153 research outputs found
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
Properties of the solitonic potentials of the heat operator
Properties of the pure solitonic -function and potential of the heat
equation are studied in detail. We describe the asymptotic behavior of the
potential and identify the ray structure of this asymptotic behavior on the
-plane in dependence on the parameters of the potential
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
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
Nonlinear Discrete Systems with Nonanalytic Dispersion Relations
A discrete system of coupled waves (with nonanalytic dispersion relation) is
derived in the context of the spectral transform theory for the Ablowitz Ladik
spectral problem (discrete version of the Zakharov-Shabat system). This 3-wave
evolution problem is a discrete version of the stimulated Raman scattering
equations, and it is shown to be solvable for arbitrary boundary value of the
two radiation fields and initial value of the medium state. The spectral
transform is constructed on the basis of the D-bar approach.Comment: RevTex file, to appear in Journ. Math. Phy
Heat operator with pure soliton potential: properties of Jost and dual Jost solutions
Properties of Jost and dual Jost solutions of the heat equation,
and , in the case of a pure solitonic potential are studied in
detail. We describe their analytical properties on the spectral parameter
and their asymptotic behavior on the -plane and we show that the values of
and the residua of at special discrete
values of are bounded functions of in a polygonal region of the
-plane. Correspondingly, we deduce that the extended version of the
heat operator with a pure solitonic potential has left and right annihilators
for belonging to these polygonal regions.Comment: 26 pages, 3 figure
- …