Volodymyr Kotlyarov

B. Verkin Institute for Low Temperature Physics and Engineering of National Academy of Sciences of Ukraine · Mathematical Physics

We study the problem of propagation of an input electromagnetic pulse through a long two-level laser amplifier under trivial initial conditions. In this paper, we consider an unstable model described by the Maxwell-Bloch equations without spectral broadening. Previously, this model was studied by S.V. Manakov in 1982 and together with V.Yu. Novoksh...

The complete information about this 1976 publication can be found in arXiv:1401.4410

extended abstract for this paper can be upload: New_abstract_ to_NLS1986.pdf

We consider the problem of the propagation of an electric field generated by periodic pumping in a stable medium of two-level atoms as the mixed problem for the Maxwell–Bloch equations without spectrum broadening. An approach to the study of such a problem is proposed. We use the inverse scattering transform method in the form of the matrix Riemann...

We consider dispersive shock waves of the focusing nonlinear Schrödinger equation generated by discontinuous initial conditions which are periodic or quasiperiodic on the left semiaxis and zero on the right semiaxis. As an initial function, we use a finite-gap potential of the Dirac operator given in an explicit form through hyperelliptic theta-fun...

We consider dispersive shock wave to the focusing nonlinear Schr\"odinger equation generated by a discontinuous initial condition which is periodic or quasi-periodic on the left semi-axis and zero on the right semi-axis. As an initial function we use a finite-gap potential of the Dirac operator given in an explicit form through hyper-elliptic theta...

The Baker-Akhiezer function theory was successfully developed in the middle of 70-th. This theory concerns of spectral theory and completely integrable nonlinear equations such as Korteweg de Vries equation, nonlinear Schrodinger equation, sine-Gordon equation, Kadomtsev-Petviashvili equation, etc. Later the theory was reproduced for the Ablowitz-K...

A mixed initial-boundary value problem for nonlinear Maxwell-Bloch (MB) equations without spectral broadening is studied by using the inverse scattering transform in the form of the matrix Riemann-Hilbert (RH) problem. We use transformation operators whose existence is closely related with the Goursat problems with nontrivial characteristics. We al...

We present the construction of the planar Baker-Akhiezer (BA) function forthe nonlinear Schr ̈odinger equation (NLS), corresponding to the finite-genus solutions ofthe NLS equation. The BA function is described as the unimodular solution of a matrixRiemann–Hilbert problem in the complex plane with piecewise constant jumps across a setof arcs. (PD...

Explicit formulas for matrix elements of the Hermitian matrix are found through a spectrum of this matrix and spectra of some number of its perturbations. A dependence of sufficient number of perturbations from the structure of the matrix and the kind of perturbations is established. It is shown that for arbitrary matrix needed number of perturbati...

We study the long-time asymptotic behavior of the Cauchy problem for the modified Korteweg—de Vries equation with an initial function of the step type. This function rapidly tends to zero as and to some positive constant c as . In 1989 Khruslov and Kotlyarov have found (Khruslov and Kotlyarov 1989 Inverse Problems 5 1075–88) that for a large time t...

The Maxwell-Bloch equations have been intensively studied by many authors. The main results are based on the inverse scattering transform and the Marchenko integral equations. However this method is not acceptable for mixed problems. In the paper, we develop a method allowing to linearize mixed problems. It is based on simultaneous spectral analysi...

The paper offers the method of discovering of some class of solutions for the nonlinear Schroedinger equation. An algorithm of constructive solving of the Cauchy periodic problem with a finite-gap initial condition was also obtained.

This paper contains first results on the finite-gap integration of the Sine-Gordon equation. They were published on Russian in 1976. The papers \cite{Koz}, \cite{KK}, \cite{KK02} have been rewritten in the English language with small modifications for a convenience. Such a translation was made due to requests of some interested readers. In those pa...

This paper contains first results on the finite-gap integration of the Sine-Gordon equation. They were published on Russian in 1976. The papers [9], [11], [20] have been rewritten in the English language with small modifications for a convenience. Such a translation was made due to requests of some interested readers. In those papers, the method of...

The Maxwell-Bloch equations have been intensively studied by many authors. The main results are based on the inverse scattering transform and the Marchenko integral equations. However this method is not acceptable for mixed problems. In the paper, we develop a method allowing to linearize mixed problems. It is based on simultaneous spectral analysi...

Considered in this paper the Maxwell-Bloch (MB) equations became known after Lamb [1-4]. In [5] Ablowitz, Kaup and Newell proposed the inverse scattering transform (IST) to the Maxwell-Bloch equations for studying a physical phenomenon known as the self-induced transparency. A description of general solutions to the MB equations and their classific...

The prominent mathematician Vladimir Aleksandrovich Marchenko is the author of more than 130 publications, including 12 monographs. He has obtained fundamental results in harmonic analysis and the theory of almost periodic functions, the spectral theory of differential and finite-difference operators, the theory of inverse problems of spectral anal...

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation with steplike initial data.

We produce a family of matrix Riemann–Hilbert (RH) problems parametrized by a given scalar function r(k). The jump matrix of the RH problem depends on r(k) and external real parameters x and t. We prove the unique solvability of such a problem. A solution M(x, t, k) of this matrix RH problem gives a smooth solution q(x, t) of the modified Korteweg–...

The modified Korteweg-de Vries equation on the line is considered. The initial function is a discontinuous and piecewise constant step function, i.e. q(x,0)=c r for x≥0 and q(x,0)=c l for x<0, where c l , c r are real numbers which satisfy c l >c r >0. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value p...

We consider the modified Korteveg–de Vries equation on the line. The initial data are the pure step function, i.e., q(x,0) = 0 for x ≥ 0 and q(x,0) = c for x<0, where c is an arbitrary real number. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as t→∞. Using the steepest descent method and th...

The long time behavior of solutions of (Dirichlet) initial boundary value prob-lems for the focusing nonlinear Schrödinger equation in the case of (asymptotically) periodic boundary conditions of special (one-frequency) structure is considered both theoretically and numerically. The results of numerical simulations are shown to confirm the theoreti...

We consider the initial value problem for the focusing nonlinear Schrödinger equation with “step-like” initial data: q(x, 0) = 0 for x ≤ 0 and q(x, 0) = Aexp(−2iBx) for x > 0, where A > 0 and B ∈ ℝ are constants. The paper aims at studying the long-time asymptotics of the solution to this problem. We show that there are three regions in the half-pl...

We study the long-time asymptotic behavior of the solution to the initial-boundary value (IBV) problem in the quarter plane (x > 0, t > 0) for nonlinear integrable equations of stimulated Raman scattering. We consider the case of zero initial condition and periodic boundary data (p eiωt). Using the steepest descent method for oscillatory matrix Rie...

21 августа 2007 года исполнилось 70 лет выдающемуся математику Леониду Андреевичу Пастуру. Л. А. Пастур родился в 1937 г. в Винницкой области. В 1955 г. по окончании средней школы в г. Мариуполе он приезжает в Харьков и поступает на инженерно-физический факультет Харьковского политехнического института. На этом факультете курсы математики и теорети...

We consider the focusing nonlinear Schrödinger equation on the quarter plane. The initial data are vanishing at infinity while the boundary data are time- periodic, of the form ae\ia e2\iwt{a{\rm e}^{\i\alpha} {\rm e}^{2\i\omega t}} . The goal of this paper is to study the asymptotic behavior of the solution of this initial-boundary-value problem...

We consider the focusing nonlinear Schrödinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic. The main goal of this paper is to introduce a Riemannboundary-value problem. This is a preliminary step to obtain uniform long-time asymptotics for the solution of this equation using both the stationa...

The Riemann–Hilbert problem proposed in [2] for the integrable stimulated Raman scattering (SRS) model was shown to be solvable under an additional condition: the boundary data have to be chosen in such a way that a corresponding spectral problem has no spectral singularities. In the general case, it can be shown that a spectral singularity occurs...

In this paper we describe characteristic properties of the scattering data of the compatible eigenvalue problem for the pair of differential equations related to the modified Korteweg-de Vries (mKdV) equation whose solution is defined in some half-strip or in the quarter plane (0<x<∞)×[0,T), T≤∞. We suppose that this solution has a C ∞ initial func...

This article is about the focusing nonlinear Schrödinger equation on the half-line. The initial function vanishes at infinity while boundary data are local perturbations of periodic or quasi-periodic (finite-gap) functions. We study the corresponding scattering problem for the Zakharov–Shabat compatible differential equations, the representation of...

We consider an integrable model of stimulated Raman scattering. The corresponding hyperbolic partial differential equations are referred to as SRS nonlinear equations. We study the initial-boundary value Goursat problem for these equations in the quarter of (x,t)-plane. The initial function vanishes at infinity while boundary data are local perturb...

We construct non-localized, real global solutions of the Kadomtsev-Petviashvili-I equation which vanish for $x\to-\infty$ and study their large time asymptotic behavior. We prove that such solutions eject (for $t\to\infty$) a train of curved asymptotic solitons which move behind the basic wave packet.

This paper is concerned with the direct and inverse scattering problems for compatible differential equations connected with the nonlinear Schrödinger equation (NLSE) on the semi-axis. The corresponding initial boundary value problem (x,t+) was studied recently by Fokas and Its. They found that the key to this problem is to linearize the initial bo...

The equations of dynamics for an easy-plane anisotropy uniaxial ferromagnet St-S*Sxx+ beta (S.n)S*n=0 S2=1 n=(0,0,1) beta >0 is shown to be gauge equivalent to the equation (vt+vxx-2/v/2v+2/vx/2v/( beta -4/v/2)-(vx/2) delta ln( beta -4/v)2)/ delta x=0. The formulae that relate the solutions of these equations for natural boundary conditions are der...

An asymptotic analysis of the Marchenko integral equation for the sine-Gordon equation is presented. The results are used for a construction of soliton asymptotics of decreasing and some non-decreasing solutions of the sine-Gordon equation. The soliton phases are shown to have an additional shift with respect to the reflectionless case caused by th...

The paper considers the Cauchy problem for the equation upsilon t+6 upsilon 2 upsilon x+ upsilon xxx=0 with step-like initial data. It is proved by the inverse scattering method that for t to infinity the solution upsilon (x,t) in the neighbourhood of the leading edge breaks up into an infinite series of non-interacting solitons generated by the co...

The Cauchy problem for the sinh-Gordon equation on the whole axis with non-zero boundary values at infinity is studied. The main part of the paper is devoted to investigation of the direct and inverse problems for the associated Lax operator. The characteristic properties of the corresponding scattering data are obtained as well as their evolution...

We prove the existence of solutions of the Kadomtsev—Petviashvili equation whose temporal asymptotics can be represented as a superposition of curved solitons.

Necessary and sufficient conditions characterizing the scattering data of Schrödinger and Dirac operators whose coefficients tend to different constant limits as x→±∞ are given.

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Foreword to arXiv: 1401.4445v1 These are the English translation of the papers published only on Russian: V.P. Kotlyarov, Periodic problem for the Schr¨odinger nonlinear equation, In: Voprosy matematicheskoi fiziki i funkcionalnogo analiza, 1, Naukova Dumka, Kiev, 1976, pp.121-131 and A.R.Its, V.P.Kotlyarov, Explicit formulas for solutions of the...

The paper offers the method of discovering of finite gap solutions fto the nonlinear Schrodinger equation and it lays the foundations for the finite-gap theory of the focusing nonlinear Schrödinger equation. By using the shifted periodic in space solution of the nonlinear Schrodinger equation and the corresponding monodromy matrix of the Dirac oper...