455 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 $N$ 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 $N$ 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

### 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

### 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
$N$ 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

### 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

### Multidimensional Localized Solitons

Recently it has been discovered that some nonlinear evolution equations in
2+1 dimensions, which are integrable by the use of the Spectral Transform,
admit localized (in the space) soliton solutions. This article briefly reviews
some of the main results obtained in the last five years thanks to the renewed
interest in soliton theory due to this discovery. The theoretical tools needed
to understand the unexpected richness of behaviour of multidimensional
localized solitons during their mutual scattering are furnished. Analogies and
especially discrepancies with the unidimensional case are stressed

### Solutions of the Kpi Equation with Smooth Initial Data

The solution $u(t,x,y)$ of the Kadomtsev--Petviashvili I (KPI) equation with
given initial data $u(0,x,y)$ belonging to the Schwartz space is considered. No
additional special constraints, usually considered in literature, as
$\int\!dx\,u(0,x,y)=0$ 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 $u(t,x,y)$ satisfies a special evolution version of the
KPI equation and that, in general, $\partial_t u(t,x,y)$ has different left and
right limits at the initial time $t=0$. The conditions of the type
$\int\!dx\,u(t,x,y)=0$, $\int\!dx\,xu_y(t,x,y)=0$ and so on (first, second,
etc. `constraints') are dynamically generated by the evolution equation for
$t\not=0$. On the other side $\int\!dx\!\!\int\!dy\,u(t,x,y)$ 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

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