# A. K. PogrebkovRussian Academy of Sciences | RAS · Steklov Mathematical Institute

A. K. Pogrebkov

Professor

## About

146

Publications

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1,124

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Introduction

Additional affiliations

September 2012 - present

January 1973 - present

## Publications

Publications (146)

We use example of the Davey-Stewartson hierarchy to show that in addition to the standard equations given by Lax operator and evolutions of times with positive numbers, one can consider time evolutions with negative numbers and the same Lax operator. We derive corresponding Lax pairs and integrable equations.

The Kadomtsev–Petviashvili equation is known to be the leading term of a semi-infinite hierarchy of integrable equations with evolutions given by times with positive numbers. Here, we introduce new hierarchy directed to negative numbers of times. The derivation of such systems, as well as the corresponding hierarchy, is based on the commutator iden...

We use example of the Davey--Stewartson hierarchy to show that in addition to the standard equations given by Lax operator and evolutions of times with positive numbers, one can consider time evolutions with negative numbers and the same Lax operator. We derive corresponding Lax pairs and integrable equations.

Подход, основанный на коммутаторных тождествах элементов ассоциативных алгебр, был ранее эффективно использован при исследовании ($2+1$)-мерных интегрируемых систем. Этот подход развит для исследования интегрируемых иерархий и их связей.

Пример индуцированной динамики реализован посредством новых мультипликативных детерминантных соотношений, корни которых задают положения частиц. Приведены как общая схема описания вполне интегрируемой динамической системы, параметризованной произвольной $(N\times{N})$-матрицей импульсов, так и явная модель, интерполирующая между гиперболическими си...

Construction of new integrable systems and methods of their investigation is one of the main directions of development of the modern mathematical physics. Here we present an approach based on the study of behavior of roots of functions of canonical variables with respect to a parameter of simultaneous shift of space variables. Dynamics of singulari...

We considered the relation between two famous integrable equations: The Hirota difference equation (HDE) and the Darboux system that describes conjugate curvilinear systems of coordinates in R 3. We demonstrated that specific properties of solutions of the HDE with respect to independent variables enabled introduction of an infinite set of discrete...

Induced dynamics is defined as dynamics of real zeros with respect to $x$ of equation $f(q_1-x,\ldots,q_N-x,p_1,\ldots,p_N)=0$, where $f$ is a function, and $q_i$ and $p_j$ are canonical variables obeying some (free) evolution. Identifying zero level lines with the world lines of particles, we show that the resulting dynamical system demonstrates h...

We discuss the problems of the connections of the modern theory of integrability and the corresponding overdetermined linear systems with works of geometers of the late nineteenth century. One of these questions is the generalization of the theory of Darboux–Laplace transforms for second-order equations with two independent variables to the case of...

We previously proposed an approach for constructing integrable equations based on the dynamics in associative algebras given by commutator relations. In the framework of this approach, evolution equations determined by commutators of (or similarity transformations with) functions of the same operator are compatible by construction. Linear equations...

For a Darboux system in ℝ³, we introduce a class of solutions for which an auxiliary second-order linear problem satisfies the factorization condition. We show that this reduction provides the (local) solvability of the Darboux system, and present an explicit solution to this problem for two types of dependent variables. We also construct explicit...

Continuous symmetries of the Hirota difference equation, commuting with shifts of independent variables, are derived by means of the dressing procedure. Action of these symmetries on the dependent variables of the equation is presented. Commutativity of these symmetries enables interpretation of their parameters as “times” of the nonlinear integrab...

Direct definition of the Cauchy-Jost (known also as Cauchy-Baker-Akhiezer) function in the case of pure solitonic solution is given and properties of this function are discussed in detail using the Kadomtsev-Petviashvili II equation as example. This enables formulation of the Darboux transformations in terms of the Cauchy-Jost function and classifi...

We show that the non-Abelian Hirota difference equation is directly related to a commutator identity on an associative algebra. Evolutions generated by similarity transformations of elements of this algebra lead to a linear difference equation. We develop a special dressing procedure that results in an integrable non-Abelian Hirota difference equat...

We consider the direct and inverse problems for the Hirota difference equation. We introduce the Jost solutions and scattering data and describe their properties. In a special case, we show that the Darboux transformation allows finding the evolution in discrete time and obtaining a recursive procedure for sequentially constructing the Jost solutio...

Properties of the Cauchy--Jost (known also as Cauchy--Baker--Akhiezer)
function of the KPII equation are described. By means of the
$\bar\partial$-problem for this function it is shown that all equations of the
KPII hierarchy are given in a compact and explicit form, including equations on
the Cauchy--Jost function itself, time evolutions of the Jo...

Direct and inverse problems for the Hirota difference equation are
considered. Jost solutions and scattering data are introduced and their
properties are presented. Darboux transformation in a special case is shown to
give evolution with respect to discrete time and a recursion procedure for
consequent construction of the Jost solution at arbitrary...

The Direct and the Inverse Scattering Problems for the heat operator with a
potential being a perturbation of an arbitrary $N$ soliton potential are
formulated. We introduce Jost solutions and spectral data and present their
properties. Then, giving the time evolution of the spectral data, the initial
value problem of the Kadomtsev-Petviashvili II...

The heat operator with a general multisoliton potential is considered and its
extended resolvent, depending on a parameter $q\in\R^2$ is derived. Its
boundedness properties in all variables and its discontinuities in the
parameter $q$ are given. As the result, the Green's functions are introduced
and their properties are studied in detail.

The heat operator with a pure soliton potential is considered and its Green's
function, depending on a complex spectral parameter k, is derived. Its
boundedness properties in all variables and its singularities in the spectral
parameter k are studied. A generalization of the Green's function, the extended
resolvent, is also given.

Properties of Jost and dual Jost solutions of the heat equation, $\Phi(x,k)$
and $\Psi(x,k)$, 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
$e^{-qx}\Phi(x,k)$ and the residua of $e^{q...

Properties of the pure solitonic $\tau$-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
$x$-plane in dependence on the parameters of the potential.

The unexpectedly rich structure of the multisoliton solutions of the KPII equation has previously been explored using different
approaches ranging from the dressing method to twisting transformations and the τ-function formulation. All these approaches
proved useful for displaying different properties of these solutions and the corresponding Jost s...

In earlier papers of the author it was shown that some simple commutator identities on an associative algebra generate integrable nonlinear equations. Here, this observation is generalized to the case of difference nonlinear equations. The identity under study leads, under a special realization of the elements of the associative algebra, to the fam...

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...

We show that commutator identities on associative algebras generate solutions of the linearized versions of integrable equations.
In addition, we introduce a special dressing procedure in a class of integral operators that allows deriving both the nonlinear
integrable equation itself and its Lax pair from such a commutator identity. The problem of...

Developing observation made in \cite{commut} we show that simple identity of the commutator type on an associative algebra is in one-to-one correspondence to 2D (infinite) Toda chain. We introduce representation of elements of associative algebra that, under some generic conditions, enables derivation of the Toda chain equation and its Lax pair fro...

This paper is devoted to a detailed description of the notion of boson-fermion correspondence introduced by Coleman and Mandelstam and to applications of this correspondence to integrable and related models. An explicit formulation of this correspondence in terms of massless fermionic fields is given, and properties of the resulting scalar field ar...

Realizing bosonic field v(x) as current of massless (chiral) fermions we derive hierarchy of quantum polynomial interactions
of the field v(x) that are completely integrable and lead to linear evolutions for the fermionic field. It is proved that
in the classical limit this hierarchy reduces to the dispersionless KdV hierarchy. Application of our c...

In the framework of the extended resolvent approach the direct and inverse scattering problems for the nonstationary Schrödinger equation with a potential being a perturbation of the N-soliton potential by means of a generic bidimensional smooth function decaying at large spaces are introduced and investigated. The initial value problem of the Kado...

The singular solutions of these equations are intimately related to the solutions. Constructing the singular solutions we introduce explicitly the characteristics of these objects - coordinates and velocities - as parameters of singular initial data. The construction allows a detailed description of the solutions: in fact equations of motion for si...

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...

We consider the nonstationary Schrodinger equation with the potential being a perturbation of a generic one-dimensional potential
by means of a decaying two-dimensional function in the framework of the extended resolvent approach. We give the corresponding
modification of the Jost and advanced/retarded solutions and spectral data and present relati...

We derive and describe in detail the extension of the inverse scattering transform method to the case of linear spectral problems with potentials that do not decay in some space directions. Our presentation is based on the extended resolvent approach. As a basic example, we consider the nonstationary Schrödinger equation with a potential that is a...

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 Schrödinger equation with potential being a perturbation of the one-soliton potential by means of a decaying two-dimensional function. Modification of...

Reconsidering the IST method for the KPII equation, we prove that
condition of existence of the generator of integrals of motion leads to
nonlinear equation on the spectral data of the associated linear problem
(the heat equation). This equation is preserved under time evolution and
leads to a specific behavior of the spectral data on the complex p...

We study the initial value problem of the Kadomtsev–Petviashvili I (KPI) equation with initial data u(x1,x2,0) = u1(x1)+u2(x1,x2), where u1(x1) is the one-soliton solution of the Korteweg–de Vries equation evaluated at zero time and u2(x1,x2) decays sufficiently rapidly on the (x1,x2)-plane. This involves the analysis of the nonstationary Schröding...

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 si...

We consider the quantization procedure for the Gardner–Zakharov–Faddeev and Magri brackets using the fermionic representation for the KdV field. In both cases, the corresponding Hamiltonians are sums of two well-defined operators. Each operator is bilinear and diagonal with respect to either fermion or boson (current) creation/annihilation operator...

Quantization procedure of the Gardner-Zakharov-Faddeev and Magri brackets by means of the fermionic representation for the KdV field is considered. It is shown that in both cases the corresponding Hamiltonians are given as sums of two well defined operators. Each of them is bilinear and diagonal with respect to either fermion, or boson (current) cr...

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.

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 B...

Potentials of the heat conduction operator constructed by means of two binary Bäcklund 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.

We consider the class of nondecaying potentials obtained by superimposing, via binary Bäcklund transformations, N solitons to an arbitrary decaying background. We introduce an exact recursion procedure for an arbitrary number of binary Bäcklund transformations and corresponding Darboux transformations for Jost solutions and solutions of the discret...

Potentials of the nonstationary Schr\"{o}dinger operator constructed by means of $n$ 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 transformatio...

The set of integrable symmetries of the nonstationary Schr\"{o}dinger equation is shown to admit a natural decomposition into subsets of mutually commuting symmetries. Hierarchies of time evolutions associated with each of these subsets ultimately lead to nonlinear (possibly, operator) equations of the Kadomtsev--Petviashvili I type or its higher a...

We try to generalize the inverse scattering transform (IST) for
the Kadomtsev-Petviashvili (KPI) equation to the case of potentials with
“ray” type behavior, that is non-decaying along a finite
number of directions in the plane. We present here the special but
rather wide subclass of such potentials obtained by applying recursively
N binary Backlun...

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. Phys

Some accurate definitions and statements are formulated in order to resolve several contradictions mentioned in the literature on the applications of the Painleve method to the analysis of partial differential equations.

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...

The general solution for the Liouville equation with any given number N of singularities is considered. With the help of the inverse scattering transform method (IST) the set of regular continuous and discrete canonical variables is derived. The dynamical generators of Poincare and dilatation groups and N-soliton solutions are constructed in terms...

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 t...

The set of integrable symmetries of the nonstationary Schr\"{o}dinger equation is shown to admit a natural decomposition into subsets of mutually commuting symmetries. Hierarchies of time evolutions associated with each of these subsets ultimately lead to nonlinear (possibly, operator) equations of the Kadomtsev--Petviashvili I type or its higher a...

The discrete Schr\"{o}dinger equation with potential belonging to $\F_{2}$ is solved explicitly. On this base the associated (1+1)-dimensional cellular automaton is examined and corresponding set of integrals of motions is constructed.

The inverse scattering method is considered for the nonstationary Schrödinger equation with the potentialu (x
1,x
2) nondecaying in a finite number of directions in thex plane. The general resolvent approach, which is particularly convenient for this problem, is tested using a potential that
is the Bäcklund transformation of an arbitrary decaying p...

The heat equation is used as an example to formulate dressing of a nontrivial potential. Corresponding transformation of spectral data and generalization of inverse problem are given. The formulation based on the resolvent approach is shown to be ultimately connected with the original ideas of Zakharov--Shabat dressing procedure. Construction of in...

The solutionu(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 the literature as ∝dxu(0,x,y)=0 are required to be satisfied by the initial data. The spectral theory associated with KPI is studied in the sp...

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 R dx u(0, x, y) = 0 are required to be satisfied by the initial data. The spectral theory associated to KPI is studied in the...

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 ∫ dxu(0, x, y) = 0 are required to be satisfied by the initial data. The problem is completely solved in the framework of the...

In the framework of the inverse scattering method, the solution of the Kadomtsev--Petviashvili equation in its version called KPI is considered. The spectral theory is extended to the case in which the initial data u(x; y) are not vanishing along a finite number of directions at large distances in the plane. The Kadomtsev--Petviashvili equation, th...

The general resolvent scheme for solving nonlinear integrable evolution equations is formulated. Special attention is paid for the problem of nontrivial dressing and corresponding transformation of spectral data. Kadomtsev--Petviashvili equation is considered as a useful laboratory for experimenting these theoretical tools able to handle the specif...

The Kadomtsev--Petviashvili I (KPI) is considered as a useful laboratory for experimenting 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 s...

The discrete Schr\"{o}dinger equation with potential belonging to $\F_{2}$ is solved explicitly. On this base the associated (1+1)-dimensional cellular automaton is examined and corresponding set of integrals of motions is constructed.

The resolvent operator corresponding to the Ablowitz-Ladik linear system with a generic potential is determined. We present
the properties of this operator and, using them, we introduce Jost solutions, different from the ones known in the literature.

The Cauchy problem for the (2+1)-dimensional nonlinear Boiti-Leon-Pempinelli (BLP) equation is studied within the framework of the inverse problem method. Evolution equations generated by the system of BLP equations under study are derived for the resolvent, Jost solutions, and scattering data for the two-dimensional Klein-Gordon differential opera...

Explicit formulae for the Coleman-Mandelstam bosonization procedure for the massless fermionic fields are presented. The construction for the fields defined on the infinite line is performed by means of the conformal transformation of the corresponding representation for the periodic fields. Properties of the resulting massless scalar field are giv...

The scattering problem for the two-dimensional Klein --- Gordon differential operator with variable coefficients is studied in the framework of the resolvent approach. Jost solutions, retarded and advanced solutions, and spectral data are introduced, and their properties are described. The inverse scattering problem is formulated.

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. Properti...

Explicit formulae for bosonization of massless fermionic fields in the periodic case are presented. The scalar field which
results in this procedure is constructed in terms of fermionic operators. It is shown that this scalar field is non-Weyl and
vacuum expectation values of products of these fields do not exist. An interpretation of the bosonizat...

The general resolvent scheme for solving nonlinear integrable evolution equations is formulated. Special attention is paid to the problem of nontrivial dressing and the corresponding transformation of spectral data. The Kadomtsev-Petviashvili equation is considered as the standard example of integrable models in 2+1 dimensions. Properties of the so...

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