A. M. Kamchatnov

Publications (48)37.31 Total impact

• Article: Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation

  A. M. Kamchatnov, Y. -H. Kuo, T. -C. Lin, T. -L. Horng, S. -C. Gou, R. Clift, G. A. El, R. H. J. Grimshaw
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  ABSTRACT: Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg--de Vries (eKdV), or Gardner, equation. We consider both possible signs for the cubic nonlinear term in the Gardner equation corresponding to different fluid density stratification profiles. We identify the range of the input parameters: the oncoming flow speed (the Froude number) and the topographic amplitude, for which the obstacle supports a stationary localised hydraulic transition from the subcritical flow upstream to the supercritical flow downstream. Such a localised transcritical flow is resolved back into the equilibrium flow state away from the obstacle with the aid of unsteady coherent nonlinear wave structures propagating upstream and downstream. Along with the regular, cnoidal undular bores occurring in the analogous problem for the single-layer flow modeled by the forced KdV equation, the transcritical internal wave flows support a diverse family of upstream and downstream wave structures, including solibores, rarefaction waves, reversed and trigonometric undular bores, which we describe using the recent development of the nonlinear modulation theory for the (unforced) Gardner equation. The predictions of the developed analytic construction are confirmed by direct numerical simulations of the forced Gardner equation for a broad range of input parameters.
  05/2013;
• Article: Undular bore theory for the Gardner equation.

  A M Kamchatnov, Y-H Kuo, T-C Lin, T-L Horng, S-C Gou, R Clift, G A El, R H J Grimshaw
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  ABSTRACT: We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg-de Vries (KdV), equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg-de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV-type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves, and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.
  Physical Review E 09/2012; 86(3-2):036605. · 2.26 Impact Factor
• Source

  Available from: 158.125.1.136

  Article: Two-dimensional supersonic nonlinear Schrödinger flow past an extended obstacle.

  G A El, A M Kamchatnov, V V Khodorovskii, E S Annibale, A Gammal
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  ABSTRACT: Supersonic flow of a superfluid past a slender impenetrable macroscopic obstacle is studied in the framework of the two-dimensional (2D) defocusing nonlinear Schrödinger (NLS) equation. This problem is of fundamental importance as a dispersive analog of the corresponding classical gas-dynamics problem. Assuming the oncoming flow speed is sufficiently high, we asymptotically reduce the original boundary-value problem for a steady flow past a slender body to the one-dimensional dispersive piston problem described by the nonstationary NLS equation, in which the role of time is played by the stretched x coordinate and the piston motion curve is defined by the spatial body profile. Two steady oblique spatial dispersive shock waves (DSWs) spreading from the pointed ends of the body are generated in both half planes. These are described analytically by constructing appropriate exact solutions of the Whitham modulation equations for the front DSW and by using a generalized Bohr-Sommerfeld quantization rule for the oblique dark soliton fan in the rear DSW. We propose an extension of the traditional modulation description of DSWs to include the linear "ship-wave" pattern forming outside the nonlinear modulation region of the front DSW. Our analytic results are supported by direct 2D unsteady numerical simulations and are relevant to recent experiments on Bose-Einstein condensates freely expanding past obstacles.
  Physical Review E 10/2009; 80(4 Pt 2):046317. · 2.26 Impact Factor
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  Available from: ArXiv

  Article: Two-dimensional supersonic nonlinear Schr\"odinger flow past an extended obstacle

  G. A. El, A. M. Kamchatnov, V. V. Khodorovskii, E. S. Annibale, A. Gammal
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  ABSTRACT: Supersonic flow of a superfluid past a slender impenetrable macroscopic obstacle is studied in the framework of the two-dimensional defocusing nonlinear Schr\"odinger (NLS) equation. This problem is of fundamental importance as a dispersive analogue of the corresponding classical gas-dynamics problem. Assuming the oncoming flow speed sufficiently high, we asymptotically reduce the original boundary-value problem for a steady flow past a slender body to the one-dimensional dispersive piston problem described by the nonstationary NLS equation, in which the role of time is played by the stretched $x$-coordinate and the piston motion curve is defined by the spatial body profile. Two steady oblique spatial dispersive shock waves (DSWs) spreading from the pointed ends of the body are generated in both half-planes. These are described analytically by constructing appropriate exact solutions of the Whitham modulation equations for the front DSW and by using a generalized Bohr-Sommerfeld quantization rule for the oblique dark soliton fan in the rear DSW. We propose an extension of the traditional modulation description of DSWs to include the linear "ship wave" pattern forming outside the nonlinear modulation region of the front DSW. Our analytic results are supported by direct 2D unsteady numerical simulations and are relevant to recent experiments on Bose-Einstein condensates freely expanding past obstacles.
  06/2009;
• Source

  Available from: ArXiv

  Article: Stationary wave patterns generated by an impurity moving with supersonic velocity through a Bose-Einstein condensate

  T. -L. Horng, S. -C. Gou, T. -C. Lin, G. A. El, A. P. Itin, A. M. Kamchatnov
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  ABSTRACT: Formation of stationary 3D wave patterns generated by a small point-like impurity moving through a Bose-Einstein condensate with supersonic velocity is studied. Asymptotic formulae for a stationary far-field density distribution are obtained. Comparison with three-dimensional numerical simulations demonstrates that these formulae are accurate enough already at distances from the obstacle equal to a few wavelengths.
  02/2009;
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  Available from: ArXiv

  Article: Transcritical flow of a Bose-Einstein condensate through a penetrable barrier

  A. M. Leszczyszyn, G. A. El, Yu. G. Gladush, A. M. Kamchatnov
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  ABSTRACT: The problem of the transcritical flow of a Bose-Einstein condensate through a wide repulsive penetrable barrier is studied analytically using the combination of the localized "hydraulic" solution of the 1D Gross-Pitaevskii equation and the solutions of the Whitham modulation equations describing the resolution of the upstream and downstream discontinuities through dispersive shocks. It is shown that within the physically reasonable range of parameters the downstream dispersive shock is attached to the barrier and effectively represents the train of very slow dark solitons, which can be observed in experiments. The rate of the soliton emission, the amplitudes of the solitons in the train and the drag force are determined in terms of the BEC oncoming flow velocity and the strength of the potential barrier. A good agreement with direct numerical solutions is demonstrated. Connection with recent experiments is discussed.
  01/2009;
• Source

  Available from: Boris A. Malomed

  Article: Wave patterns generated by a supersonic moving body in a binary Bose-Einstein condensate

  Yu. G. Gladush, A. M. Kamchatnov, Z. Shi, P. G. Kevrekidis, D. J. Frantzeskakis, B. A. Malomed
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  ABSTRACT: Generation of wave structures by a two-dimensional object (laser beam) moving in a two-dimensional two-component Bose-Einstein condensate with a velocity greater than both sound velocities of the mixture is studied by means of analytical methods and systematic simulations of the coupled Gross-Pitaevskii equations. The wave pattern features three regions separated by two Mach cones. Two branches of linear patterns similar to the so-called "ship waves" are located outside the corresponding Mach cones, and oblique dark solitons are found inside the wider cone. An analytical theory is developed for the linear patterns. A particular dark-soliton solution is also obtained, its stability is investigated, and two unstable modes of transverse perturbations are identified. It is shown that, for a sufficiently large flow velocity, this instability has a convective character in the reference frame attached to the moving body, which makes the dark soliton effectively stable. The analytical findings are corroborated by numerical simulations. Comment: 13 pages, 6 figures
  11/2008;
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  Available from: Maxim Pavlov

  Article: Kinetic equation for a soliton gas and its hydrodynamic reductions

  G. A. El, A. M. Kamchatnov, M. V. Pavlov, S. A. Zykov
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  ABSTRACT: We introduce and study a new class of kinetic equations, which arise in the description of nonequilibrium macroscopic dynamics of soliton gases with elastic collisions between solitons. These equations represent nonlinear integro-differential systems and have a novel structure, which we investigate by studying in detail the class of $N$-component `cold-gas' hydrodynamic reductions. We prove that these reductions represent integrable linearly degenerate hydrodynamic type systems for arbitrary $N$ which is a strong evidence in favour of integrability of the full kinetic equation. We derive compact explicit representations for the Riemann invariants and characteristic velocities of the hydrodynamic reductions in terms of the `cold-gas' component densities and construct a number of exact solutions having special properties (quasi-periodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed the light on the structure of a continuum limit for a large class of integrable systems of hydrodynamic type and are also relevant to the description of turbulent motion in conservative compressible flows. Comment: 39 pages, 1 figure. Extended version; to appear in Journal of Nonlinear Science
  02/2008;
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  Available from: ArXiv

  Article: Nonlinear diffraction of light beams propagating in photorefractive media with embedded reflecting wire

  E. G. Khamis, A. Gammal, G. A. El, Yu. G. Gladush, A. M. Kamchatnov
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  ABSTRACT: The theory of nonlinear diffraction of intensive light beams propagating through photorefractive media is developed. Diffraction occurs on a reflecting wire embedded in the nonlinear medium at relatively small angle with respect to the direction of the beam propagation. It is shown that this process is analogous to the generation of waves by a flow of a superfluid past an obstacle. The ``equation of state'' of such a superfluid is determined by the nonlinear properties of the medium. On the basis of this hydrodynamic analogy, the notion of the ``Mach number'' is introduced where the transverse component of the wave vector plays the role of the fluid velocity. It is found that the Mach cone separates two regions of the diffraction pattern: inside the Mach cone oblique dark solitons are generated and outside the Mach cone the region of ``ship waves'' is situated. Analytical theory of ``ship waves'' is developed and two-dimensional dark soliton solutions of the equation describing the beam propagation are found. Stability of dark solitons with respect to their decay into vortices is studied and it is shown that they are stable for large enough values of the Mach number. Comment: 18 pages
  02/2008;
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  Available from: ArXiv

  Article: The theory of optical dispersive shock waves in photorefractive media

  G. A. El, A. Gammal, E. G. Khamis, R. A. Kraenkel, A. M. Kamchatnov
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  ABSTRACT: The theory of optical dispersive shocks generated in propagation of light beams through photorefractive media is developed. Full one-dimensional analytical theory based on the Whitham modulation approach is given for the simplest case of sharp step-like initial discontinuity in a beam with one-dimensional strip-like geometry. This approach is confirmed by numerical simulations which are extended also to beams with cylindrical symmetry. The theory explains recent experiments where such dispersive shock waves have been observed.
  07/2007;
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  Available from: ArXiv

  Article: Generation of linear waves in the flow of Bose-Einstein condensate past an obstacle

  Yu. G. Gladush, A. M. Kamchatnov
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  ABSTRACT: The theory of linear wave structures generated in Bose-Einstein condensate flow past an obstacle is developed. The shape of wave crests and dependence of amplitude on coordinates far enough from the obstacle are calculated. The results are in good agreement with the results of numerical simulations obtained earlier. The theory gives a qualitative description of experiments with Bose-Einstein condensate flow past an obstacle after condensate's release from a trap. Comment: 11 pages, 3 figures, to be published in Zh. Eksp. Teor. Fiz
  05/2007;
• Source

  Available from: Roger Grimshaw

  Article: Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction

  G. A. El, R. H. J. Grimshaw, A. M. Kamchatnov
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  ABSTRACT: This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg-de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modeled by the Chezy law, but also importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media.
  05/2007;
• Article: Radiation of linear waves in the stationary flow of a Bose-Einstein condensate past an obstacle

  Yu. G. Gladush, G. A. El, A. Gammal, A. M. Kamchatnov
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  ABSTRACT: Using stationary solutions of the linearized two-dimensional Gross-Pitaevskii equation, we describe the wave pattern occurring in the supersonic flow of a Bose-Einstein condensate past an obstacle. It is shown that these waves are generated outside the Mach cone. The developed analytical theory is confirmed by numerical simulations of the flow past body problem in the frame of the full nonstationary Gross-Pitaevskii equation. Relation of the developed theory with recent experiments is discussed.
  Phys. Rev. A. 03/2007; 75(3).
• Source

  Available from: Valery Shchesnovich

  Article: Whitham method for the Benjamin-Ono-Burgers equation and dispersive shocks.

  Y Matsuno, V S Shchesnovich, A M Kamchatnov, R A Kraenkel
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  ABSTRACT: The Whitham modulation equations for the parameters of a periodic solution are derived using the generalized Lagrangian approach for the case of the damped Benjamin-Ono equation. The structure of the dispersive shock is considered in this method.
  Physical Review E 02/2007; 75(1 Pt 2):016307. · 2.26 Impact Factor
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  Available from: ArXiv

  Article: Linear "ship waves" generated in stationary flow of a Bose-Einstein condensate past an obstacle

  Yu. G. Gladush, G. A. El, A. Gammal, A. M. Kamchatnov
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  ABSTRACT: Using stationary solutions of the linearized two-dimensional Gross-Pitaevskii equation, we describe the ``ship wave'' pattern occurring in the supersonic flow of a Bose-Einstein condensate past an obstacle. It is shown that these ``ship waves'' are generated outside the Mach cone. The developed analytical theory is confirmed by numerical simulations of the flow past body problem in the frame of the full non-stationary Gross-Pitaevskii equation.
  12/2006;
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  Available from: arxiv.org

  Article: Oblique dark solitons in supersonic flow of a Bose-Einstein condensate.

  G A El, A Gammal, A M Kamchatnov
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  ABSTRACT: In the framework of the Gross-Pitaevskii mean field approach, it is shown that the supersonic flow of a Bose-Einstein condensate can support a new type of pattern--an oblique dark soliton. The corresponding exact solution of the Gross-Pitaevskii equation is obtained. It is demonstrated by numerical simulations that oblique solitons can be generated by an obstacle inserted into the flow.
  Physical Review Letters 12/2006; 97(18):180405. · 7.37 Impact Factor
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  Available from: arxiv.org

  Article: Two-dimensional periodic waves in a supersonic flow of a Bose-Einstein condensate

  G. A. El, Yu. G. Gladush, A. M. Kamchatnov
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  ABSTRACT: Stationary periodic solutions of the two-dimensional Gross-Pitaevskii equation are obtained and analyzed for different parameter values in the context of the problem of a supersonic flow of a Bose-Einstein condensate past an obstacle. The asymptotic connections with the corresponding periodic solutions of the Korteweg-de Vries and nonlinear Schr\"odinger equations are studied and typical spatial wave distributions are discussed.
  10/2006;
• Source

  Available from: isan.troitsk.ru

  Article: Propagation of a self-induced transparency pulse in a spatially dispersive medium

  Yu. G. Gladush, A. M. Kamchatnov
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  ABSTRACT: Propagation of a self-induced transparency pulse in spatially dispersive media is analyzed. A generalized model of two-level systems allowing for excitonic energy transfer is proposed. Periodic and soliton solutions to the governing equations are found. Estimates show that the spatial dispersion of a pulse propagating in a slow-light medium can be substantially enhanced and become important in the case of resonant transition and sufficiently long pulse duration.
  Journal of Experimental and Theoretical Physics 03/2006; 102(4):562-569. · 1.03 Impact Factor
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  Available from: arxiv.org

  Article: Kinetic equation for a dense soliton gas.

  G A El, A M Kamchatnov
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  ABSTRACT: We propose a general method to derive kinetic equations for dense soliton gases in physical systems described by integrable nonlinear wave equations. The kinetic equation describes evolution of the spectral distribution function of solitons due to soliton-soliton collisions. Owing to complete integrability of the soliton equations, only pairwise soliton interactions contribute to the solution, and the evolution reduces to a transport of the eigenvalues of the associated spectral problem with the corresponding soliton velocities modified by the collisions. The proposed general procedure of the derivation of the kinetic equation is illustrated by the examples of the Korteweg-de Vries and nonlinear Schrödinger (NLS) equations. As a simple physical example, we construct an explicit solution for the case of interaction of two cold NLS soliton gases.
  Physical Review Letters 12/2005; 95(20):204101. · 7.37 Impact Factor
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  Available from: Roger Grimshaw

  Article: Analytic model for a weakly dissipative shallow-water undular bore.

  G A El, R H J Grimshaw, A M Kamchatnov
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  ABSTRACT: We use the integrable Kaup-Boussinesq shallow water system, modified by a small viscous term, to model the formation of an undular bore with a steady profile. The description is made in terms of the corresponding integrable Whitham system, also appropriately modified by viscosity. This is derived in Riemann variables using a modified finite-gap integration technique for the Ablowitz-Kaup-Newell-Segur (AKNS) scheme. The Whitham system is then reduced to a simple first-order differential equation which is integrated numerically to obtain an asymptotic profile of the undular bore, with the local oscillatory structure described by the periodic solution of the unperturbed Kaup-Boussinesq system. This solution of the Whitham equations is shown to be consistent with certain jump conditions following directly from conservation laws for the original system. A comparison is made with the recently studied dissipationless case for the same system, where the undular bore is unsteady.
  Chaos An Interdisciplinary Journal of Nonlinear Science 10/2005; 15(3):37102. · 2.08 Impact Factor