# Sergey MedvedevFederal Research Center for Information and Computational Technologies

Sergey Medvedev

DrSc

## About

90

Publications

5,216

Reads

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552

Citations

Citations since 2016

Introduction

**Skills and Expertise**

Additional affiliations

September 2017 - present

November 2011 - December 2013

June 1997 - July 1998

Education

September 1980 - June 1986

## Publications

Publications (90)

We propose a new method for solving the Gelfand-Levitan-Marchenko equation (GLME) based on the block version of the Toeplitz Inner-Bordering (TIB) with an arbitrary point to start the calculation. This makes it possible to find solutions of the GLME at an arbitrary point with a cutoff of the matrix coefficient, which allows to avoid the occurrence...

We analyze a family of fourth-order non-linear diffusion models corresponding to local approximations of 4-wave kinetic equations of weak wave turbulence. We focus on a class of parameters for which a dual cascade behaviour is expected with an infrared finite-time singularity associated to inverse transfer of waveaction. This case is relevant for w...

The nonlinear Schrödinger equation (NLSE) is widely used in telecommunication applications, since it allows one to describe the propagation of pulses in an optical fiber. Recently some new approaches based on the nonlinear Fourier transform (NFT) have been actively explored to compensate for fiber nonlinearity and to exceed the limitations of nonli...

We analyze a family of fourth-order non-linear diffusion models corresponding to local approximations of 4-wave kinetic equations of weak wave turbulence. We focus on a class of parameters for which a dual cascade behaviour is expected with an infrared finite-time singularity associated to inverse transfer of waveaction. This case is relevant for w...

We study a self-similar solution of the kinetic equation describing weak wave turbulence in Bose–Einstein condensates. This solution presumably corresponds to an asymptotic behavior of a spectrum evolving from a broad class of initial data, and it features a non-equilibrium finite-time condensation of the wave spectrum n(ω) at the zero frequency ω....

We study a self-similar solution of the kinetic equation describing weak wave turbulence in Bose-Einstein condensates. This solution presumably corresponds to an asymptotic behavior of a spectrum evolving from a broad class of initial data, and it features a non-equilibrium finite-time condensation of the wave spectrum $n(\omega)$ at the zero frequ...

We propose a new method for finding discrete eigenvalues for the direct Zakharov-Shabat problem, based on moving in the complex plane along the argument jumps of the function a(ζ), the localization of which does not require great accuracy. It allows to find all discrete eigenvalues taking into account their multiplicity faster than matrix methods a...

This is a continuation of the paper [5] wherein the Hamiltonian structure together with the non-canonical singular Poisson bracket and Casimir functionals were established for two-dimensional linear elasticity model. The aim of the present work is the extension of the above-mentioned results to the three-dimension case.

Based on the generalized Cayley transform, a family of conservative one-step schemes of the sixth order of accuracy for the Zakharov-Shabat system is constructed. The exponential integrator is a special case. Schemes based on rational approximation allow the use of fast algorithms to solve the initial problem for a large number of values of the spe...

We study stationary solutions in the differential kinetic equation, which was introduced in [1] for description of a local dual cascade wave turbulence. We give a full classification of single-cascade states in which there is a finite flux of only one conserved quantity. Analysis of the steady-state spectrum is based on a phase-space analysis of or...

We study stationary solutions in the differential kinetic equation, which was introduced in for description of a local dual cascade wave turbulence. We give a full classification of single-cascade states in which there is a finite flux of only one conserved quantity. Analysis of the steady-state spectrum is based on a phase-space analysis of orbits...

We propose a new method for finding discrete eigenvalues for the direct Zakharov-Shabat problem, based on moving in the complex plane along the argument jumps of the function $a(\zeta)$, the localization of which does not require great accuracy. It allows to find all discrete eigenvalues taking into account their multiplicity faster than matrix met...

The direct Zakharov–Shabat scattering problem has recently gained significant attention in various applications of fiber optics. The development of accurate and fast algorithms with low computational complexity to solve the Zakharov–Shabat problem (ZSP) remains an urgent problem in optics. In this paper, a fourth-order multi-exponential scheme is p...

Nowadays, improving the accuracy of computational methods to solve the initial value problem of the Zakharov-Shabat system remains an urgent problem in optics. In particular, increasing the approximation order of the methods is important, especially in problems where it is necessary to analyze the structure of complex waveforms. In this work, we pr...

A fourth-order multi-exponential scheme is proposed for the Zakharov-Shabat system. The scheme represents a product of 13 exponential operators. The construction of the scheme is based on a fourth-order three-exponential scheme, which contains only one exponent with a spectral parameter. This exponent is factorized to the fourth-order with the Suzu...

We propose two finite-difference algorithms of fourth order of accuracy for solving the initial problem of the Zakharov-Shabat system. Both schemes have the exponential form and conserve quadratic invariant of Zakharov-Shabat system. The second scheme contains the spectral parameter in exponent only and allows to apply the fast computational algori...

We propose a finite-difference algorithm for solving the initial problem for the Zakharov-Shabat system. This method has the fourth order of accuracy and represents a generalization of the second-order Boffetta-Osborne scheme. Our method permits the Zakharov-Shabat spectral problem to be solved more effectively for continuous and discrete spectra.

Using the cubic Ginzburg-Landau equation as an example, we demonstrate how the inverse scattering transform can be applied to characterize coherent structures in dissipative nonlinear systems. Using this approach one can reduce the number of the effective degrees of freedom in the system when the dynamic is dominated by the coherent structures, eve...

Time-dependend evolution of hydrodynamic turbulence corresponding to
formation of a thermodynamic state at the large-scale part of the spectrum
is studied using the inviscid Leith model. In the wave vector space, the
evolution leads to shrinking of the zero-spectrum ‘hole’—the so-called
focusing problem. However, in contrast with the typical focusi...

We propose a new high-precision algorithm for solving the initial problem for the Zakharov-Shabat system. This method has the fourth order of accuracy and is a generalization of the second order Boffetta-Osborne scheme. It is allowed by our method to solve more effectively the Zakharov-Shabat spectral problem for continuous and discrete spectra.

Time-dependend evolution of hydrodynamic turbulence corresponding to formation of a thermodynamic state at the large-scale part of the spectrum is studied using the inviscid Leith model. In the wave vector space, the evolution leads to shrinking of the zero-spectrum "hole"-the so-called focusing problem. However, in contrast with the typical focusi...

In this work we applied a Hamiltonian formalism to simplify the equations of non-degenerate nonlinear four-wave mixing to the one-degree-of-freedom Hamiltonian equations with a three-parameter Hamiltonian. Thereby, a problem of signal amplification in a phase-sensitive double-pumped parametric fiber amplifier with pump depletion was reduced to a ge...

Based on the exponential representation of signal amplification along an active fibre, we construct an analytical approximation of the solution to a system of balance equations describing the dynamics of an average signal power and pump inside a linear cavity. The output power of the signal at the ends of the linear cavity is estimated. The output...

We study self-similar solutions of the kinetic equation for MHD wave turbulence derived in (Galtier S et al 2000 J. Plasma Phys. 63 447-88). Motivated by finding the asymptotic behaviour of solutions for initial value problems, we formulate a nonlinear eigenvalue problem comprising in finding a number x ∗such that the self-similar shape function f(...

Approximate analytical solutions to the time-independent two-dimensional Gross–Pitaevsky equation are obtained by a variational method. The solutions found are compared with the results of direct numerical calculations. The accuracy of analytical solutions is determined under various conditions of the problem, primarily the degree of nonlinear inte...

Вариационным методом получены приближенные аналитические решения стационарного 2D уравнения Гросса–Питаевского. Найденные решения сопоставлены с результатами прямых численных расчетов. Определена точность аналитических решений при различных условиях задачи, в первую очередь степени нелинейного взаимодействия атомов в бозе-конденсате.

Highlights of 2016
The Journal of Physics A: Mathematical and Theoretical collection of highlights showcases some of the excellent papers we published in 2016.
We would like to thank all of our authors for choosing to submit their high-quality work to the journal and thank our referees and board members for providing constructive peer review and m...

Visualisation of complex nonlinear equation solutions is a useful analysis tool for various scientific and engineering applications. We have re-examined the geometrical interpretation of the classical nonlinear four-wave mixing equations for the specific scheme of a phase sensitive one-pump fiber optical parametric amplification, which has recently...

In [1], we found that an anomalous exponent x∗ of the power-law asymptotic E(k,t) ∼ Ck−x∗
as t → t∗ for k � 1 in k-space representation occurs into the interval 5/3 < x∗ < x2, where t∗ is a finite singular time. Here 5/3 is the Kolmogorov index for the pure Kolmogorov spectrum and x2 ≈ 1.95 corresponds to a Hopf bifurcation which appears in the dyn...

We present a comprehensive study and full classification of the stationary solutions in Leith's model of turbulence with a generalised viscosity. Three typical types of boundary value problems are considered: Problems 1 and 2 with a finite positive value of the spectrum at the left (right) and zero at the right (left) boundaries of a wave number ra...

Free expansion of Bose–Einstein condensates of rubidium atoms at finite temperatures has been analyzed experimentally and theoretically. It has been shown that the interaction between condensed and noncondensed atoms is manifested most clearly by a decrease in the density of atoms in the center of the expanding cloud as compared to the theoretical...

In this study, the existence of Hamiltonian structures for a two-dimensional, linear-elastic model is considered. We show that this model admits the so-called noncanonical singular Poisson bracket. Casimir functionals are found by using the singularity properties of the Poisson bracket obtained. We also demonstrate that these functionals are conser...

Differential models for hydrodynamic, passive-scalar and wave turbulence given by nonlinear first- and second-order evolution equations for the energy spectrum in the $k$-space were analysed.
Both types of models predict formation an anomalous transient power-law spectra.
The second-order models were analysed in terms of self-similar solutions of t...

Differential models for hydrodynamic, passive-scalar and wave turbulence
given by nonlinear first- and second-order evolution equations for the energy
spectrum in the $k$-space were analysed. Both types of models predict formation
an anomalous transient power-law spectra. The second-order models were analysed
in terms of self-similar solutions of t...

Expansion of a steady state of the Gross-Pitaevskii equation after switching off the external field has been investigated. It has been shown that the evolution of the aspect ratio of the localized solution is described by the one-dimensional oscillator equation with renormalized time. The renormalization is determined by the evolution of the width...

The notoriously difficult problem of turbulence slowly reveals its secrets. The phenomenon
of explosive formation of an anomalous turbulence spectrum, which was observed in
numerical simulations, has now been obtained analytically (Self-similar solution in the
Leith model of turbulence: anomalous power law and asymptotic analysis V N Grebenev et
al...

The notoriously difficult problem of turbulence slowly reveals its secrets. The phenomenon
of explosive formation of an anomalous turbulence spectrum, which was observed in
numerical simulations, has now been obtained analytically (Self-similar solution in the
Leith model of turbulence: anomalous power law and asymptotic analysis V N Grebenev et
al...

This paper looks at the two-layer ocean model from a wave-turbulence (WT) perspective. A symmetric form of the two-layer kinetic equation for Rossby waves is derived using canonical variables, allowing the turbulent cascade of energy between the barotropic and baroclinic modes to be studied. It is already well known that in two-layers, energy is tr...

We study Leith's model of turbulence represented by a nonlinear degenerate diffusion equation (Leith 1967 Phys. Fluids 10 1409–16, Connaughton and Nazarenko 2004 Phys. Rev. Lett. 92 044501–506). The model is constructed such that in the case of vanishing viscosity, there are two steady-state solutions: the Kolmogorov spectrum that corresponds to th...

We establish an equivalence of two systems of equations of one-dimensional shallow water models describing the propagation of surface waves over even and sloping bottoms. For each of these systems, we obtain formulas for the general form of their nondegenerate solutions, which are expressible in terms of solutions of the Darboux equation. The invar...

We consider a Leith model of turbulence (Leith C 1967 Phys. Fluids 10 1409) in which the energy spectrum obeys a nonlinear diffusion equation. We analytically prove the existence of a self-similar solution with a power-law asymptotic on the low-wavenumber end and a sharp boundary on the high-wavenumber end, which propagates to infinite wavenumbers...

It is shown that the set of conservation laws for the nonlinear system of equations describing plane steady potential barotropic flow of gas is given by the set of conservation laws for the linear Chaplygin system. All the conservation laws of zero order for the Chaplygin system are found. These include both known and new nonlinear conservation law...

In this paper we look at the two-layer model for oceans and atmospheres from
a wave turbulence approach. We derive a symmetric form of the two-layer kinetic
equation using canonical variables which allows us to study the turbulent
cascade of energy between the barotropic and baroclinic modes. It turns out
that energy is transferred via local triad...

Equivalence of systems of the equations for one-dimensional shallow water models describing motion of surface waves over horizontal and inclined bottoms is established. For each of these systems general formulas for their non-degenerate solutions which are expressed by solutions of the Darboux equation are obtained. The found invariant solutions of...

We obtain point transformations for three one-dimensional systems: shallow-water equations on a flat and a sloping bottom and the system of linear equations obtained by formal linearization of shallow-water equations on a sloping bottom. The passage of these systems to the Carrier-Greenspan parametrization is also obtained. For linear shallow-water...

The problem of interaction of a plasma flow with an inhomogeneous magnetic field is considered. The field is generated by magnetic coils. Field lines along the coil axis form a channel which is used to control the plasma flow. A two-dimensional axial-symmetric plasma model and 3D hybrid code has been used. The work results can be used for a study o...

We revisit the well-known analogy between the effects of stratification and rotation in hydrodynamics, and between both and the external magnetic field in magnetohydrodynamics. After sketching the similarities among models in what concerns linear and nonlinear stationary waves, we show that the Hamiltonian structure of the 2D versions of the models...

In this chapter a development of normal form methods for special classes of partial differential equations is presented. A basic application of the methods is splitting of slow and fast wave motions and finding of equations for the slow wave motion. The rotating shallow water model is the main example for the application of the general theory.

By using the normal form of continuously stratified “primitive” equations of geophysical fluid dynamics with density (in the ocean), or potential temperature (in the atmosphere) playing the role of the vertical coordinate, we decouple vortex and wave motions in the system, introduce normal variables, and derive the effective Hamiltonian for waves w...

A steady Kolmogorov-like spectrum of turbulence is found as an exact solution of the kinetic equation for inertial-gravity waves. The spectrum obtained satisfactorily fits the results of atmospheric observations for mesoscale motions (from hundred to thousand kilometers).

We investigate theoretically and numerically properties of dispersion-managed (DM) solitons in fiber lines with dispersion
compensation period L much shorter than amplification distance Za We present the pathaveraged theory of DM transmission lines with short-scale management. Applying a quasi-identical transformation
we demonstrate that the pathav...

We derive a normal form of nonlinear equations for short equatorial
waves considered in the framework of the rotating shallow water model.
We show dynamical splitting of equatorial Rossby and inertia-gravity
waves. We derive an effective Hamiltonian for the short inertia-gravity
waves and consider their kinetics using the weak turbulence approach....

The nonlinear Schrödinger equation with periodic coefficients is analyzed under the condition of large variation in the local
dispersion. The solution after n periods is represented as the sum of the solution to the linear part of the nonlinear Schrödinger equation and the nonlinear
first-period correction multiplied by the number of periods n. An...

A nonlinear Schrödinger equation with periodic coefficients, as it appears, e.g., in nonlinear optics, is considered. The high-frequency, variable part of the dispersion may be even much larger than the mean value. The ratio of the length of the dispersion map to the period of a solution is assumed as one small parameter. The second one corresponds...