Korteweg-de Vries Solitons Research Articles

Influences of densities and thermal temperatures of electron and beam fluids on the characteristics of the electron acoustic electrostatic fields and fluid energies

A model of nonlinear electron acoustic electrostatic plasma waves having electron beams has been investigated. The structure of Korteweg–de Vries soliton energy and the corresponding electrostatic field have been discussed. The parametric regimes for wave existence of compressive and rarefactive structures have been examined via auroral zone data. The effects of densities and thermal temperature of electrons and beams on the nature of the electrostatic field and fluid energies have been taken into account.

Modified Korteweg–de Vries solitons with quartic nonlinearity in a dusty plasma

The present multicomponent dusty plasma with ions, Cairns distributed electrons and immobile dusts has been investigated first time through modified Korteweg–de Vries (mKdV) equation of quartic nonlinearity derived by reductive perturbation technique. In this new investigation, it is found that the dust ion-acoustic (DIA) solitary waves have smaller amplitudes compared to the amplitudes of mKdV-DIA compressive solitons of our previous investigation []. Roles of non-thermal parameter (β), dust to ion density ratio (σ), number of dust charge (Z d ) and initial streaming speed of ion (u i0) in the growth of amplitudes and widths of this new mKdV-DIA solitons are investigated.

Landau damping effects on dust ion-acoustic solitary waves in a nonthermal pair-ion plasma

The Landau damping effects on the nonlinear propagation of dust ion-acoustic solitary waves (DIASWs) are studied in an electron-free unmagnetized collisionless dusty pair-ion plasma. The two species of nonthermal ions (positive and negative) are described by the kinetic Vlasov equations, whereas the massive charged dust grains form only the background plasma. Using the multi-scale reductive perturbation technique generalized for the applications to the Vlasov equation, we derive a modified Korteweg–de Vries (KdV) equation that governs the evolution of DIASWs with the effects of linear resonance, in particular, linear Landau damping. The properties of the phase velocity and the Landau damping rate of DIASWs are studied with the effects of positive and negative-ion nonthermality parameters ( α p , α n ), the ratios of the positive to negative ion masses (m) and temperatures (T) as well as the dust to negative-ion number density ratio (δ). It is found that, depend on the degree of positive to negative ion mass ratios (m), both of damped and growing oscillations occur in a collisionless dusty pair-ion plasma due to the resonance phenomena between the wave and the nonthermal particles of the system. The properties of the decay rates of the amplitude of the KdV soliton with a small effect of Landau damping are also studied with the above system of parameters. The results may be useful for understanding the localization of solitary pulses and associated resonance damping and/or growing oscillations of the wave in laboratory and space plasmas, in which the number density of free electrons is much smaller than that of ions and highly charged heavy, micron seized dust grains.

Nonplanar ion-acoustic solitary and cnoidal waves in a non-Maxwellian plasma: Study on nonplanar (modified) Kawahara equation

This study aims to examine the properties of the nonplanar (cylindrical and spherical) ion-acoustic solitary waves (SWs) and cnoidal waves (CWs) in a collisionless, unmagnetized electron-ion (EI) plasma having a Cairns–Tsallis distribution for the electrons. This study is structured around two main lines. The first trend involves deriving the nonplanar Korteweg-de Vries (KdV) equation by utilizing the method of reductive perturbation (MRP). This equation describes small-amplitude (non)planar acoustic waves (AWs). Furthermore, the nonplanar Kawahara equation (KE) is formulated to examine the significant magnitude of planar and nonplanar SWs and CWs. The current plasma model supports compressive and rarefactive IA solitary and cnoidal structures, depending upon the associated physical factors such as the nonextensive parameter (nonextensivity) and nonthermal parameter (nonthermality). When the plasma compositions reach some critical values, such as the critical value of nonthermality, the coefficient of the quadratic nonlinear term vanishes. Hence, both nonplanar modified KdV (mKdV) equation and modified KE (mKE) with cubic nonlinearity are derived to accurately depict the dynamics of both small and large amplitudes of nonplanar SWs and CWs and any other structures related to this family of evolution equations. The influences of the nonextensivity and nonthermality on the profile of (non)planar KdV soliton and the (non)planar Kawahara SWs and CWs are numerically examined using some semi-analytical and numerical approximations. Also, the impact of the nonextensivity on the profile of (non)planar mKdV soliton and the (non)planar modified Kawahara SWs and CWs is reported. It is tracked down that the variety of different plasma parameters significantly alters the characteristic properties of the small and large amplitude ion-acoustic waves (IAWs) discussed by the nonplanar KdV-type equations.

Existence of Korteweg-de Vries solitons and relevance of relativistic effects in a dusty electron-ion plasma

Nonlinear effects in the propagation of perturbations in a dusty electron-ion plasma are studied, considering fully relativistic wave motion. A multifluid model is considered for the particles, from which a KdV equation can be derived. In general, two different soliton solutions are found depending on the kind of dispersion of the KdV equation. We study when the dispersion coefficient of this equation is positive. In this case, two kinds of behavior are possible, one associated with a slow wave mode, another with a fast wave mode. It is shown that, depending on the value of the system parameters, compressive and/or rarefactive solitons, or no soliton at all, can be found and that relativistic effects for ions are much more relevant than for electrons. It is also found that relativistic effects can strongly decrease the soliton amplitude for the slow mode, whereas for the fast mode they can lead to compressive-rarefactive soliton transitions and vice versa, depending on the dust charge density in both modes.

Propagation of generalized Korteweg-de Vries solitons along large-scale waves.

We consider propagation of solitons along large-scale background waves in the generalized Korteweg-de Vries (gKdV) equationtheory when the width of the soliton is much smaller than the characteristic size of the background wave. Due to this difference in scales, the soliton's motion does not affect the dispersionless evolution of the background wave. We obtained the Hamilton equationsfor soliton's motion and derived simple relationships which express the soliton's velocity in terms of a local value of the background wave. Solitons' paths obtained by integration of these relationships agree very well with the exact numerical solutions of the gKdV equation.

Ion acoustic soliton with thermal ions and non-thermal electrons in a high-relativistic electron-positron-ion plasma

In this investigation, both compressive and rarefactive solitons are shown to exist in electron-positron-ion plasma consisting of thermal positrons, non-thermal electrons, and high relativistic thermal ions. The reductive perturbation method (RPM) is used to derive the nonlinear Korteweg-de Vries (KdV) equation from the governing normalized basic set of equations. It is shown that only the fast ion-acoustic mode may produce the small amplitude rarefactive and compressive KdV solitons. It has been observed that the addition of variable ion species temperature dramatically alters the fundamental characteristics of amplitude and width. Moreover, the amplitude decreases as the ion to electron temperature ratio rises. The soliton's amplitude is also increased by the relativistic factor. High-relativistic electron-positron-ion plasmas are found in extreme astrophysical environments like pulsar magnetospheres, gamma-ray bursts, and active galactic nuclei. These plasmas’ wave and soliton activity can shed light on the dynamics of these astrophysical systems.

Head-on collision of ion-acoustic (modified) Korteweg–de Vries solitons in Saturn's magnetosphere plasmas with two temperature superthermal electrons

In view of the recent observations by plasma science-spacecraft-voyager and Cassini plasma spectrometer of Saturn's magnetosphere, the interaction between two counter-propagating ion-acoustic (IA) solitons is studied in an unmagnetized plasma consisting of warm adiabatic ions in addition to hot and cold electrons following kappa distribution. The head-on collision of the IA solitons is investigated using the extended Poincare–Lighthill–Kuo technique. Since this model supports both compressive and rarefactive solitons, therefore, the soliton collisions for both Korteweg–de Vries (KdV) and the modified KdV (mKdV) equations are investigated. The corresponding phase shifts after the collision for both these equations are also derived and examined. Furthermore, the effects of different plasma parameters (corresponding to Saturn's magnetosphere), including superthermality, density, and temperature on the colliding soliton profiles and their phase shifts, are examined. It is concluded that the phase shift is smaller when both hot and cold electrons are Maxwellian by comparison with the superthermal case.

Ion acoustic solitons at the acoustic speed in a warm negative ion plasma with superthermal electrons

Using the Sagdeev pseudopotential method, the existence of Korteweg–de Vries (KdV) and non-KdV solitons is investigated in a negative ion plasma comprising adiabatic positive and negative ions and kappa distributed electrons. For some plasma parameter values, the plasma model supports the coexistence of solitons of both polarities. Positive KdV solitons coexist with negative non-KdV solitons at low values of negative to positive ion density ratio, and positive non-KdV solitons coexist with negative KdV solitons at higher values. There is therefore a switch in polarity between positive KdV and negative KdV solitons at a critical value of negative to positive ion density ratio and a switch in polarity between negative non-KdV and positive non-KdV solitons at the same point. At the critical point, there is no soliton at the acoustic speed, although there is coexistence at larger Mach numbers. This confirms that the existence of a soliton at acoustic speed is not a necessary condition for the coexistence of solitons of both polarities. When electrons are strongly non-thermal and the ion temperatures are important, the coexistence region vanishes and the non-KdV solitons disappear with it. It was also found that there is a forbidden region in terms of negative (positive) ion temperatures when the negative (positive) ion temperature increases with the other plasma parameters held fixed.

Partial degeneration of finite gap solutions to the Korteweg–de Vries equation: soliton gas and scattering on elliptic backgrounds

We obtain Fredholm type formulas for partial degenerations of theta functions on (irreducible) nodal curves of arbitrary genus, with emphasis on nodal curves of genus one. An application is the study of ‘many-soliton’ solutions on an elliptic (cnoidal) background standing wave for the Korteweg–de Vries (KdV) equation starting from a formula that is reminiscent of the classical Kay–Moses formula for N-solitons. In particular, we represent such a solution as a sum of the following two terms: a ‘shifted’ elliptic (cnoidal) background wave and a Kay–Moses type determinant containing Jacobi theta functions for the solitonic content, which can be viewed as a collection of solitary disturbances on the cnoidal background. The expressions for the travelling (group) speed of these solitary disturbances, as well as for the interaction kernel describing the scattering of pairs of such solitary disturbances, are obtained explicitly in terms of Jacobi theta functions. We also show that genus N + 1 finite gap solutions with random initial phases converge in probability to the deterministic cnoidal wave solution as N bands degenerate to a nodal curve of genus one. Finally, we derive the nonlinear dispersion relations and the equation of states for the KdV soliton gas on the residual elliptic background.

Creation of solitons and density cavities by lower hybrid waves

Formation of Korteweg-de Vries (KdV) solitons by nonlinear lower hybrid waves (LHWs) is not possible in usual electron ion plasma because a dispersive term does not have a suitable form. It is pointed out that the dispersion characteristics of electrostatic LHWs are modified in the presence of field-aligned shear flow to produce KdV solitons with negative electrostatic potential as it has been observed by satellites in the upper ionosphere. Plasma density decreases within the solitary structures. The parallel electron velocity shear ve0=ve0(x)ẑ also gives rise to unstable waves in the intermediate frequency range in linear limit whereas the ambient magnetic field B0=B0ẑ is assumed to be constant. The instability and structure size depend upon the electron parallel velocity shear parameter Se=1Ωedve0(x)dx (where Ωe is the electron gyrofrequency) and the propagation direction with respect to the ambient magnetic field B0. The theoretical model is applied to the upper ionosphere, and the estimated width of the structures turns out to be of the order of 70 m, which is closer to the observations.

Using the Nonequilibrium Hydrodynamic Approach to Describe Collisions between Atomic Nuclei as Collisions between Korteweg–de Vries Solitons

Using the Nonequilibrium Hydrodynamic Approach to Describe Collisions between Atomic Nuclei as Collisions between Korteweg–de Vries Solitons

Bright, dark and breather soliton solutions of the generalized long-wave short-wave resonance interaction system

In this paper, a generalized long-wave short-wave resonance interaction system, which describes the nonlinear interaction between a short-wave and a long-wave in fluid dynamics, plasma physics and nonlinear optics, is considered. Using the Hirota bilinear method, the general N-bright and N-dark soliton solutions are deduced and their Gram determinant forms are obtained. A special feature of the fundamental bright soliton solution is that, in general, it behaves like the Korteweg-deVries soliton. However, under a special condition, it also behaves akin to the nonlinear Schrödinger soliton when it loses the amplitude-dependent velocity property. The fundamental dark-soliton solution admits anti-dark, gray, and completely black soliton profiles, in the short-wave component, depending on the choice of wave parameters. On the other hand, a bright soliton-like profile always occurs in the long-wave component. The asymptotic analysis shows that both the bright and dark solitons undergo an elastic collision with a finite phase shift. In addition to these, by tuning the phase shift regime, we point out the existence of resonance interactions among the bright solitons. Furthermore, under a special velocity resonance condition, we bring out the various types of bright and dark soliton bound states. Also, by fixing the phase factor and the system parameter \(\beta \), corresponding to the interaction between long and short wave components, the different types of profiles associated with the obtained breather solution are demonstrated.

Weak Relativistic Effect in the Formation of Ion-Acoustic Solitary Waves in Dusty Plasma

Role of streaming speeds of ions ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$u_{i0}$ </tex-math></inline-formula> ) and relativistic electrons ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$u_{e0}$ </tex-math></inline-formula> ) together with the immobile dust charge ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Z_{d}$ </tex-math></inline-formula> ) to form dust-ion acoustic (IA) compressive and rarefactive relativistic solitons in this multispecies plasma model for immobile dusty plasma are investigated. In this new investigation of dust-IA waves with positive ions and relativistic electrons in presence of immobile dust, it is noteworthy to mention that only compressive Korteweg-de Vries (KdV) solitons are seen to produce for some pairs of initial streaming of ions and relativistic electrons. Whereas for some other pairs of ion and relativistic electron streaming speeds, rarefactive solitons of either increasing or decreasing character are seen to produce. The presence of dust charge of immobile dust, present in the plasma, plays a very crucial role to form compressive and rarefactive KdV solitons which is a salient feature of this theoretical investigation. The presence of massive dust particles in the stationary background of the plasma, andthe lighter ions and relativistic electrons get appreciable initial drifts which makes a great change in the growth of solitons which is a salient feature of this new investigation. The results of this finding may be great useful in space plasma and applications of this plasma model in space plasmas are also discussed in brief.

Forced KdV and Envelope Soliton in Magnetoplasma With Kappa Distributed Ions

In this article, we have studied the electron acoustic waves in a magnetized plasma containing Kappa distributed ions which are streaming with high velocity within the plasma and under the action of external force. We employed the homotopy perturbation method (HPM) to obtain the graphical plots of solitary structures and their evolution into an envelope soliton. Analytically, we obtained the linear dispersion relation and studied its characteristics. Further, we derived the Korteweg–De Vries (KdV) equation using the reductive perturbation technique and studied the parametric dependence on the KdV solitary profile. These findings augment the homotopy results. The analytical and simulation results thus obtained will be helpful in interpretating and identifying electron acoustic wave (EAW) modes in such a plasma system.

Collisional positron acoustic soliton and double layer in an unmagnetized plasma having multi-species

This paper explores the head-on collision between two-counter propagating positron acoustic solitons and double layers (DLs) in an unmagnetized collisionless plasma having mobile cold positrons fluid, immobile positive ions and (r,;q)-distributed hot positrons, and hot electrons. By employing the extended Poincaré–Lighthill–Kuo method, the coupled Korteweg–de Vries (KdV), modified KdV (mKdV) and Gardner equations are derived to archive this goal. The effect of dimensionless parameters on the propagation characteristics of interacting KdV solitons (KdVSs), mKdV solitons (mKdVSs), Gardner solitons (GSs) and DLs are examined in detail by considering the limiting cases of (r,;q)-distribution. It is noted that the interaction of GSs and DLs are reported for the first time. The outcomes might be comprehended and beneficial not only in space and astrophysical environments but also in laboratory studies.

Quantum Electron Acoustic Solitons and Double Layers with κ-deformed Kaniadakis Distributed Electrons

The present investigation is dedicated to the study of propagation characteristics of electron acoustic (EA) waves and double layers in the quantum plasma system containing inertialess hot electrons and inertial cold electrons with stationary ions forming the charge neutralizing background. It is assumed that hot electrons follow κ-deformed Kaniadakis distribution as governed by the parameter κ. Using the appropriate stretched coordinates and reductive perturbation method (RPM) the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations have been derived. For the sake of analysis, a limit of range of deformation parameter (κ) has been set as -0.4≤κ≤0.4. For the defined range, it has been observed that plasma system supports rarefactive solitary structures. The amplitude and width of KdV soliton have been significantly affected by the quantum effects and remains unaffected by the deformation parameter (κ). The analysis was further extended to the derivation of mKdV equation to investigate the existence of small amplitude double layers (DLs). Only negative potential DLs are found to exist whose dynamics significantly depends on deformation parameter (κ), quantum effects (H) and hot to cold electron density ratio (α). The outcome of the present discourse may be helpful to understand the use of generalized entropies in the environment of plasma physics.

Self-interacting Stationary Formations in Plasmas under Externally Controlled Fields

The formation and evolution of a Korteweg–de Vries (KdV) soliton in a dense quantum plasma consisting of electrons and ions is studied. The solitary profile is first obtained and the external force is applied to study the effect of force on the stationary formation. Further, a new technique based on the MacCormack Predictor Corrector scheme is designed to study the plasma species individual behavior in external field and their collective effect in the KdV soliton formations. The results thus obtained will help in future understanding of the plasma particles’ behavior under a variety of external forces.

Universal reductions and solitary waves of weakly nonlocal defocusing nonlinear Schrödinger equations

We study asymptotic reductions and solitary waves of a weakly nonlocal defocusing nonlinear Schrödinger (NLS) model. The hydrodynamic form of the latter is analyzed by means of multiscale expansion methods. To the leading-order of approximation (where only the first of the moments of the response function is present), we show that solitary waves, in the form of dark solitons, are governed by an effective Boussinesq/Benney–Luke (BBL) equation, which describes bidirectional waves in shallow water. Then, for long times, we reduce the BBL equation to a pair of Korteweg–de Vries (KdV) equations for right- and left-going waves, and show that the BBL solitary wave transforms into a KdV soliton. In addition, to the next order of approximation (where both the first and second moment of the response function are present), we find that dark solitons are governed by a higher-order perturbed KdV (pKdV) equation, which has been used to describe ion-acoustic solitons in plasmas and water waves in the presence of higher-order effects. The pKdV equation is approximated by a higher-order integrable system and, as a result, only insubstantial changes in the soliton shape and velocity are found, while no radiation tails (in this effective KdV picture) are produced.

Evolution of truncated and bent gravity wave solitons: the Mach expansion problem

Abstract