High-frequency surface waves at a plasma-metal interface: II. Dispersion of a nonlinear wave

A study is made of the dispersion properties of nonlinear surface waves propagating along a plasma-metal interface under conditions corresponding to the formation of a space charge sheath that equalizes the electron and ion fluxes to the wall. Oscillations of the plasma boundary under the action of the surface wave field are taken into account. It is shown that these oscillations are the main nonlinear mechanism for generating wave field harmonics and are analogous to the nonlinearity in the current-voltage characteristic of the space charge sheath. The effect of the nonlinearity on the dispersion properties of surface waves due to the relationship between the sheath thickness and wave amplitude is calculated with allowance for harmonic generation. The energy transported by surface waves under conditions typical of RF and microwave discharges is calculated.     

Ion-acoustic solitons in Bi-Ion dusty plasma

The propagation of ion-acoustic solitons in a warm dusty plasma containing two ion species is investigated theoretically. Using an approach based on the Korteveg de Vries equation, it is shown that the critical value of the negative ion density that separates the domains of existence of compression and rarefaction solitons depends continuously on the dust density. A modified Korteveg de Vries equation for the critical density is derived in the higher order of the expansion in the small parameter. It is found that the nonlinear coefficient of this equation is positive for any values of the dust density and the masses of positive and negative ions. For the case where the negative ion density is close to its critical value, a soliton solution is found that takes into account both the quadratic and cubic nonlinearities. The propagation of a solitary wave of arbitrary amplitude is investigated by the quasi-potential method. It is shown that the range of dust densities around the critical value within which solitary waves with positive and negative potentials can exist simultaneously is relatively wide.     

Nonlinear dynamics of drift structures in a magnetized dissipative plasma

A study is made of the nonlinear dynamics of solitary vortex structures in an inhomogeneous magnetized dissipative plasma. A nonlinear transport equation for long-wavelength drift wave structures is derived with allowance for the nonuniformity of the plasma density and temperature equilibria, as well as the magnetic and collisional viscosity of the medium and its friction. The dynamic equation describes two types of nonlinearity: scalar (due to the temperature inhomogeneity) and vector (due to the convectively polarized motion of the particles of the medium). The equation is fourth order in the spatial derivatives, in contrast to the second-order Hasegawa-Mima equations. An analytic steady solution to the nonlinear equation is obtained that describes a new type of solitary dipole vortex. The nonlinear dynamic equation is integrated numerically. A new algorithm and a new finite difference scheme for solving the equation are proposed, and it is proved that the solution so obtained is unique. The equation is used to investigate how the initially steady dipole vortex constructed here behaves unsteadily under the action of the factors just mentioned. Numerical simulations revealed that the role of the vector nonlinearity is twofold: it helps the dispersion or the scalar nonlinearity (depending on their magnitude) to ensure the mutual equilibrium and, thereby, promote self-organization of the vortical structures. It is shown that dispersion breaks the initial dipole vortex into a set of tightly packed, smaller scale, less intense monopole vortices-alternating cyclones and anticyclones. When the dispersion of the evolving initial dipole vortex is weak, the scalar nonlinearity symmetrically breaks a cyclone-anticyclone pair into a cyclone and an anticyclone, which are independent of one another and have essentially the same intensity, shape, and size. The stronger the dispersion, the more anisotropic the process whereby the structures break: the anticyclone is more intense and localized, while the cyclone is less intense and has a larger size. In the course of further evolution, the cyclone persists for a relatively longer time, while the anticyclone breaks into small-scale vortices and dissipation hastens this process. It is found that the relaxation of the vortex by viscous dissipation differs in character from that by the frictional force. The time scale on which the vortex is damped depends strongly on its typical size: larger scale vortices are longer lived structures. It is shown that, as the instability develops, the initial vortex is amplified and the lifetime of the dipole pair components-cyclone and anticyclone-becomes longer. As time elapses, small-scale noise is generated in the system, and the spatial structure of the perturbation potential becomes irregular. The pattern of interaction of solitary vortex structures among themselves and with the medium shows that they can take part in strong drift turbulence and anomalous transport of heat and matter in an inhomogeneous magnetized plasma.     

Nonlinear equation for Farley–Buneman waves in multispecies plasma

The nonlinear equation describing the Farley–Buneman (FB) waves in multispecies collisional plasmas is derived by employing the multiple-scale reduction analysis. It is shown that the presence of several ion species with different collisionalities and different ion masses removes the degeneracy of the nonlinear equation and generates the nonlinear terms resulting in wave steepening and wave breaking. This effect may be responsible for formation of one-dimensional coherent FB waves of a finite amplitude.     

Effect of dust size distribution on ion-acoustic solitons in dusty plasmas with different dust grains

Theoretical studies are carried out for ion acoustic solitons in multicomponent nonuniform plasma considering the dust size distribution. The Korteweg?de Vries equation for ion acoustic solitons is given by using the reductive perturbation technique. Two special dust size distributions are considered. The dependences of the width and amplitude of solitons on dust size parameters are shown. It is found that the properties of a solitary wave depend on the shape of the size distribution function of dust grains.     

Ion flux associated with an ion-acoustic cnoidal wave in magnetized dusty plasma

The oblique propagation of nonlinear periodic ion-acoustic waves in magnetized dusty plasma is investigated. The equations describing the dynamics of the wave potential in the first and second orders of the perturbation theory are derived, and their nonsecular periodic solutions are found. The average nonlinear ion flux caused by the propagation of a cnoidal wave is estimated. The magnitude and direction of the ion flux are analyzed as functions of the dust charge density and the angle between the wave propagation direction and the magnetic field.     

Microdamage evaluation in human trabecular bone based on nonlinear ultrasound vibro-modulation (NUVM)

The primary aim of this work is to investigate the potential of nonlinear ultrasound for microdamage detection in human bone. Microdamage evaluation in human bone is of great importance, because it is considered a significant parameter for characterizing fracture risk. Experiments employing nonlinear acoustic vibro-modulation were carried out in human femoral trabecular specimens removed during surgery. A frequency mixing (inter-modulation) was observed between an ultrasound wave, propagating in the bone, and a low-frequency vibration applied directly to the bone specimens. The appearance of side frequencies, which are related to the vibrational excitation, around the fundamental ultrasound frequency manifests the modulation nonlinear phenomenon. Instead of inducing microdamage by mechanical fatigue loading, specimens with different degree of osteoporosis were used. The experiments demonstrated that osteoporotic bone exhibits stronger nonlinearity compared to healthy bone presenting significant increase of the modulation amplitude with increasing degree of osteoporosis. The obtained results indicate that, in contrast to conventional hysteretic nonlinearity, dissipative acoustic nonlinearity can be of significance in the generation of nonlinear modulation effects. In the proposed technique the size and the shape of samples are not crucial compared to nonlinear resonant ultrasound spectroscopy (NRUS). Furthermore, the method is sensitive to the presence of microdamage, non-invasive, easy to implement and most important, it can be proved valuable tool for in vivo bone damage characterization.     

Dust−ion acoustic freak wave propagation in a nonthermal mesospheric dusty plasma

Nonlinear properties of dust?ion acoustic freak waves have been studied in homogeneous unmagnetized dusty plasmas consisting of ions, nonthermal fast electrons, and positive and negative dust grains. By using derivative expansion method under the assumption of strongly dispersive medium, the basic equations are reduced to nonlinear Schrödinger equation (NLSE). One of NLSE solutions in the unstable region is the rational one which is responsible for creation of the freak waves. The dependence of the freak wave profile on the dust grain charge, carrier wavenumber, and energetic nonthermal electron population is discussed.     

Nonlinear properties of two-dimensional wake waves excited by relativistic electron bunches in plasma

The properties of a nonlinear plasma wake wave excited by an axially symmetric relativistic electron bunch are studied. It is shown that the nonlinear dependence of the wake wavelength on the transverse coordinate leads to distortion of the phase front of the wake wave and to steepening and oscillations of the transverse profile of the wakefield. The magnetic field of the wake wave is nonzero and oscillates at a frequency higher than the plasma electron frequency. Because of nonlinearity, the amplitude of the excited wake wave changes with distance from the bunch. The increase in nonlinearity leads to the development of turbulence and chaotization of the wakefield and results in the switching-on of the thermal effects and plasma heating.     

Weakly ionized plasma in a supersonic plasma flow

The structure of the ion acoustic precursor of a shock wave in a weakly ionized collision-dominated plasma is studied numerically. It is shown that the simultaneous action of the nonlinearity, dispersion, and dissipation leads to the formation of an oscillating profile of the ion density in the precursor. There exist regimes in which the charged-particle density decreases abruptly and simultaneously the number of maxima in its profile within the precursor becomes smaller as the shock wave velocity increases in a jumplike manner. This effect is analogous to the corresponding hydrodynamic effect in narrow shallow channels (the “Houston's horse” effect). In the stage preceding this jumplike process, local regions may appear in which the degree of plasma ionization is elevated. Such plasma “bunches” give rise to the strong reverse action of the charged particles on the neutral component, resulting in the “stretching” of the precursor. This phenomenon is resonant in character and occurs in a narrow range of shock wave velocities.     

Nonlinear ion acoustic waves in a quantum degenerate warm plasma with dust grains

A study is made of the propagation of ion acoustic waves in a collisionless unmagnetized dusty plasma containing degenerate ion and electron gases at nonzero temperatures. In linear theory, a dispersion relation for isothermal ion acoustic waves is derived and an exact expression for the linear ion acoustic velocity is obtained. The dependence of the linear ion acoustic velocity on the dust density in a plasma is calculated. An analysis of the dispersion relation reveals parameter ranges in which the problem has soliton solutions. In nonlinear theory, an exact solution to the basic equations is found and examined. The analysis is carried out by Bernoulli’s pseudopotential method. The ranges of the phase velocities of periodic ion acoustic waves and the velocities of solitons are determined. It is shown that these ranges do not overlap and that the soliton velocity cannot be lower than the linear ion acoustic velocity. The profiles of the physical quantities in a periodic wave and in a soliton are evaluated, as well as the dependence of the critical velocity of solitons on the dust density in a plasma.     

Development of an Exactly Solvable Model of Resonance Tunneling of Electromagnetic Waves through Gradient Barriers in a Nonuniform Magnetoactive Plasma

An exactly solvable one-dimensional model describing resonance tunneling (reflectionless transmission) of a transverse electromagnetic wave through wide layers of magnetoactive plasma is developed on the basis of the Helmholtz equation. The plasma layers include a set of spatially localized density structures the amplitudes and thicknesses of which are such that approximate methods are inapplicable for their analysis. The profiles of the plasma density structures strongly depend on the choice of the free parameters of the problem that determine the amplitudes of plasma density modulation, characteristic scale lengths of the density structures, their number, and the total thickness of the nonuniform plasma layer. The plasma layers can also include a set of random inhomogeneities. The propagation of electromagnetic waves through such complicated plasma inhomogeneities is analyzed numerically within the proposed exactly solvable model. According to calculations, there are a wide set of inhomogeneous structures for which an electromagnetic wave incident from vacuum can propagate through the plasma layer without reflection, i.e., the complete tunneling of thick plasma barriers takes place. The model also allows one to exactly solve a one-dimensional problem on the nonlinear transillumination of a nonuniform plasma layer in the presence of cubic nonlinearity. It is important that, due to nonlinearity, the thicknesses of the evanescent plasma regions can decrease substantially and, at a sufficiently strong nonlinearity, such regions will disappear completely. The problem of resonance tunneling of electromagnetic radiation through gradient wave barriers is of interest for various applications, such as efficient heating of dense plasma by electromagnetic radiation and transmission of electromagnetic signals from a source located in the near-Earth plasma or deep in the plasma of an astrophysical object through the surrounding evanescent regions.     

Nonlinear ion flux caused by a periodic ion-acoustic wave in plasma with two-temperature electrons

The propagation of periodic ion-acoustic waves in plasma with two-temperature electrons and cold ions is analyzed. The equations for the wave potential are derived in the first- and second-orders of the perturbation theory, and their nonsecular periodic solutions are obtained. The average nonlinear ion flux is determined, and its properties are studied as functions of the ratios between the densities and temperatures of the cold and hot electron components. The conditions are analyzed under which the ion flux is co- or counter-directed to the wave propagation direction. For the case in which, depending on the plasma parameters, the ion flux at a given wave amplitude can be either positive or negative, the domains of existence of positive and negative ion fluxes in the “temperature ratio-density ratio” plane are determined.     

Nonlinear theory of the collective Cherenkov interaction of a dense relativistic electron beam with a dense plasma in a waveguide

A nonlinear theory of the instability of a straight relativistic dense electron beam in a plasma waveguide is derived for conditions of the stimulated collective Cherenkov effect. A study is made of a waveguide with a dense plasma such that the plasma wave excited by the beam during the instability can be escribed, with a good degree of accuracy, as a potential wave. General relativistic nonlinear equations are btained that describe the temporal dynamics of beam-plasma instabilities with allowance for plasma nonlinearity and the generation of harmonics of the initial perturbation. Under the assumption that the resonant interaction between the beam waves and the plasma waves is weak, the general equations are reduced to relativistic equations with cubic nonlinearities by using the method of expansion in small perturbations of the trajectories and momenta of the beam and plasma electrons. The reduced equations are solved analytically, the time scales on which the instability saturates are determined, and the nonlinear saturation amplitudes are obtained. A comparison between analytical solutions to the reduced equations and numerical solutions to the general nonlinear equations shows them to be in good agreement. Nonlinear processes caused by the relativistic nature of the beam are found to prevent stochastization of the system in the nonlinear stage of the well-developed instability. In contrast, a nonrelativistic electron beam is found to be subject to significant anomalous nonlinear stochastization.     

Stability of dust acoustic wavepackets suffering from polarization force due to the presence of trapped ions

The combined effects of the polarization force, free and trapped ions, and dust charge variation are incorporated in a rigorous study of the nonlinear dust acoustic waves (DAWs) propagating in an unmagnetized dusty plasma. Owing to the departure from the Boltzmann ion distribution, it is found that the nonlinear DAWs are governed by a modified Korteweg?de Vries (mKdV) equation. The association between the mKdV solitary wave and the DAW envelope in the system under consideration is discussed. A modified nonlinear Schrödinger equation appropriate for describing the modulated DAWs is derived. The modulation instability (MI) and the dependence of the system physical parameters on the polarization force, trapped ions, and dust charge variation have been analyzed. It is found that the critical curve separating the stable/unstable regions is strongly influenced by both of the polarization and the ion trapping parameters. Moreover, increasing the polarization leads to an increase of the critical wave number, while increasing the trapping parameter yields the opposite effect. The MI maximum growth rate decreases (increases) as the polarization (trapped ion) increases. The obtained results may be helpful in better understanding of space observations of the solar energetic particle flows in interplanetary space and the energetic particle events in the Earth’s magnetosphere.     

Bifurcations of axisymmetric plasma equilibrium in a tokamak

Bifurcation of solutions to the Grad–Shafranov-type equation for helically symmetric plasma near the threshold for tearing instability are analyzed. Quadratic and cubic nonlinearities were added to the linear dependence of the current density on the helical flux. Depending on the character of nonlinearity, two types of bifurcation can be observed, the “small” and the “large” ones. The small bifurcation is typical of cubic nonlinearity and reveals itself in the growth of the magnetic island from zero as the profile parameter increases above the instability threshold. The large bifurcation is typical of quadratic nonlinearity and causes jumplike formation of a large-scale magnetic island upon exceeding the instability threshold. As the profile parameter decreases below the instability threshold, the large-scale island continues to persist for some time (the hysteresis effect) and then suddenly disappears.     

Plasma Approach to Describing the Electric Dynamics of a Neuron

The electric excitation of a neuron is interpreted as the formation of a nonlinear solitary ion acoustic wave of the charge density of sodium and hydrogen ions in an electrolytic intracellular fluid, which is treated as a dense plasma. It is shown that such a wave can be described by the coupled sine-Gordon and Korteweg-de Vries equations, having a solution in the form of a soliton whose internal vibrational structure is described by the Fermi-Pasta-Ulam spectrum. It is concluded that a nerve impulse can be interpreted as a low-frequency solitary wave of the charge density of sodium ions with a trapped high-frequency charge density wave of protons.     

Nonlinear isothermal waves in a degenerate electron plasma

A nonlinear differential equation describing oscillations of the chemical potential in a one-dimensional steady-state wave propagating in a degenerate electron gas against an immobile neutralizing ion background is derived, investigated, and solved exactly. It is found that the wave phase velocity is bounded below by a critical velocity, whose exact value is obtained.     

Experimental and numerical exploration of intrinsic localized modes in an atomic lattice

This review focuses attention on the experimental studies of intrinsic localized modes (ILMs) produced in driven atomic lattices. Production methods involve the application of modulational instability under carefully controlled conditions. One experimental approach is to drive the atomic lattice far from equilibrium to produce ILMs, the second is to apply a driver of only modest strength but nearby in frequency to a plane wave mode so that a slow transformation from large amplitude standing waves to ILMs takes place. Since, in either case, the number of ILMs produced is small, the experimental observation tool appropriate for this task is four-wave mixing. This nonlinear detection technique makes use of the nonlinearity associated with an ILM to enhance its signal over that produced by the more numerous, but linear, spin waves. The final topic deals with numerical simulations of a nonlinear nanoscale atomic lattice where the new feature is running ILMs.     

Nonlinear nonstationary Alfvén resonance in a finite-pressure plasma

A nonlinear equation for a resonant Alfvén disturbance in a finite-pressure plasma is derived and matching conditions for a fast magnetosonic wave are obtained. The evolution of the resonant disturbance and the rate of resonant energy absorption are calculated for two cases: when the source is switched-on exponentially and instantaneously. It is shown that the evolution to a nonlinear regime is accompanied by the stratification of the resonant disturbance and the average plasma parameters at a progressively decreasing spatial scale. Essential properties of the nonlinear resonant disturbance are the nonlinear saturation of its amplitude, the displacement of the resonance layer, and the disappearance of resonant energy absorption.