O. M. Kiselev

Ufa Scientific Center of the Russian Academy of Science, Oufa, Bashkortostan, Russia

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Publications (40)19.68 Total impact

  • Oleg Kiselev, Nikolai Tarkhanov
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    ABSTRACT: The subject of this paper is solutions of an autoresonance equation. We look for a connection between the parameters of the solution bounded as t→-∞, and the parameters of two two-parameter families of solutions as t→∞. One family consists of the solutions which are not captured into resonance, and another of those increasing solutions which are captured into resonance. In this way we describe the transition through the separatrix for equations with slowly varying parameters and get an estimate for parameters before the resonance of those solutions which may be captured into autoresonance. ©2014 American Institute of Physics
    Journal of Mathematical Physics 06/2014; 55(6). DOI:10.1063/1.4875105 · 1.18 Impact Factor
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    ABSTRACT: We consider a resonantly perturbed system of coupled nonlinear oscillators with small dissipation and outer periodic perturbation. We show that for the large time $t \sim \epsilon^{-2}$ one component of the system is described in the main by the inhomogeneous Mathieu equation while the other component represents pulsation of large amplitude. A Hamiltonian system is obtained which describes the behaviour of the envelope in the main. The analytic results agree to numerical simulations.
    Chaos (Woodbury, N.Y.) 06/2011; 21(2):023109. DOI:10.1063/1.3578047 · 1.76 Impact Factor
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    ABSTRACT: We consider a solution of the nonlinear Klein–Gordon equation perturbed by a parametric driver. The frequency of parametric perturbation varies slowly and passes through a resonant value, which leads to a solution change. We obtain a new connection formula for the asymptotic solution before and after the resonance.
    Studies in Applied Mathematics 01/2010; 124(1):19-37. DOI:10.1111/j.1467-9590.2009.00460.x · 1.15 Impact Factor
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    ABSTRACT: We study the autoresonant solution of Duffing's equation in the presence of dissipation. This solution is proved to be an attracting set. We evaluate the maximal amplitude of the autoresonant solution and the time of transition from autoresonant growth of the amplitude to the mode of fast oscillations. Analytical results are illustrated by numerical simulations.
    Journal of Physics A Mathematical and Theoretical 12/2009; DOI:10.1088/1751-8113/43/21/215203 · 1.69 Impact Factor
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    Oleg Kiselev, Sergei Glebov
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    ABSTRACT: We study an initial stage of autoresonant growth of a solution in a dissipative system. We construct an asymptotic formula of an autoresonant germ that is an attractor for autoresonant solutions. We present a moment of a fall and a maximum value of the amplitude for the germ. Numerical simulations are done.
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    ABSTRACT: In this paper we present the solution of local parametric resonance equation in terms of parabolic cylinder functions and solve the scattering problem for this equation.
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    S. G. Glebov, O. M. Kiselev, V. A. Lazarev
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    ABSTRACT: A system of two weakly coupled oscillators is investigated. It is shown that under an external periodic perturbation a capture into resonance may occur. A description of this effect by the methods of asymptotic analysis, as well as a numerical simulation, is presented. An explicit formula for the threshold value of the perturbation amplitude at which the resonance occurs is obtained.
    Proceedings of the Steklov Institute of Mathematics 11/2007; 259:S111-S123. DOI:10.1134/S0081543807060077 · 0.23 Impact Factor
  • O M Kiselev
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    ABSTRACT: An asymptotic expansion is constructed and substantiated for the solution of the boundary-value problem for the two-dimensional elliptic system of Dirac equations with rapidly oscillating coefficients, which holds uniformly with respect to the complex variable and the two real variables.
    Russian Academy of Sciences Sbornik Mathematics 10/2007; 190(2):233. DOI:10.1070/SM1999v190n02ABEH000384 · 0.50 Impact Factor
  • O. M. Kiselev
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    ABSTRACT: This paper considers asymptotics of solutions of higher-dimensional nonlinear integrable equations (such as the Kadomtsev-Petviashvili equation, the Davey-Stewartson equations, etc.) and also that of their perturbations.
    Journal of Mathematical Sciences 11/2006; 138(6):6067-6230. DOI:10.1007/s10958-006-0347-8
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    ABSTRACT: We construct a special asymptotic solution for the forced Boussinesq equation. The perturbation is small and oscillates with a slowly varied frequency. The slow passage through the resonance generates waves with the finite amplitude. This phenomenon is described in details.
    05/2006; DOI:10.1063/1.2205803
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    ABSTRACT: We construct a special asymptotic solution for the forced Boussinesq equation. The perturbation is small and oscillates with a slowly varied frequency. The slow passage through the resonance generates waves with the finite amplitude. This phenomenon is described in details.
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    O. M. Kiselev, S. G. Glebov
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    ABSTRACT: In this work we show that the capture into parametric resonance may be explained as the pitchfork bifurcation in the primary parametric resonance equation. We prove that the solution close to the moment of the capture is described by the Painleve-2 equation. We obtain the connection formulas for the asymptotic solution of the primary parametric resonance equation before and after the capture using the matching of the asymptotic expansions.
    Nonlinear Dynamics 12/2005; DOI:10.1007/s11071-006-9084-2 · 2.42 Impact Factor
  • Sergei Glebov, Oleg Kiselev
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    ABSTRACT: We investigate the propagation of solitons for the nonlinear Schrödinger equation under a small driving force i∂ t Ψ+∂ x 2 Ψ+|Ψ| 2 Ψ=ε 2 fe iS/ε 2 ,0<ε≪1· The driving force passes through a resonance. The process of scattering on the resonance leads to changing the number of solitons. After the resonance the number of solitons depends on the amplitude of the driving force.
    Journal of Nonlinear Mathematical Physics 08/2005; 12(3):330-. DOI:10.2991/jnmp.2005.12.3.2 · 0.61 Impact Factor
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    S. G. Glebov, O. M. Kiselev
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    ABSTRACT: Solution of the nonlinear Klein-Gordon equation perturbed by small external force is investigated. The perturbation is represented by finite collections of harmonics. The frequencies of the perturbation vary slowly and pass through the resonant values consecutively. The resonances lead to the sequence of the wave packets with the different fast oscillated carriers. Full asymptotic description of this process is presented.
    Dynamics of partial differential equations 04/2005; DOI:10.4310/DPDE.2005.v2.n3.a2 · 1.23 Impact Factor
  • O. M. Kiselev
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    ABSTRACT: The survey is devoted to the study of solutions of (2 + 1)-dimensional (two spatial variables and time) integrable equations decreasing in spatial directions. As main representatives of these equations, the author considers the Kadomtsev-Petviashvili, Davey-Stewartson, and Ishimori equations.
    Journal of Mathematical Sciences 02/2005; 125(5):689-716. DOI:10.1007/s10958-005-0085-3
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    ABSTRACT: A solution of the nonlinear Klein-Gordon equation perturbed by a small external force is investigated. The frequency of the perturbation varies slowly and passes through a resonance. The resonance generates solitary packets of waves. The full asymptotic description of this process is presented.
    SIAM Journal on Applied Mathematics 01/2005; 65(6):2158-2177. DOI:10.1137/040618084 · 1.41 Impact Factor
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    O. M. Kiselev, S. G. Glebov, V. A. Lazarev
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    ABSTRACT: Solution of the nonlinear Klein-Gordon equation perturbed by small external force is investigated. The frequency of perturbation varies slowly and passes through a resonance. The resonance generates a solitary packets of waves. Full asymptotic description of this process is presented.
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    O. M. Kiselev, S. G. Glebov
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    ABSTRACT: We investigate a propagation of solitons for nonlinear Schr\"odinger equation under small driving force. The driving force passes through the resonance. The process of scattering on the resonance leads to changing of number of solitons. After the resonance the number of solitons depends on the amplitude of the driving force. The analytical results were obtained by WKB and matching method. We bring two examples of numeric simulations for verifying obtained analytical formulas.
  • O M Kiselev, S G Glebov
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    ABSTRACT: We investigate a propagation of solitons for nonlinear Schrodinger equation under small driving force. The driving force passes the res- onance. The process of scattering on the resonance leads to changing of number of solitons. After the resonance the number of solitons depends on the amplitude of the driving force. Nonlinear Schrodinger equation (NLSE) is a mathematical model for wide class of wave phenomenons from signal propagation into optical fibre (1, 2) to surface wave propagation (3). This equation is integrable by inverse scattering transform method (4) and can be considered as an ideal model equation. The perturbations of this ideal model lead to nonintegrable equations. Here we consider such nonintegrable example which is NLSE perturbed driving force. The most known class of the solutions of NLSE is solitons (4). The struc- ture of this kind of solutions is not changed in a case of nonperturbed NLSE. The perturbations usually lead to modulation of parameters of solitons (5, 6). Number of solitons does not change. In this work we investigate a new effect called scattering of solitons on resonance. We consider the process of scattering in detail and obtain the connection formula between pre-resonance and post-resonance solutions. In general case the passage through resonance leads to changing of the number of solitons. This effect is based on the soliton generation due to passage through resonance by external driving force (7). We found that the scattering of solitary waves on resonance is a general effect for nonlinear equations described the wave propagation. In this work we investigate this effect for the simplest model. It allows to show the essence of this effect without unnecessary details.
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    S. G. Glebov, O. M. Kiselev, V. A. Lazarev
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    ABSTRACT: In this paper we show a mechanism of the soliton generation in nonlinear Schrodinger equation due to a small fast oscillating driving force.