and to generate the corresponding Jost
solutions. These twisting operators are also used to study the existence of the
related extended resolvent. Existence and uniqueness of the extended resolvent
in the case of solitons with N "ingoing" rays and one "outgoing" ray is
studied in details.Comment: 15 pages, 2 figure
Towards an Inverse Scattering theory for non decaying potentials of the heat equation
The resolvent approach is applied to the spectral analysis of the heat
equation with non decaying potentials. The special case of potentials with
spectral data obtained by a rational similarity transformation of the spectral
data of a generic decaying potential is considered. It is shown that these
potentials describe solitons superimposed by Backlund transformations to a
generic background. Dressing operators and Jost solutions are constructed by
solving a DBAR-problem explicitly in terms of the corresponding objects
associated to the original potential. Regularity conditions of the potential in
the cases N=1 and N=2 are investigated in details. The singularities of the
resolvent for the case N=1 are studied, opening the way to a correct definition
of the spectral data for a generically perturbed soliton.Comment: 22 pages, submitted to Inverse Problem
An integrable discretization of KdV at large times
An "exact discretization" of the Schroedinger operator is considered and its
direct and inverse scattering problems are solved. It is shown that a
differential-difference nonlinear evolution equation depending on two arbitrary
constants can be solved by using this spectral transform and that for a special
choice of the constants it can be considered an integrable discretization of
the KdV equation at large times. An integrable difference-difference equation
is also obtained.Comment: 12 page
On the equivalence of different approaches for generating multisoliton solutions of the KPII equation
The unexpectedly rich structure of the multisoliton solutions of the KPII
equation has been explored by using different approaches, running from dressing
method to twisting transformations and to the tau-function formulation. All
these approaches proved to be useful in order to display different properties
of these solutions and their related Jost solutions. The aim of this paper is
to establish the explicit formulae relating all these approaches. In addition
some hidden invariance properties of these multisoliton solutions are
discussed
A discrete Schrodinger spectral problem and associated evolution equations
A recently proposed discrete version of the Schrodinger spectral problem is
considered. The whole hierarchy of differential-difference nonlinear evolution
equations associated to this spectral problem is derived. It is shown that a
discrete version of the KdV, sine-Gordon and Liouville equations are included
and that the so called `inverse' class in the hierarchy is local. The whole
class of related Darboux and Backlund transformations is also exhibited.Comment: 14 pages, LaTeX2
On the extended resolvent of the Nonstationary Schrodingher operator for a Darboux transformed potential
In the framework of the resolvent approach it is introduced a so called
twisting operator that is able, at the same time, to superimpose \`a la Darboux
solitons to a generic smooth decaying potential of the Nonstationary
Schr\"odinger operator and to generate the corresponding Jost solutions. This
twisting operator is also used to construct an explicit bilinear representation
in terms of the Jost solutions of the related extended resolvent. The main
properties of the Jost and auxiliary Jost solutions and of the resolvent are
discussed.Comment: 24 pages, class files from IO
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