dV Equation Research Articles

Study of the dynamical nonlinear modified Korteweg–de Vries equation arising in plasma physics and its analytical wave solutions

In this article we have discussed the analytical analysis of two dimensional modified Korteweg–de Vries (mK–dV) equation arising in plasma physics that governs the ion-acoustic solitary waves for their asymptotic behavior because of the trapping of electrons using auxiliary equation mapping method. By using this technique we have obtained some quite general and new variety of exact traveling wave solutions which are collecting some kind of semi half bright, bright, dark, semi half dark, doubly periodic, combined, periodic, half hark and half bright via three parametric values which is the primary key point of difference of our technique. These results are highly applicable to develop new theories of quantum mechanics, biomedical problems, soliton dynamics, plasma physics, nuclear physics, optical physics, fluid dynamics, electromagnetism, industrial studies, mathematical physics, biomedical problems, and in many other natural and physical sciences. For detailed physical dynamical representation of our results we have shown them with graphs in different dimensions via Mathematica 10.4 to get more understanding of different new dynamical shapes of solutions.

Effect of external oblique magnetic field on the nonextensive dust acoustic soliton energy

A theoretical study has been carried out of obliquely propagating dust acoustic solitary waves in a magnetized multi-component dusty plasma which consists of fluid dust grains, nonextensive ions, and Maxwellian electrons. Our plasma model is inspired from the experimental study of Bandyopadhyay et al. (Phys Rev Lett 101:065006, 2008). The reductive perturbation method has been used to establish the Korteweg–de Vries (K–dV) equation which admits a nonlinear solitary wave solution for small but finite amplitude limit. Our results reveal that the main quantities of DA solitons are drastically affected by the nonextensivity, obliqueness, and static magnetic field effects. In particular, it may be noted that an addition of a degree of ion nonextensivity leads to an increase in the amplitude and width of DA soliton. This effect becomes more important when the DA soliton propagates more obliquely. The role the nonextensivity, obliqueness, and magnetic intensity can play in the energy engendered by the DA soliton is also analyzed. Interestingly, it is shown that the presence of the strong magnetic field causes a dissipation of the energy carried by the DA soliton. This dissipation becomes less important as the ions correlation becomes stronger. For a good understanding, the theoretical results of our model are compared to their experimental counterparts.

Excitations of low-frequency solitary waves in collisional magnetized plasmas with two kappa-distributed electron populations

The electrostatic excitations of low-frequency solitary waves are considered in a collisional magnetized plasma consisting of two-electron populations with kappa distribution. A nonlinear damped Korteweg–de Vries (dK–dV) equation is derived by using the reductive perturbation method. In the present plasma model, both positive and negative potential solitons can be supported. The numerical results reveal that ion-neutral collision frequency modifies features of the electrostatic solitary waves significantly. The effects of wave propagation angle, spectral index of cold (hot) electrons, temperature ratio of cold to hot electron, ion temperature and the strength of the external magnetic field on electrostatic solitary wave structures are also discussed here.

Nonlinear propagation of ion acoustic waves in quantum plasma in the presence of an ion beam

AbstractNonlinear propagation of ion acoustic waves has been studied in unmagnetized quantum (degenerate) plasma in the presence of an ion beam using the one-dimensional quantum hydrodynamic model. The Korteweg–de Vries (K–dV) equation has been derived by using the reductive perturbation technique. The solution of ion acoustic solitary waves is obtained from the K–dV equation. The theoretical results have been analyzed numerically for different values of plasma parameters and the results are presented graphically. It is seen that the formation and structure of solitary waves are significantly affected by the ion beam in quantum plasma. The solitary waves will be compressive or rarefactive depending upon the values of velocity, concentration, and temperature of the ion beam. The critical value of ion beam density for the nonexistence of solitary wave has been numerically estimated, and its variation with velocity and temperature of ion beam has been discussed graphically. The results are new and would be very useful for understanding the beam–plasma interactions and the formation of nonlinear wave structures in dense quantum plasma.

Dust Acoustic Solitary Waves in a Five Component Plasma-Application to Comets

We have studied the characteristics of the Dust Acoustic Solitary Wave (DASW) in a five component cometary plasma composed of two components of electrons (one hotter and one colder), hydrogen ions and a pair of oppositely charged dust particles (heavier ions). The Korteweg – deVries wave equation has been derived for the system and its solitary wave solution obtained. The dust acoustic solitary wave amplitude is almost a constant with varying temperatures of the hot electron component but decreases with increasing hydrogen ion densities. A comparison with a complementary nonlinear wave, the Dust Ion Acoustic Solitary Wave, has also been made. Keywords: Dust acoustic solitary wave, K dV equation, kappa distributed ions and electrons Cite this Article Neethu Theresa Willington, Sijo Sebastian, Chandu Venugopal. Dust Acoustic Solitary Waves in a Five Component Plasma-Application to Comets. Research & Reviews: Journal of Physics . 2019; 8(2): 44–51p.

New travelling wave solutions for plasma model of extended K–dV equation

In this paper we aims to obtain new exact and analytical solutions for extended Korteweg–de Vries (K–dV) equation by using of new modified method named Riccati equation method. We introduce the extended K–dV equation with help of the reductive perturbation method, which admits a double layer structure in current plasma model. The obtained solutions are expressed by the hyperbolic, trigonometric and rational functions. Our calculations show that, considered method is straightforward to solve nonlinear partial differential equations. This method is not only powerful mathematical tool for generating more solutions of nonlinear partial differential equations but also can be applied to nonlinear partial differential equations.

Dust acoustic solitons in a dusty plasma with Cairns–Gurevich distributed ions

The problem of dust acoustic solitons is addressed in an unmagnetized collisionless dusty plasma with ions satisfying a Cairns–Gurevich distribution. A new type of density that describes, simultaneously, the evolution of the energetic ions and those trapped in the plasma potential well is outlined. Both highly and weakly nonlinear waves are investigated by deriving the Sagdeev potential for the large amplitude limit, and establishing the nonlinear partial differential equations (mK–dV equation) for the small but finite amplitude limit. It is shown that the effect of the nonthermal ion trapping on DA waves can be quite important. In particular, an increase of nonthermal character of ions following the Cairns–Gurevich distribution lead to an increase of the main quantities (amplitude and width) of dust acoustic solitons. Our investigation will be helpful in understanding the nonlinear DA waves in the presence of nonthermal trapped ions which may exist in space.

Effects of ionization and ion loss on dust ion-acoustic solitary waves in a collisional dusty plasma with suprathermal electrons

The combined effects of ionization, ion loss, and electron suprathermality on dust ion-acoustic solitary waves in a collisional dusty plasma are examined. Carrying out a small but finite amplitude analysis, a damped Korteweg-de Vries (dK–dV) equation is derived. The damping term decreases with the increase of the spectral index and saturates for Maxwellian electrons. Choosing typical plasma parameters, the analytical approximate solution of the dK-dV equation is numerically analyzed. We first neglect the ionization and ion loss effects and account only for collisions to estimate the relative importance between these damping terms which can act concurrently. Interestingly, we found that as the suprathermal character of the electrons becomes important, the strength of the collisions related dissipation becomes more important and causes the dust ion-acoustic solitary wave amplitude to decay more rapidly. Moreover, the collisional damping may largely prevail over the ionization and ion loss related damping. The latter becomes more effective as the electrons evolve far away from their thermal equilibrium. Our results complement and provide new insights into previously published work on this problem.

Ion-acoustic soliton energy in a plasma with nonextensive electrons

The role electron nonextensivity may play on the energy carried by the ion-acoustic (IA) solitons is examined. A Korteweg–de Vries (K–dV) equation is derived. It is found that an increase of the nonextensive parameter q leads to a decrease of the soliton amplitude and width. The soliton energy is depleted as q increases.

Cylindrical and spherical Gardner solitons and double layers in a dusty electronegative non-thermal plasma with two-temperature electrons

A precise theoretical investigation has been made on the cylindrical and spherical (nonplanar) Gardner solitons (GSs) and double layers (DLs) in a dusty electronegative plasma (composed of inertial positive and negative ions, Maxwellian cold electrons, non-thermal hot electrons, and negatively charged static dust). The reductive perturbation method has been used in derivation of the modified Gardner (MG) equation describing the nonlinear propagation of the dust ion-acoustic (DIA) waves. The MG equation admits solitary waves (SWs) and DLs solutions for σ around its critical value σc (where σc is the value of σ corresponding to the vanishing of the nonlinear coefficient of the Korteweg de-Vries (K−dV) equation). The nonplanar SWs and DLs solutions are numerically analyzed and the parametric regimes for the existence of the positive as well as negative SWs and negative DLs are obtained. The basic features of nonplanar DIA SWs and DLs, which are found to be different from planar ones, are also identified. The implications of our results to different space and laboratory dusty plasma situations, are discussed.

A new pressure relaxation closure model for one-dimensional two-material Lagrangian hydrodynamics

We present a new model for closing a system of Lagrangian hydrodynamics equations for a two-material cell with a single velocity model. We describe a new approach that is motivated by earlier work of Delov and Sadchikov and of Goncharov and Yanilkin. Using a linearized Riemann problem to initialize volume fraction changes, we require that each material satisfy its own p dV equation, which breaks the overall energy balance in the mixed cell. To enforce this balance, we redistribute the energy discrepancy by assuming that the corresponding pressure change in each material is equal. This multiple-material model is packaged as part of a two-step time integration scheme. We compare results of our approach with other models and with corresponding pure-material calculations, on two-material test problems with ideal-gas or stiffened-gas equations of state.

The tanh and the sine-cosine methods for the complex modified K dV and the generalized K dV equations

The complex modified K dV (CMK dV) equation and the generalized K dV equation are investigated by using the tanh method and the sine-cosine method. A variety of exact travelling wave solutions with compact and noncompact structures are formally obtained for each equation. The study reveals the power of the two schemes where each method complements the other.

Nonlinear Wave Behavior of the Sinuous Mode on a Free Liquid Sheet with Variable Surface Tension and Viscosity

We investigate nonlinear wave behavior of a free liquid sheet subject to a temperature difference between both surfaces. Considering the temperature dependence of the surface tension and viscosity, we analytically obtain a nonlinear evolution equation for the sinuous (antisymmetric) mode under the membrane approximation. For infinitesimal disturbances or without temperature difference, the equation is reduced to the linearized K–dV equation in which the mode is neutrally stable. When weak nonlinearity is considered in the equation, however, the mode is modulationally unstable for the temperature difference above a critical value. Numerical analysis on the equation shows that the sheet becomes unstable with steepening and distortion above the critical temperature difference. In particular, for large temperature difference, the bulge (symmetric) mode is additionally induced on the sinuous mode and this induced mode is expected to cause the breakup of the sheet. On the other hand, the temperature dependence ...

Spherical and cylindrical dust acoustic solitary waves

Abstract The nonlinear propagation of cylindrical and spherical dust acoustic (DA) waves in an unmagnetized dusty plasma is considered. By employing the reductive perturbation technique, a modified Korteweg–de Vries (K–dV) equation is derived from the dust continuity and momentum equations, Poisson's equation as well as Boltzmann electrons and ions. Numerical solutions of the modified K–dV equation are obtained. It has been found that the propagation characteristics of the cylindrical and spherical DA solitary waves significantly differ from those of one-dimensional DA solitary waves.

Evolution of Nonlinear Sloshing in a Tank Near Half the Fundamental Resonance

This article describes the evolution of shallow water waves in a tank that is closed at one end and is periodically forced at half the fundamental frequency at the other end. The nonlinear response occurs at the same order as the linear response and is governed by a forced Korteweg–de Vries (K dV) equation. Unlike the corresponding problem for a gas (or the hydraulic limit), there may be nonperiodic (beating) solutions and multiple steady solutions for the same frequency. The addition of a component at the fundamental frequency to the piston input can be used to cancel the nonlinear effects and leave only the linear response in the steady state.

On the prolongation structure and integrability of HNLS equation

Higher order nonlinear Schrödinger equation occurring in the field of nonlinear optics is analysed from the view point of prolongation theory and Painléve analysis. The gauge equivalence of the equation to the complex modified K dV equation is never used. But from our direct analysis we can sort out the various relations as existing between the parameters of the equation for which it is integrable and one can have a Lax pair.

Evolution of ion-acoustic solitary waves in an inhomogeneous discharge plasma

The physical processes in the generation of a solitary wave through a special nonlinear wave equation, called the Korteweg–deVries (K–dV) equation [D. J. Korteweg and G. deVries, Philos. Mag. 39, 422 (1895)], have been studied in an ideal inhomogeneous plasma incorporating the electric field and low current discharges. It is concluded that the usual procedure, because of the plasma inhomogeneity and ionization, is not suitable to exhibit soliton propagation unless an assumption for a mathematical simplification is made. Despite that, the soliton solution does not satisfactorily correspond to experiments. A new approach has been applied for finding the exact soliton propagation with a view to relating the observation to that expected exactly in an experiment. The results let one evaluate the formation of a tailing structure, which follows the main soliton with a growing nature as a function of the plasma configuration. Finally, the theoretical observations seem to significantly stimulate the observations in laboratory and space plasmas.

Obliquely Propagating Electrostatic Solitary Structures in aHot Magnetized Dusty Plasma

A rigorous theoretical investigation has been made of obliquelypropagating electrostatic solitary structures in a hot magnetizeddusty plasma which consists of a negatively charged, extremelymassive, hot, dust fluid and Maxwellian distributed ions andelectrons. The reductive perturbation method has been employed toderive the Korteweg–deVries (K–dV) equation which admits asolitary wave solution for small but finite amplitude limit. It hasbeen shown that the effects of dust-temperature, obliqueness, magneticfield and free electrons have changed the nature of the electro-staticsolitary structures. The result of the present investigation forunmagnetized case, parallel propagation, no free electron and colddust fluid completely agrees with the existing published results(Mamun et al. [4]).

Time-reverse acoustic array focusing in the presence of a soliton-type internal wave

Time-reverse or phase-conjugate arrays (TRAs or PCAs) have been shown to retrofocus sound to the location(s) of its origin even when the acoustic medium and its boundaries are unknown. This retrofocusing is degraded by temporal variations in the acoustic environment. In this study, analytical predictions are made for the focal properties of a linear vertical time-reversing array in a stably stratified acoustic medium having sharp thermocline separating regions with two different sound speeds. The medium is assumed to be quiescent except for the presence of a deterministic soliton internal wave traveling along the thermocline. The soliton wave shape and propagation speed are obtained from the K–dV equation and relevant shallow-water oceanic parameters. Soliton motion is incorporated in the necessary two-way acoustic propagation formalism with the single-scattering approximation. Depending on the acoustic frequency and source-array range, current results suggest there is ample opportunity for successful TRA deployment in the presence of soliton internal waves. [Work sponsored by the Office of Naval Research.]

A Partial Differential Equation Arising in a 1D Model for the 3D Vorticity Equation

We continue the study of the one-dimensional model for the vorticity equation considered in [4]. The partial differential equation σ yt + σσ xy = σ x σ y + vσ yxx is deduced, which appears as a generalization of the Burgers' equation, with possibly some connection also to the K dV equation. Some properties of this equation are given and propagating solutions are found which are of soliton type, both with non-compact and compact support.