Kepler’s celestial two-body equation: a second attempt to establish a continuous and integrable solution via the <Emphasis Type="Italic">BPES</Emphasis>: Boubaker Polynomials Expansion Scheme

The propagation of nonlinear waves in warm dusty plasmas with variable dust charge, two-temperature ions, and nonthermal electrons is studied. By using the reductive perturbation theory, the Kadomtsev–Petviashivili (KP) equation is derived. The energy of the soliton has been calculated. By using standard normal modes analysis a linear dispersion relation has been obtained. The effects of variable dust charge on the energy of the soliton and the angular frequency of the linear wave are also discussed. It is shown that the amplitude of solitary waves of the KP equation diverges at the critical values of plasma parameters. We derive solitons of a modified KP equation with finite amplitude in this situation.     

Bifurcations of dust ion acoustic travelling waves in a magnetized quantum dusty plasma

Bifurcation behavior of nonlinear dust ion acoustic travelling waves in a magnetized quantum dusty plasma has been studied. Applying the reductive perturbation technique (RPT), we have derived a Kadomtsev-Petviashili (KP) equation for dust ion acoustic waves (DIAWs) in a magnetized quantum dusty plasma. By using the bifurcation theory of planar dynamical systems to the KP equation, we have proved that our model has solitary wave solutions and periodic travelling wave solutions. We have derived two exact explicit solutions of the above travelling waves depending on different parameters.     

Effect of nonextensive electron and ion on dust acoustic rogue waves in dusty plasma of opposite polarity

Propagation of nonlinear dust-acoustic waves in a magnetized collisionless plasma having positively, negatively charged dust grains and nonextensive distributed electrons and ions has been investigated. A reductive perturbation method is used to obtain a nonlinear Korteweg-de Vries (KdV) equation describing the model. The dynamics of the modulational instability gives rise to the formation of rogue waves that is described by a nonlinear Schrödinger equation. The dependence of rogue waves profiles on positive and negative charged dust cyclotron frequencies, nonextensive parameters of electrons and ions is investigated numerically. The result of the present investigation may be applicable to some plasma environments, such as cometary tails and upper mesosphere.     

Investigation of non-isothermal electron effects on the dust acoustic waves in four components dusty plasma

The time fractional modified KdV, the so-called TFMKdV equation is solved to study the nonlinear propagation of the dust acoustic (DA) solitary waves in un-magnetized four components dusty plasma. This plasma consists of positively charged warm adiabatic dust, negatively charged cold dust, non-isothermal electrons and Maxwellian ions. The TFMKdV equation is derived by using semi-inverse and Agrawal’s method and solved by the Laplace Adomian decomposition method (LADM). The effects of the time fractional order (β), the ratio of dust to ion temperature (δ _d ), the time (τ), the mass and charge ratio (α), the non-isothermal parameter (γ) and wave velocity (v) on the DA solitary wave are studied. Our results show that the variations of the amplitude of DA solitary wave versus (γ) are in agreement with the results obtained previously. Moreover, the time fractional order plays a role of higher order perturbation in modulating the soliton shape. The achievements of this research for the DA solitary waves may be applicable in space plasma environments and laboratory plasmas.     

Formation of obliquely propagating dust-ion-acoustic shock waves due to dust charge fluctuation in magnetized nonthermal dusty plasma

A theoretical investigation is made on the formation as well as basic properties of dust-ion-acoustic (DIA) shock waves in a magnetized nonthermal dusty plasma consisting of immobile charge fluctuating dust, inertial ion fluid and nonthermal electrons. The reductive perturbation method is employed to derive the Korteweg-de Vries-Burgers equation governing the DIA shock waves. The combined effects of external static magnetic field, obliqueness, nonthermal electron distribution and dust charge fluctuation on the DIA shock waves are also investigated. It is shown that the dust charge fluctuation is a source of dissipation, and is responsible for the formation of the DIA shock waves. It is also observed that the combined effects of obliqueness, nonthermal electron distribution and dust charge fluctuation significantly modify the basic properties of the DIA shock waves. The implications of our results in space and laboratory dusty plasma situations are briefly discussed.     

Propagation of two dimensional cylindrical fast magnetoacoustic solitary waves in a warm dust plasma

A set of multi-fluid equations and Maxwell’s equations are carried out to investigate the properties of nonlinear fast magnetoacoustic solitary waves with the combined effects of dusty plasma pressure and transverse perturbation in the bounded cylindrical geometry. The reductive perturbation method has been applied to the dynamical system causeway and the derived two dimensional cylindrical Kadomtsev–Petviashvili equation (CKP) predicts different natures of solitons in complex plasma. Under a suitable coordinate transformation the CKP equation can be solved analytically. The change in the soliton structure due to mass of dust, ion temperature, ion density, and dust temperature is studied by numerical calculation of the CKP equation. It is noted that the dust cylindrical fast magnetoacoustic solitary waves in warm plasmas may disappear slowly because of an increase in dust mass. The present analysis could be helpful for understanding the nonlinear ion-acoustic solitary waves propagating in interstellar medium and pulsar wind,which contain an excess of superthermal particles.     

Effect of electron nonextensivity on oblique propagation of arbitrary ion acoustic waves in a magnetized plasma

The combined effects of the obliqueness and nonextensive electrons are incorporated in the study of ion acoustic (IA) waves in a magnetized plasma. The propagation properties of two possible modes (in the linear regime) are investigated. It is found that the electron nonextensivity decreases the phase velocities of both two modes. Also obliqueness leads to increase of separation between two modes. The nonlinear evolution of IA solitary waves is governed by an energy-like equation. The influence of electron nonextensivity, obliqueness and electron population on the existence domain of solitary waves and the soliton characteristics are examined. It is shown that the existence domain of the IA soliton and its profile is significantly depended on the deviation of electrons from thermodynamic equilibrium and obliqueness. Interestingly, the present model supports compressive as well as rarefactive IA solitary waves. Our finding should elucidate the nonlinear electrostatic structures that propagate in astrophysical and cosmological plasma scenarios where nonextensive and magnetized plasma can exist; like instellar plasma stellar polytropes, solar neutrino problem, peculiar velocities of galaxy clusters, dark-matter halos, protoneutron stars, hadronic matter, quark-gluon plasma, and magnetosphere, etc.     

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. This revised version was published online in July 2006 with corrections to the Cover Date.     

Higher-order Zakharov–Kuznetsov equation for dust-acoustic solitary waves with dust size distribution

Nonlinear propagating dust-acoustic solitary waves (DASWs) in a warm magnetized dusty plasma containing different size and mass negatively charged dust particles, isothermal electrons, high- and low-temperature ions are investigated. For this purpose, a reasonable normalization of the hydrodynamic and Poisson equations is used to derive the Zakharov–Kuznetsov (ZK) equation for the first-order perturbed potential. As the wave amplitude increases, the width and the velocity of the solitons deviate from the prediction of the ZK equation, i.e., the breakdown of the ZK approximation. To describe the soliton of larger amplitude, a linear inhomogeneous Zakharov-Kuznetsov-type (ZK-type) equation for the second-order perturbed potential is derived. Stationary solutions of both equations are obtained using the renormalization method. Numerically, the effect of power law distribution on the higher-order corrections is examined. It is found that the soliton amplitude in case of power law distribution is smaller than that of monosized dust grains. The higher-order corrections play a role to reduce the strength of the nonlinearity for power law distribution case. The relevance of the present investigation to Saturn's F-ring and laboratory experiment is discussed.     

Dust ion acoustic travelling waves in the framework of a modified Kadomtsev-Petviashvili equation in a magnetized dusty plasma with superthermal electrons

For the critical values of the parameters q and V, the work (Samanta et al. in Phys. Plasma 20:022111, 2013b) is unable to describe the nonlinear wave features in magnetized dusty plasma with superthermal electrons. To describe the nonlinear wave features for critical values of the parameters q and V, we extend the work (Samanta et al. in Phys. Plasma 20:022111, 2013b). To extend the work, we derive the modified Kadomtsev-Petviashvili (MKP) equation for dust ion acoustic waves in a magnetized dusty plasma with q-nonextensive velocity distributed electrons by considering higher order coefficients of ?. By applying the bifurcation theory of planar dynamical systems to this MKP equation, the existence of solitary wave solutions of both types rarefactive and compressive, periodic travelling wave solutions and kink and anti-kink wave solutions is proved. Three exact solutions of these above waves are determined. The present study could be helpful for understanding the nonlinear travelling waves propagating in mercury, solar wind, Saturn and in magnetosphere of the Earth.     

Effects of dust charge fluctuations and deviations from isothermality of electrons on nonlinear dust ion-acoustic waves

The effects of dust charge fluctuations and deviations from isothermality of electrons are incorporated in the study of nonlinear dust ion-acoustic waves. Deviations from isothermality of electrons are included in this model as a result of nonlinear resonant interaction of the electrostatic wave potential with electrons during its evolution. The basic properties of stationary structures are studied by employing the reductive perturbation method, and conditions for the formation of small but finite amplitude dust ion-acoustic solitary waves in the space dusty plasma situations are clearly explained. It is shown that a more depletion of the background free electrons owing to the attachment of these electrons to the surface of the dust grains during the charging process can lead to the formation of solitary waves with smaller amplitude. Furthermore, effects of the dust charge fluctuation and deviations from isothermality of electrons show a non-uniform behavior for the amplitude of solitary waves in transition from the Boltzmann electron distribution to a trapped electron one. It is also found that the dust charge fluctuation caused by trapped as well as free electrons is a source of dissipation, and is responsible for the formation of the dust ion-acoustic shock waves.     

Higher order corrections to dust-acoustic ZK-solitons in a magnetized quantum dusty plasma

Nonlinear propagation of two dimensional dust-acoustic solitary waves in a magnetized quantum dusty plasma whose constituents are electrons, ions, and negatively charged heavy dust particles are investigated using quantum hydrodynamic model. The Zakharov-Kuznetsov (ZK) equation is derived by using reductive perturbation technique (RPT). The higher order inhomogeneous ZK-type differential equation is obtained for the correction to ZK- soliton. The dynamical equation for dressed soliton is solved by using renormalization method. The effects of obliqueness (l _x ) of the wave vector, magnetic field strength (B ₀), quantum parameter for ions (H _i ), soliton velocity (θ) and Fermi temperature ratio (σ) on amplitudes and widths of the ZK-soliton and as well as of the dressed soliton are investigated. The conditions for the validity of the higher order correction are described. Suitable parameter ranges for the existence of compressive and rarefactive dressed solitons are also discussed.     

Dust-ion-acoustic solitary waves and their multi-dimensional instability in a magnetized nonthermal dusty electronegative plasma

A rigorous theoretical investigation has been made on multi-dimensional instability of obliquely propagating electrostatic dust-ion-acoustic (DIA) solitary structures in a magnetized dusty electronegative plasma which consists of Boltzmann electrons, nonthermal negative ions, cold mobile positive ions, and arbitrarily charged stationary dust. The Zakharov-Kuznetsov (ZK) equation is derived by the reductive perturbation method, and its solitary wave solution is analyzed for the study of the DIA solitary structures, which are found to exist in such a dusty plasma. The multi-dimensional instability of these solitary structures is also studied by the small-k (long wave-length plane wave) perturbation expansion technique. The combined effects of the external magnetic field, obliqueness, and nonthermal distribution of negative ions, which are found to significantly modify the basic properties of small but finite-amplitude DIA solitary waves, are examined. The external magnetic field and the propagation directions of both the nonlinear waves and their perturbation modes are found to play a very important role in changing the instability criterion and the growth rate of the unstable DIA solitary waves. The basic features (viz. speed, amplitude, width, instability, etc.) and the underlying physics of the DIA solitary waves, which are relevant to many astrophysical situations (especially, auroral plasma, Saturn’s E-ring and F-ring, Halley’s comet, etc.) and laboratory dusty plasma situations, are briefly discussed.     

Dust-ion-acoustic solitary waves and their multi-dimensional instability in a magnetized dusty electronegative plasma with trapped negative ions

A theoretical investigation has been made on obliquely propagating dust-ion-acoustic solitary waves (DIASWs) in magnetized dusty electronegative plasma containing Boltzmann electrons, trapped negative ions, cold mobile positive ions, and arbitrarily charged stationary dust. The reductive perturbation method has been employed to derive the modified Zakharov-Kuznetsov (MZK) equation which admits solitary wave solution under certain conditions. The multi-dimensional instability of these solitary waves is also studied by the small-k (long wavelength plane wave) perturbation-expansion technique. The basic properties (speed, amplitude, width, instability, etc.) of small but finite amplitude DIASWs are significantly modified by the effects of external magnetic field, obliqueness, polarity of dust, and trapped negative ions. The implications of our results in space and laboratory plasmas are briefly discussed.     

Compressive and Rarefactive Dust-acoustic Solitary Structures in a Magnetized Two-ion-temperature Dusty Plasma

A rigorous theoretical investigation has been made of obliquely propagating dust-acoustic solitary structures in a cold magnetized two-ion-temperature dusty plasma consisting of a negatively charged, extremely massive, cold dust fluid and ions of two different temperatures. The reductive perturbation method has been employed to derive the Korteweg-de Vries (K-dV) equation which admits a solitary wave solution for small but finite amplitude limit. It has been shown that the presence of second component of ions modifies the nature of dust-acoustic solitary structures and may allow rarefactive dust-acoustic solitary waves (solitary waves with density dip) to exist in such a dusty plasma system. The effects of obliqueness and external magnetic field on the properties of these dust-acoustic solitary structures are also briefly discussed.     

Planar and nonplanar quantum dust ion-acoustic Gardner double layers in multi-ion dusty plasma

The longitudinal fast solitary waves induced by weakly relativistic positron showers of astrophysical origin are studied in a plasma system contaminated with some massive impurities in presence of superthermal effects. The superthermal effects are due to the high energy electrons. The impurities are dust corpuscles with positive and negative charges. It is noticed that increase in the kappa parameter of electrons and relativistic streaming factor of weakly relativistic positron shower, negative dust concentration invoke an enhancement in the strength of solitary wave. On the other hand increase in the shower’s temperature as well as positive dust concentration diminish the solitary hump strength. It is worth to mention that only hump type compressive fast solitary waves are predicted by our model, for the given set of plasma parameters, because the convective coefficient of the nonlinear governing equation for solitary wave remains positive in considered regime of interaction for plasma and positron shower. Our calculations in linear regime predict both the fast and slow positron shower induced longitudinal, electrostatic perturbations. Our results may be of importance in understanding the nonlinear propagation of waves in doped astrophysical superthermal plasmas with relativistic positron showers.     

Arbitrary amplitude electron acoustic waves in a magnetized nonextensive plasma

Arbitrary amplitude electron acoustic (EA) solitary waves in a magnetized nonextensive plasma comprising of cool fluid electrons, hot nonextensive electrons, and immobile ions are investigated. The linear dispersion properties of EA waves are discussed. We find that the electron nonextensivity reduces the phase velocities of both modes in the linear regime: similarly the nonextensive electron population leads to decrease of the EA wave frequency. The Sagdeev pseudopotential analysis shows that an energy-like equation describes the nonlinear evolution of EA solitary waves in the present model. The effects of the obliqueness, electron nonextensivity, hot electron temperature, and electron population are incorporated in the study of the existence domain of solitary waves and the soliton characteristics. It is shown that the boundary values of the permitted Mach number decreases with the nonextensive electron population, as well as with the electron nonextensivity index, q. It is also found that an increase in the electron nonextensivity index results in an increase of the soliton amplitude. A comparison with the Vikong Satellite observations in the dayside auroral zone is also taken into account.     

Fully nonlinear ion-acoustic solitary waves in a warm magnetized plasma with electron inertia

Ion-acoustic solitary waves in a warm, magnetized plasma with electron inertia have been investigated through Sagdeev pseudopotential method. It has been established the existence of both compressive supersonic solitons, and rarefactive subsonic and supersonic solitons within the parametric domains. The effect of the external magnetic field for generation of the supersonic compressive solitons of constant amplitudes appears to be passive after some critical direction of propagation of the wave. However, up to the critical direction of propagation, the magnetic resistance is found to be quite active to drastically reduce the soliton amplitudes. The generation of rarefactive solitons in this warm magnetized plasma is rather more feasible to be supersonic without electron inertia.     

Nonlinear propagation of dust-acoustic waves in a magnetized nonextensive dusty plasma

A theoretical investigation has been made of obliquely propagating dust-acoustic solitary waves in a magnetized three-component dusty plasma, which consists of a negatively charged dust fluid, ions, and nonextensive electrons. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation which admits a solitary wave solution. It has been shown that the combined effects of external magnetic field (obliqueness), ions, and electron nonextensivity change the behavior of these electrostatic solitary structures that have been found to exist with positive and negative potential in this dusty plasma model. The implications of our results in astrophysical and cosmological scenarios like vicinity of the Moon, magnetospheres of Jupiter and Saturn, dark-matter halos, hadronic matter, quark-gluon plasma, protoneutron stars, stellar polytropes etc. have been mentioned.     

Modulational instability of nonlinear waves in the relativistic plasma with account of the nonlinear Landau damping

The modulational instability of the weakly nonlinear longitudinal Langmuir as well as the transverse electromagnetic waves, propagation in the relativistic plasma without the static fields is described. The nonlinear Schrödinger equation taking account of the nonlinear Landau damping for these waves has been derived by means of the relativistic Vlasov and Maxwell equations. The plasma with the weakly relativistic temperature and that with an ultrarelativistic one has been investigated. In the first case, for the electron-proton plasma with the temperature more than 2.3 KeV we found the regional change of the wave numbers for which the soliton of two types, subsonic and supersonic, can exist. The soliton of the transverse waves can exist when the group velocity of the waves is between the thermal velocity of the electron and ion and the length of the linear waves is less than 2c/ _pi .In the second case the regions of the wave numbers, with the solitons of the Langmuir and transverse waves have been determined.The nonlinear waves in the electron-positron plasma and the waves with the phase velocity, which is about the light one, are also considered in the following paper.