Small Amplitude Approximation Research Articles

Nonlinear ion acoustic waves in dense magnetoplasmas: Analyzing interaction solutions of the KdV equation using Wronskian formalism for electron trapping with Landau diamagnetism and thermal excitations

Nonlinear ion acoustic waves in the presence of electron trapping with Landau quantization and thermal excitations induced smearing effects of the Fermi step function are investigated using the two-fluid theory in the vicinity of white dwarfs. In spite of the fact that the Kortweg de Vries (KdV) equation is derived within the confines of small amplitude approximation used in the reductive perturbation theory (RPT), it is found to admit solutions that exhibit outstanding capability of wave enhancement and have applications in the nonlinear wave propagation. It is found that enhancing the quantizing magnetic field and the electron thermal effects modify the spatial scale of the formation of the novel nonlinear structures reported for our model. Interestingly, it is found that, unlike the solitary and periodic structures, the choice of space and time coordinates affects the structure of these solutions. Interaction of these solutions with a stable nonlinear structure are also studied.

Reductive perturbation method in magnetized plasma and role of negative ions

An analysis of reductive perturbation method (RPM) is presented to show why the solitary structures of non-linear ion acoustic waves (IAWs) cannot be obtained in magnetized electron ion plasma by employing this technique. In RPM, the non-linear Korteweg–de Vries equation is derived using stretched co-ordinates in the reference frame of the wave phase speed, considering the dispersion to be a higher-order effect that balances the non-linearity to produce a solitary structure. The maximum amplitude |Φm| of the non-linear solitary wave turns out to be larger than one that contradicts the small amplitude approximation. In the presence of negative ions, the maximum amplitude satisfies the condition |Φm|<1. To elaborate these points, the results have been applied to an experimental plasma consisting of positive ions of xenon (Xe+) and negative ions of fluorene (F−) along with electrons. The amplitude and width of the solitary structures depend upon the ratio of the electron to positive ion density (ne0ni0). Since the non-linear coefficient turns out to be negative, rarefied (dip) solitons are formed in the magnetized Xe+−F−−e plasma.

Inertial motion on the earth's spheroidal surface.

As seen by an observer in the rotating frame, the earth's small spheroidal deformations neutralize the centrifugal force, leaving only the smaller Coriolis force to govern the "inertial" motion of objects that move on its surface, assumed smooth and frictionless. Previous studies of inertial motion employ weakly spheroidal equations of motion that ignore the influence of the centrifugal force and yet treat the earth as a sphere. The latitude dependence of these equations renders them strongly nonlinear. We derive and justify these equations and use them to identify, classify, name, describe, and illustrate all possible classes of inertial motion, including a new class of motion called circumpolar waves, which encircle both poles during each cycle of the motion. We illustrate these classes using CorioVis, our freely available Coriolis visualization software. We identify a rotational/time-reversal symmetry for motion on the earth's surface and use this symmetry to develop and validate closed-form small-amplitude approximations for the four main classes and one degenerate class of inertial motion. For these five classes, we supply calculations of experimentally relevant frequencies, zonal drifts, and latitude ranges.

Long-wavelength limit for the Green–Naghdi equations

<abstract><p>This paper studies the long-wavelength limit for the one-dimensional Green–Naghdi (GN) equations, which are often used to describe the propagation of fully nonlinear waves in coastal oceanography. We prove that, under the long-wavelength, small-amplitude approximation, the formal Korteweg–de Vries (KdV) equation for the GN equations is mathematically valid in the time interval for which the KdV dynamics survive. The main idea in the proof is to apply the Gardner–Morikawa transform, the reductive perturbation method, and some error energy estimates. The main novelties of this paper are the construction of valid approximate solutions of the GN equations with respect to the small wave amplitude parameter and global uniform energy estimates for the error system.</p></abstract>

Hamiltonian description of internal ocean waves with Coriolis force

<p style="text-indent:20px;">The interfacial internal waves are formed at the pycnocline or thermocline in the ocean and are influenced by the Coriolis force due to the Earth's rotation. A derivation of the model equations for the internal wave propagation taking into account the Coriolis effect is proposed. It is based on the Hamiltonian formulation of the internal wave dynamics in the irrotational case, appropriately extended to a <em>nearly</em> Hamiltonian formulation which incorporates the Coriolis forces. Two propagation regimes are examined, the long-wave and the intermediate long-wave propagation with a small amplitude approximation for certain geophysical scales of the physical variables. The obtained models are of the type of the well-known Ostrovsky equation and describe the wave propagation over the two spatial horizontal dimensions of the ocean surface.</p>

The existence and propagation of dust acoustic waves in quantum four-component plasma

The existence and propagation of dust acoustic waves in quantum four-component plasma are investigated by employing Sagdeev pseudo-potential. The plasma under consideration consists of both quantum electrons and ions beside negative and positive dust particulates. The general determining equation of the Sagdeev pseudo-potential is obtained, and it is found that the equation is very difficult to integrate. Due to the complexity involved in the structure of Sagdeev potential, this equation can only be solved in some limiting cases, such as semi-classical limit and small amplitude approximation. Our findings support the existence of compressive, rarefactive solitary waves and double layers in our plasma system. The influence of the relevant plasma parameters on the existence and profile of the obtained quantum nonlinear waves has been studied. The implications of the obtained results have wide relevance in the study of space and laboratory quantum dusty plasmas situations such as Jupiter’s magnetosphere, upper and lower mesosphere, and comet tails, as well as in condensed matter systems.

Scattering of exchange spin waves from a helimagnetic layer sandwiched between two semi-infinite ferromagnetic media

We have calculated the scattering (reflection and transmission) coefficients of linear exchange spin waves normally incident upon a helimagnetic layer sandwiched between two semi-infinite ferromagnetic media. Our calculations show that, despite the helimagnetic order induced in the layer by the Dzyaloshinskii-Moriya interaction (DMI), the scattering is reciprocal and insensitive to the presence of the helimagnetic order in the layer. This comes as a result of the disappearance of the DMI from the boundary conditions in the considered geometry under the small-amplitude approximation and from the specific form of the nonreciprocity of the spin-wave dispersion relation in the helimagnetic material. We show that the helimagnetic layer's interfaces act as a system of two semicrossed polarizers for the circularly polarized spin waves incident from the ferromagnetic media. This results from the ellipticity of the magnetic precession induced by the easy-plane anisotropy in the helimagnetic layer. Our calculations also reveal the importance of evanescent solutions to correctly describe the spin-wave scattering in samples with elliptical precession. Our findings will aid development of magnonic devices containing helimagnetic constituents.

Navigation control of flagellated magnetic microswimmer by parametric excitation

The paper presents a novel mechanism for controlling swimming behavior of a flagellated microswimmer by parametric excitation. The microswimmer design consists of a spherical rigid magnetic head attached to a slender rigid link (tail) via torsional spring joint. The swimmer is actuated by an oscillating magnetic field that produces undulatory coplanar motion. The parametric excitation consists of harmonic modulation of elastic stiffness of the spring joint. The equation of motion is described by the balance of forces and torques at low Reynolds number. The analytical solutions were obtained by harmonic balance method under small amplitude approximation. It is found that the parametric excitation modifies the symmetry and amplitude of beating patterns of the tail. The influence of these on microswimmer performance (time-averaged displacement and propulsion velocity along longitudinal and transverse directions) is analyzed. A comparison with the performance of a microswimmer without parametric excitation shows that the propulsion velocity can be enhanced or suppressed by tuning the phase difference between parametric excitation and magnetic actuation. Also, not only the longitudinal propulsion but a simultaneous transverse propulsion is generated by the parametric excitation. This facilitates the navigational control of the microswimmer by parametric modulation.

Gravitoelectrostatic excitations in an opposite polarity complex plasma

The linear and nonlinear properties of gravitoelectrostatic mode, in a plasma system consisting of inertial opposite polarity charged dust grains as well as inertialess nonextensively q-distributed ions and electrons (including the effect of polarization force on the massively charged dust grains), have been investigated. A general dispersion relation has been derived yielding only one eigen wave mode. It is found that the polarization force and the nonextensively distributed ions effects play directly a destabilizing role in Jeans instability. Moreover, a new pair of gravitoelectrostatically coupled energy integral equation has been obtained by applying the Sagdeev pseudopotential technique. Also, a small-amplitude approximation is considered for the self-gravitational potential. It is reported that the fluctuations dynamics of the dust grains evolve self-gravitational rarefactive soliton pulses and electrostatic compressive soliton-like patterns. Applying the phase plane analysis, the phase portraits of the dynamical system have been presented and also the corresponding wave solutions. Our results could be applicable for different space and astrophysical plasma systems, particularly for dust molecular clouds of H-II region.

Movement of spherical colloid particles carried by flow in tubes of periodically varying diameter.

We provide analytical formulas for the movement of spherical particles in a corrugated tube, in the approximation of small amplitude of the tube diameter variation. We calculate how the particle is pushed toward the wall at some places and pulled off the wall at others. We show that this effect causes rectification of the particle movement, when the direction of the fluid flow is alternated, thus leading to the hydrodynamic ratchet effect. We propose such scheme as a particle-separation device.

Obliquely Propagating Electron Acoustic Shock Waves in Magnetized Plasma

Obliquely propagating electron acoustic shock waves in plasma with stationary ions, cold and superthermal hot electrons are investigated in magnetized plasma. Employing reductive perturbation method, Korteweg-de Vries-Burgers equation (KdVB) is derived in the small amplitude approximation limit. The analytical and numerical calculations of the KdVB equation show the variation of shock waves structure (amplitude, velocity, and width) with different plasma parameters. Particle density (α), superthermal parameter (κ), electron temperature ratio (𝜃), kinetic viscosity (η0), obliqueness (kz), and strength of magnetic field (ωc) significantly modify the properties of the shock waves structures. The present investigation is useful to understand dissipative structures observed in space or laboratory plasma where multielectrons population with superthermal electrons are prevalent.

Filament instability under constant loads

Buckling of semi-flexible filaments appears in different systems and scales. Some examples are: fibers in geophysical applications, microtubules in the cytoplasm of eukaryotic cells and deformation of polymers freely suspended in a flow. In these examples, instabilities arise when a system’s parameter exceeds a critical value, being the Euler force the most known. However, the complete time evolution and wavelength of buckling processes are not fully understood. In this work we solve analytically the time evolution of a filament under a constant compressive force in the small amplitude approximation. This gives an insight into the variable force scenario in terms of normal modes. The evolution is highly sensitive to the initial configuration and to the magnitude of the compressive load. This model can be a suitable approach to many different real situations.

Effect of electron temperature on small-amplitude electron acoustic solitary waves in non-planar geometry

Effects of electron temperature on the propagation of electron acoustic solitary waves in plasma with stationary ions, cold and superthermal hot electrons is investigated in non-planar geometry employing reductive perturbation method. Modified Korteweg–de Vries equation is derived in the small amplitude approximation limit. The analytical and numerical calculations of the KdV equation reveal that the phase velocity of the electron acoustic waves increases as one goes from planar to non planar geometry. It is shown that the electron temperature ratio changes the width and amplitude of the solitary waves and when electron temperature is not taken into account,our results completely agree with the results of Javidan & Pakzad (2012). It is found that at small values of \(\tau \), solitary wave structures behave differently in cylindrical (\(\text {m} = 1\)), spherical (\(\text {m} = 2\)) and planar geometry (\(\text {m} = 0\)) but looks similar at large values of \(\tau \). These results may be useful to understand the solitary wave characteristics in laboratory and space environments where the plasma have multiple temperature electrons.

Quantum and FLR effects on the Rayleigh Taylor instability of stratified plasmas

The combined effects of quantum corrections and finite Larmor radius (FLR) have been investigated on the linear hydrodynamic Rayleigh-Taylor (RT) instability of an incompressible stratified plasma. The basic quantum magnetohydrodynamic equations incorporating quantum and FLR effects are constructed and linearized under the small amplitude approximation. The general dispersion relation is derived considering appropriate magnetic field and density profiles using the normal mode analysis. It is analyzed numerically to study the effects of quantum and FLR corrections on the growth rate of RT instability in the stratified hydrodynamic fluids. It is found that the cut-off wavenumber (kmax) and critical wavenumber (kc) determining the growth rate of RT instability are modified due to the presence of FLR corrections and quantum effects. The FLR and quantum corrections have a stabilizing effect on the growth rate of RT instability of a stratified plasma.

Effect of tapering on elastic filament microswimming under planar body actuation

This paper is concerned with microswimming of elastic filaments at low Reynolds number. The microswimmer is composed of a rigid spherical head attached to an elastic filament tail of varying cross section (taper). The swimming is actuated by periodic planar oscillation of slope at the head–tail junction. The actuation mechanism is considered to be embedded in the head body sending flexural waves traveling down the filament length. This causes elastohydrodynamic interactions between the filament and the fluid producing a net propulsive thrust on the head. The swimming speed is calculated by balancing this thrust against the viscous drag on the swimmer head. The paper analyzes effects of filament tapering on the propulsive thrust, swimming speed and energy efficiency. Three types of taper functions (linear, quadratic and exponential) are analyzed within the small amplitude approximation of the governing equations assuming filament inextensibility. The simulation results are generated with respect to the variations of filament length, taper ratio and actuation frequency. The analysis concludes that the nonlinear taper (quadratic and exponential) enhances the performance of microswimmer by 1 to 2 orders of magnitude in comparison to the linearly tapered and the uniform cross section (non-tapered) filaments.

A Novel Envelope Soliton Solution to the Granular Crystal Model**Supported by the National Natural Science Foundation of China under Grant Nos. 1604121 and 11464012

A theoretical work on envelope solitons to a one-dimensional granular chain model is reported. In the small amplitude approximation, we analytically solve the equation of motion with the help of the semidiscrete multiple-scale method. Our results show that the granular chain model can support an asymmetric high-order envelope soliton under the certain condition. It is found that the second-harmonic term of this high-order envelope soliton has an additional phase. In addition, the influence of both the material parameter and the static load on the localized features of the high-order envelope soliton is discussed.

Non linear analysis of obliquely propagating spin electron acoustic wave in a partially spin polarized degenerate plasma

By employing the separated spin evolution quantum hydrodynamic model, non-linear evolution of obliquely propagating spin electron acoustic wave (SEAW) is presented. The solitary structures of SEAW is investigated through the Korteweg–de Vries (KdV) equation derived using reductive perturbation method. From the first order perturbations we derive the dispersion relation of SEAW and find that both the spin polarization and the propagation angle reduce the phase velocity while the electron streaming enhances it. Using small amplitude approximation, the solitary structure of SEAW is analyzed and the effects of spin polarization, propagation angle and electron streaming on the SEA soliton are studied. Our numerical results demonstrate that the spin polarization and the propagation angle play a balancing act on the soliton structures. The possible applications of our investigation to the astrophysical environments like white dwarfs is also discussed.

Obliquely propagating ion acoustic solitary structures in the presence of quantized magnetic field

The effect of linear and nonlinear propagation of electrostatic waves have been studied in degenerate magnetoplasma taking into account the effect of electron trapping and finite temperature with quantizing magnetic field. The formation of solitary structures has been investigated by employing the small amplitude approximation both for fully and partially degenerate quantum plasma. It is observed that the inclusion of quantizing magnetic field significantly affects the propagation characteristics of the solitary wave. Importantly, the Zakharov-Kuznetsov equation under consideration has been found to allow the formation of compressive solitary structures only. The present investigation may be beneficial to understand the propagation of nonlinear electrostatic structures in dense astrophysical environments such as those found in white dwarfs.

Nonlinear dissipative and dispersive electrostatic structures in unmagnetized relativistic electron-ion plasma with warm ions and trapped electrons

In this paper, we have investigated electrostatic solitary and shock waves in an unmagnetized relativistic electron-ion (ei) plasma in the presence of warm ions and trapped electrons. In this regard, we have derived the trapped Korteweg-de Vries Burgers (TKdVB) equation using the small amplitude approximation method, which to the best of our knowledge has not been investigated in plasmas. Since the TKdVB equation involves fractional nonlinearity on account of trapped electrons, we have employed a smartly crafted extension of the tangent hyperbolic method and presented the solution of the TKdVB equation in this paper. The limiting cases of the TKdVB equation yield trapped Burgers (TB) and trapped Korteweg-de Vries (TKdV) equations. We have also presented the solutions of TB and TKdV equations. We have also explored how the plasma parameters affect the propagation characteristics of the nonlinear structures obtained for these modified nonlinear partial differential equations. We hope that the present work will open new vistas of research in the nonlinear plasma theory both in classical and quantum plasmas.

On the modulation of ionic velocity in electron–positron–ion plasmas

Ion-acoustic structures in electron–positron–ion (e-p-i) plasma are examined by algebraic analysis of model equations. The dynamic of solitary waves are studied through direct approach. The system energy is obtained in terms of velocity of ions. It is founded that, ion-acoustic solitary waves existence are sensitive to densities and temperatures ratio of electrons and positrons. A small but finite amplitude approximation for soliton and shock waves are studied. The effect of positron parameters on solitonic wave and shocks are also inspected.