A. K. Pogrebkov

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 -problem explicitly in terms of the corresponding objects associated with the original potential. Regularity conditions of the potential in the cases N = 1 and 2 are investigated in detail. 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.

Properties of solutions of the Kadomtsev–Petviashvili I equation

The Kadomtsev–Petviashvili I (KPI) equation is considered as a useful laboratory for experimenting with new theoretical tools able to handle the specific features of integrable models in 2+1 dimensions. The linearized version of the KPI equation is first considered by solving the initial value problem for different classes of initial data. Properties of the solutions in different cases are analyzed in details. The obtained results are used as a guideline for studying the properties of 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. The spectral theory associated to KPI is studied in the space of the Fourier transform of the solutions. The variables p={p1,p2} of the Fourier space are shown to be the most convenient spectral variables to use for spectral data. Spectral data are shown to decay rapidly at large p but to be discontinuous at p=0. Direct and inverse problems are solved with special attention to the behavior of...

Resolvent approach for the nonstationary Schrodinger equation

The spectral transform for the nonstationary Schrodinger equation is considered. The resolvent operator of the Schrodinger equation is introduced and the Fourier transform of its kernel (called the resolvent function) is studied. It is shown that it can be used to construct a generalized version of the theory of the spectral transform which enables one to handle also potentials approaching zero in every direction except a finite number, which corresponds to the physical situation of long waves mutually interacting in the plane.

Inverse scattering theory of the heat equation for a perturbed one-soliton potential

The inverse scattering theory of the heat equation is developed for a special subclass of potentials nondecaying at space infinity—perturbations of the one-soliton potential by means of decaying two-dimensional functions. Extended resolvent, Green’s functions, and Jost solutions are introduced and their properties are investigated in detail. The singularity structure of the spectral data is given and then the inverse problem is formulated in an exact distributional sense.

Extended resolvent and inverse scattering with an application to KPI

We present in detail an extended resolvent approach for investigating linear problems associated to 2+1 dimensional integrable equations. Our presentation is based as an example on the nonstationary Schrodinger equation with potential being a perturbation of the one-soliton potential by means of a decaying two-dimensional function. Modification of the inverse scattering theory as well as properties of the Jost solutions and spectral data as follows from the resolvent approach are given.

Solving the Kadomtsev - Petviashvili equation with initial data not vanishing at large distances

We consider, in the framework of the inverse scattering method, the solution of the Kadomtsev - Petviashvili equation in its version called KPI. The spectral theory is extended to the case in which the initial data are not vanishing along a finite number of directions at large distances on the plane.

New features of Backlund and Darboux transformations in 2 + 1 dimensions

Backlund transformations of the time-dependent Schrodinger equation which transform a real potential into another real potential are constructed, as well as their Darboux versions. The iterated application of these Backlund transformations to a generic potential is considered and the obtained recursion relations are explicitly solved. It is shown that the dressing of the generic potential can be obtained by taking the continuous limit of this infinite sequence of Backlund transformations, and that the delta -bar integral equations, which solve the inverse spectral problem, can be obtained as the continuous limit of the recurrence equation defining the sequence of the Darboux transformations.

Heat operator with pure soliton potential: Properties of Jost and dual Jost solutions

Properties of Jost and dual Jost solutions of the heat equation, Φ(x, k) and Ψ(x, k), in the case of a pure solitonic potential are studied in detail. We describe their analytical properties on the spectral parameter k and their asymptotic behavior on the x-plane and we show that the values of e−qxΦ(x, k) and the residues of eqxΨ(x, k) at special discrete values of k are bounded functions of x in a polygonal region of the q-plane. Correspondingly, we deduce that the extended version L(q) of the heat operator with a pure solitonic potential has left and right annihilators for q belonging to these polygonal regions.

On the extended resolvent of the nonstationary Schrödinger operator for a Darboux transformed potential

In the framework of the resolvent approach, a so-called twisting operator is introduced that is able, at the same time, to superimpose a la Darboux N solitons to a generic smooth decaying potential of the nonstationary Schrodinger 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.

Inverse scattering transform for the perturbed 1-soliton potential of the heat equation ☆

Abstract Inverse scattering theory of the heat equation with potential being a perturbation of the one-soliton potential by means of a decaying two-dimensional function is presented. Modification of the Jost solutions and scattering data are given.