167 research outputs found
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
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
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
Commutator identities on associative algebras and integrability of nonlinear pde's
It is shown that commutator identities on associative algebras generate
solutions of linearized integrable equations. Next, a special kind of the
dressing procedure is suggested that in a special class of integral operators
enables to associate to such commutator identity both nonlinear equation and
its Lax pair. Thus problem of construction of new integrable pde's reduces to
construction of commutator identities on associative algebras.Comment: 12 page
Solutions of the Kpi Equation with Smooth Initial Data
The solution of the Kadomtsev--Petviashvili I (KPI) equation with
given initial data belonging to the Schwartz space is considered. No
additional special constraints, usually considered in literature, as
are required to be satisfied by the initial data. The
problem is completely solved in the framework of the spectral transform theory
and it is shown that satisfies a special evolution version of the
KPI equation and that, in general, has different left and
right limits at the initial time . The conditions of the type
, and so on (first, second,
etc. `constraints') are dynamically generated by the evolution equation for
. On the other side with prescribed
order of integrations is not necessarily equal to zero and gives a nontrivial
integral of motion.Comment: 17 pages, 23 June 1993, LaTex fil
A nonlinear transformation of the dispersive long wave equations in (2+1) dimensions and its applications
A nonlinear transformation of the dispersive long wave equations in (2+1)
dimensions is derived by using the homogeneous balance method. With the aid of
the transformation given here, exact solutions of the equations are obtained
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
Ablowitz-Ladik system with discrete potential. I. Extended resolvent
Ablowitz-Ladik linear system with range of potential equal to {0,1} is
considered. The extended resolvent operator of this system is constructed and
the singularities of this operator are analyzed in detail.Comment: To be published in Theor. Math. Phy
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
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