Reductive Theory Research Articles

Dent patterns on the surface of a longitudinally compressed, non-linear elastic cylindrical shell

A novel version of reductive perturbation theory is proposed for analysing the dynamics of bends in a longitudinally compressed, non-linear elastic cylindrical shell near the stability threshold given by the linear theory. Soliton-like annular folds and patterns of diamond-shaped dents on the shell surface are predicted and analytically described. Similar formations, which are both stress concentrators and precursors of plastic flow of the material, contain information on the precritical stress state of the shell. It is shown that a shell with dents supports an external load, which is tens of percent less than the upper critical load in the linear theory of shells. The conditions for the formation of and explicit expressions for solitary waves that propagate along the generatrix of the shell on a background of arrays of folds and dents are found.

Korteweg-de Vries description of Helmholtz–Kerr dark solitons

A wide variety of different physical systems can be described by a relatively small set of universal equations. For example, small-amplitude nonlinear Schrodinger dark solitons can be described by a Korteweg-de Vries (KdV) equation. Reductive perturbation theory, based on linear boosts and Gallilean transformations, is often employed to establish connections to and between such universal equations. Here, a novel analytical approach reveals that the evolution of small-amplitude Helmholtz–Kerr dark solitons is also governed by a KdV equation. This broadens the class of nonlinear systems that are known to possess KdV soliton solutions, and provides a framework for perturbative analyses when propagation angles are not negligibly small. The derivation of this KdV equation involves an element that appears new to weakly nonlinear analyses, since transformations are required to preserve the rotational symmetry inherent to Helmholtz-type equations.

Zakharov–Kuznetsov–Burgers equation for dust ion acoustic waves

The nonlinear wave structures of small, but finite amplitude dust ion acoustic waves in a magnetized dusty plasma consisting of cold positive ions, isothermal electrons and variable charged stationary dust particles are investigated using reductive perturbation theory. The basic set of fluid equations is reduced to Zakharov–Kuznetsov–Burgers (ZKB) equation. The presence of charging process give rise to three cases. The first case arises when the charging process lead to originate anomalous dissipation, which makes possible existence of a new kind of shocks related to this dissipation. Case two, in the absence of dissipation (or if the dissipation is weak) the balance is then between nonlinear and dispersion effects, which can result in the formation of a symmetrical solitary waves. Case three considers the dissipation and dispersion at the same footing, i.e. we cannot neglect either dissipation or dispersion. Exact solution of the ZKB equation is obtained, for the first time, using a improved modified extended tanh-function method. Then, all possible cases of ZKB equation are covered.

Effects of two-temperature ions, magnetic field, and higher-order nonlinearity on the existence and stability of dust-acoustic solitary waves in Saturn’s F ring

Nonlinear propagation of dust-acoustic solitary waves (DASWs) in a strong magnetized dusty plasma comprising warm adiabatic variable-charged dust particles, isothermal electrons, and two-temperature ions is investigated. Applying a reductive perturbation theory, a nonlinear Zakharov-Kuznetsov (ZK) equation for the first-order perturbed potential and a linear inhomogeneous ZK-type equation for the second-order perturbed potential are derived. However, at a certain value of high-temperature ion density, the coefficient of the nonlinear terms of both ZK and ZK-type equations vanishes. Therefore, a new set of expansion physical parameters and stretched coordinates are then used to derive a modified Zakharov-Kuznetsov (mZK) equation for the first-order perturbed potential and a mZK-type equation for the second-order perturbed potential. Stationary solutions of these equations are obtained using a renormalization method. A condition for two-temperature ions assumption is examined for various cosmic dust-laden plasma systems. It is found that this condition is satisfied for Saturn’s F ring. The effects of two-temperature ions, magnetic field, and higher-order nonlinearity on the behavior of the DASWs are discussed. To obtain the stability condition of the waves, a method based on energy consideration is used and the condition for stable solitons is derived.

Making It Totally Explicit

This paper begins by isolating the reductive component of Brandom's inferentialism. In order to assess to what extent that reductive component is supported by the considerations Brandom marshals in its defense, I assess the comparative degree of support those considerations provide a non-reductive counterpart of Brandom's original, reductive theory. One of the central claims here is that once the reductive and non-reductive theories are placed side-by-side, it is clear that, save one, all of the considerations Brandom marshals in defense of inferentialism equally well support its non-reductive counterpart. This shows that those considerations offer no support for the reductions at inferentialism's heart. What the considerations raised here ultimately show is that Brandom's defense of the reductive core of his theory ultimately rests on the simple fact that it has a certain feature, namely, that it is reductive in the sense reserved here. I close with a brief discussion of some advantages that some reductive theories have over non-reductive ones, but show how none of these advantages are had by Brandom's theory in particular.

Nonlinear electron-acoustic waves with vortex-like electron distribution and electron beam in a strongly magnetized plasma

Abstract The nonlinear properties of solitary waves structure in a strongly magnetized plasma consisting of cold electron fluid, hot electrons obeying trapped/vortex-like distribution, warm electron beam, and stationary ions, are investigated. Applying a reductive perturbation theory (RPT), a nonlinear modified Korteweg–de Vries (mKdV) equation and a linear inhomogeneous mKdV-type equation are derived. The stationary solutions of these coupled equations are obtained using a renormalization method. However, both of the mKdV- and mKdV-type solutions present a positive potential, that corresponds to a hole (hump) in the cold (hot) electron number density. The dependence of the wave amplitude, width and velocity on the system parameters is investigated. The application of our results in interpreting the salient features of the broadband electrostatic noise (BEN) in the dayside auroral zone of the Earth’s magnetosphere is discussed.

Dust-acoustic solitary waves in an inhomogeneous magnetized hot dusty plasma with dust charge fluctuations

Using a standard reductive perturbation theory, a Zakharov-Kuznnetsov (ZK) equation is derived in an inhomogeneous dusty plasma comprised of negatively charged dust grains of equal radii, Boltzmann distributed electrons and positive ions. The effects of dust thermal pressure, dust charge variation, dust-neutral collision, and the external magnetic field are taken into account. Either compressive or rarefective solitons are shown to exist depending on the critical value of the phase velocity which in turn, depends on the dust temperature and pressure. An analytic soliton solution is obtained and discussed. The behaviors of the soliton amplitude, width, and the Mach number are investigated numerically for different plasma parameters.

Effect of two-temperature trapped electrons to nonlinear dust-ion-acoustic solitons

Propagation of three-dimensional dust-ion-acoustic solitons is investigated in a dusty plasma consisting of positive ions, negatively variable-charged dust particles, and two-temperature trapped electrons. We use the reductive perturbation theory to reduce the basic set of fluid equations to one evolution equation called damped modified Kadontsev-Petviashivili equation. Exact solution of this equation is not possible, so we obtain the time evolution solitary wave form approximate solution. It is found that only compressive soliton can propagate in this system. We develop a theoretical estimate condition under which the solitons can propagate. It is found that this condition is satisfied for Saturn’s F ring. It is found also that low electron temperature has a role on the behavior of the soliton width, i.e., for lower (higher) range of low electron temperature the soliton width decreases (increases). However, high electron temperature decreases the width. The trapped electrons have no effect on the soliton width. The ratio of free low (high) to trapped low (high) electron temperatures increases the soliton amplitude. Also, the amplitude increases with free low and free high electron temperatures. To investigate the stabilty of the waves, we used a method based on energy consideration to obtain a condition for stable solitons. It is found that this condition depends on dust charge variation, streaming velocity, directional cosine of the wave vector k along the x axis, and temperatures of dust particles, ions, and free electrons.

Dust–ion–acoustic solitons and shocks in dusty plasmas

Abstract Propagation of nonlinear dust–ion–acoustic waves (DIAWs) has been investigated considering the effects of charge fluctuation dynamics of stationary dust particles, transverse perturbations, ion streaming velocity and ions to electrons temperature ratio. By using the reductive perturbation theory, Kadomtsev–Petviashivili–Burgers (KPB) equation is derived. It is found that both solitons or shocks may appear in the system depending on the anomalous dissipation from dust particles charging process, whereas transverse perturbations, ion streaming velocity and ions to electrons temperature ratio are responsible for the stability of the solitons.

Higher-order contribution to obliquely nonlinear electron-acoustic waves with electron beam in a magnetized plasma

Propagation of electron-acoustic waves in a strongly magnetized four-component plasma consisting of cold and hot electrons, a warm electron beam, and stationary ions is investigated. The present model considered weakly dispersive and strongly magnetized plasma in the limit of long wavelengths. The introduction of an electron beam allows the existence of electron-acoustic solitons with velocity related to the beam velocity. With increasing the beam velocity and the beam temperature, both the soliton amplitude and the width increase. Applying a reductive perturbation theory, a nonlinear Zakharov-Kuznetsov (ZK) equation for the first-order perturbed potential and a linear inhomogeneous Zakharov-Kuznetsov (ZK-type) equation for the second-order perturbed potential are derived. Stationary solutions of these coupled equations are obtained using a renormalization method. These solutions admit either compressive or rarefactive soliton type depending on the electron-beam parameters. Moreover, the dependence of the solution on the beam parameters, obliqueness on the magnetic field, and the magnetic field itself is also investigated. The application of the present investigation to the broadband electrostatic noise in the dayside auroral zone of the Earth’s magnetosphere is considered.

Dust-acoustic solitary waves and double layers in a magnetized dusty plasma with nonthermal ions and dust charge variation

The effect of nonthermal ions and variable dust charge on small-amplitude nonlinear dust-acoustic (DA) waves is investigated. It is found that both compressive and rarefactive solitons exist and depend on the nonthermal parameter a. Using a reductive perturbation theory, a Zakharov–Kuznetsov (ZK) equation is derived. At critical value of a, ac, a modified ZK equation with third- and fourth-order nonlinearities, is obtained. Depending on a, the solution of the evolution equation reveals whether there is coexistence of both compressive and rarefactive solitary waves or double layers (DLs) with the possibility of their two kinds. In addition, for certain plasma parameters, the solitary wave disappears and a DL is expected. The variation of dust charge number, wave velocity, and soliton amplitude and its width against system parameters is investigated for the DA solitary waves. It is shown that the incorporation of both the adiabatic dust-charge variation and the nonthermal distributed ions modifies significantly the nature of DA solitary waves and DA DLs. The findings of this investigation may be useful in understanding the ion acceleration mechanisms close to the Moon and also enhances our knowledge on pickup ions around unmagnetized bodies, such as comets, Mars, and Venus.

Critical density solitary waves structures in a hot magnetized dusty plasma with vortexlike ion distribution in phase space

The nonlinear properties of solitary waves structures in a hot magnetized dusty plasma consisting of isothermal hot electrons, nonisothermal ions, and high negatively charged massive dust grains are reported. A modified Korteweg–de Vries (modified KdV) equation, which admits a solitary waves solution, for small but finite amplitude, is derived using a reductive perturbation theory. A nonisothermal ion distribution provides the possibility of existence of rarefactive solitary waves. On the other hand, the dynamics of solitary waves at a critical ion density is governed by KdV equation. The modification in the amplitude and width of the solitary waves structures due to the inclusion of obliqueness and external magnetic field are also investigated.

Higher-order nonlinearity of electron-acoustic solitary waves with vortex-like electron distribution and electron beam

The nonlinear wave structure of small-amplitude electron-acoustic solitary waves (EASWs) is investigated in a four-component plasma consisting of cold electron fluid, hot electrons obeying vortex-like distribution traversed by a warm electron beam and stationary ions. The streaming velocity of the beam, uo, plays the dominant role in determining the roots of the linear dispersion relation associated with the system. Using the reductive perturbation theory, the basic set of equations is reduced to a modified Korteweg–de Vries (mKdV) equation. With the inclusion of higher-order nonlinearity, a linear inhomogeneous mKdV type equation with fifth-order dispersion term is derived and the higher-order solution is obtained using a renormalization method. However, both mKdV and mKdV-type solutions present a positive potential, which corresponds to a hole (hump) in the cold (hot) electron number density. The mKdV-type solution has a smaller energy amplitude and a wider width than that of mKdV solution. The dependence of the energy amplitude, the width, and the velocity on the system parameters is investigated. The findings of this investigation are used to interpret the electrostatic solitary waves observed by the Geotail spacecraft in the plasma sheet boundary layer of the Earth’s magnetosphere.

Dust-ion-acoustic solitary waves in a hot magnetized dusty plasma with charge fluctuations

The effects of dust grain charge fluctuation, dust temperature, ion temperature, obliqueness and external magnetic field on the nonlinear properties of dust-ion-acoustic (DIA) solitary waves are studied in a hot magnetized dusty plasma consisting of a negatively charged, extremely massive hot dust fluid, positively charged hot ion fluid and vortex-like distributed electrons. A modified Korteweg de Vries equation (mKdV), which admits a solitary wave solution for small but finite amplitude is derived using a reductive perturbation theory. The numerical results are applied to Saturn’s F-ring.

Higher order solution of the dust ion acoustic solitons in a warm dusty plasma with vortex-like electron distribution

Abstract The ratios of dust to free electron and free to trapped electron temperatures are examined in warm dusty plasmas with vortex-like electron distribution through the derivation of a modified Korteweg–de Vries (MKdV) equation using a reductive perturbation theory. As the wave amplitude increases, the width and velocity of the soliton deviate from the prediction of the MKdV equation, i.e., the breakdown of the MKdV approximation. To describe the soliton of larger amplitude, the MKdV equation with the fifth-order dispersion term is employed and its higher-order solutions are obtained.

On the higher-order solution of the dust-acoustic solitary waves in a warm magnetized dusty plasma with dust charge variation

The higher-order contribution in reductive perturbation theory is studied for small-butfinite-amplitude dust-acoustic solitary waves in warm magnetized three-component dusty plasmas comprised of variational charged dust grains, isothermal ions, and electrons. The basic set of fluid equations is reduced to the Zakharov–Kuznetsov equation for the first-order perturbed potential and a linear inhomogeneous Zakharov–Kuznetsov-type equation for the second-order perturbed potential. Stationary solutions of both equations are obtained using a renormalization method. The effects of the higher-order contribution, external magnetic field, dust charge variation, dust grain temperature, ratios of temperature and density of positive ions to electrons, and directional cosine of the wave vector k along the x axis on the nature of the solitary waves are investigated.

Nonlinear Evolution Equations for Parallel Electromagnetic Multispecies Plasma Modes: Reductive Perturbation Theory in the Linear Wave Frame and Electric Field Generation

A variant of the reductive perturbation theory is introduced, describing nonlinear phenomena in a frame moving with the linear phase velocity. The stretching is simpler and less terms are to be carried in expanding the basic equations to different orders, but the method only works well when the displacement current can be neglected, for plasmas that are not too strongly magnetized. Within this framework, the derivative nonlinear Schrödinger (DNLS) equation for parallel propagating electromagnetic waves has been (re)derived in ordinary multi-ion plasmas, with due attention to the stage at which a parallel electric field is generated and charge neutrality no longer holds. A simple reasoning shows that the DNLS equation cannot have stationary solitary waves, only envelope solitons. For critical plasma compositions the coefficient of the dispersive term in the DNLS vanishes, necessitating a different stretching and leading to a vector modified Korteweg-de Vries (VmKdV) equation. The latter is encountered not only in electron-positron plasmas, but other plasmas at critical densities are possible, e.g. dusty plasmas. Although the VmKdV equation is nonintegrable, in contrast to the DNLS equation, it can have stationary solitary solutions with linearly polarized magnetic fields.

Obliquely propagating electrostatistic solitary structures in a hot magnetized dusty plasma with vortex-like electron distribution

The nonlinear properties of solitary waves structure in a hot magnetized dusty plasma consisting of a negatively charged, extremely massive hot dust fluid, positively charged hot ion fluid and vortex-like distributed electrons, are reported. A modified Korteweg de Vries equation (mKdV) which admits a solitary wave solution for small but finite amplitude is derived using a reductive perturbation theory. The modifications in the amplitude and width of the solitary wave structures due to the inclusion of an external magnetic field and dust and ions temperature are investigated.

Modulational instability of a weakly relativistic ion acoustic wave in a warm plasma with nonthermal electrons

An investigation has been made of modulational instability of a nonlinear ion acoustic wave in a weakly relativistic warm unmagnetized nonthermal plasma whose constituents are an inertial ion fluid and nonthermally distributed electrons. Up to the second order of the perturbation theory, a nonlinear Schrodinger type (NST) equation for the complex amplitude of the perturbed ion density is obtained. The coefficients of this equation show that the relativistic effect, the finite ion temperature and the nonthermal electrons modify the condition of the modulational stability. The association between the small-wavenumber limit of the NST equation and the oscillatory solution of the Korteweg-de Varies equation, obtained by a reductive perturbation theory, is satisfied.

Bright–Dark Vector Solitons in Chiral Media

We study wave propagation in nonlinear chiral media. In the weak dispersion, low chirality limit the dynamics of the waves are reduced to the dynamics of a set of coupled partial differential equations of the NLS type. Using reductive perturbation theory we show the existence of bright–dark chiral vector solitons.