Characteristics of nonlinear dust acoustic waves in a lorentzian dusty plasma with effect of adiabatic and nonadiabatic grain charge fluctuation

Characteristics of nonlinear dust acoustic waves in a Lorentzian dusty plasma with effect of adiabatic and nonadiabatic grain charge fluctuation Characteristics of nonlinear dust acoustic waves in a L[.]

Characteristics of nonlinear dust acoustic waves in a Lorentzian dusty plasma with effect of adiabatic and nonadiabatic grain charge fluctuation Raicharan Denra, Samit Paul, and Susmita Sarkar Citation: AIP Advances 6, 125045 (2016); doi: 10.1063/1.4972520 View online: http://dx.doi.org/10.1063/1.4972520 View Table of Contents: http://aip.scitation.org/toc/adv/6/12 Published by the American Institute of Physics AIP ADVANCES 6, 125045 (2016) Characteristics of nonlinear dust acoustic waves in a Lorentzian dusty plasma with effect of adiabatic and nonadiabatic grain charge fluctuation Raicharan Denra,a Samit Paul,b and Susmita Sarkarc Department of Applied Mathematics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata 700009, India (Received 11 September 2016; accepted December 2016; published online 21 December 2016) In this paper, characteristics of small amplitude nonlinear dust acoustic wave have been investigated in a unmagnetized, collisionless, Lorentzian dusty plasma where electrons and ions are inertialess and modeled by generalized Lorentzian Kappa distribution Dust grains are inertial and equilibrium dust charge is negative Both adiabatic and nonadiabatic fluctuation of charges on dust grains have been taken under consideration For adiabatic dust charge variation reductive perturbation analysis gives rise to a KdV equation that governs the nonlinear propagation of dust acoustic waves having soliton solutions For nonadiabatic dust charge variation nonlinear propagation of dust acoustic wave obeys KdV-Burger equation and gives rise to dust acoustic shock waves Numerical estimation for adiabatic grain charge variation shows the existence of rarefied soliton whose amplitude and width varies with grain charges Amplitude and width of the soliton have been plotted for different electron Kappa indices keeping ion velocity distribution Maxwellian For non adiabatic dust charge variation, ratio of the coefficients of Burger term and dispersion term have been plotted against charge fluctuation for different kappa indices All these results approach to the results of Maxwellian plasma if both electron and ion kappa tends to infinity © 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4972520] I INTRODUCTION Dust grains exist in a wide range of space and astrophysical plasmas like cometary tails, interstellar clouds, earth’s mesosphere, ionosphere, Saturn’s ring, the gossamer ring of Jupiter, and also in laboratory experiments.1,2 Interaction of these dust grains with plasma environment alter the collective plasma behavior and gives birth of new kind of waves In unmagnetized dusty plasma dust ion acoustic and dust acoustic waves are two such low frequency waves whose experimental and theoretical studies have become important focus of plasma research for last three decades In laboratory experiments low frequency fluctuations with a typical frequency of 12 Hz and a wavelength of 0.5 cm were observed by Chu et al3 which later interpreted by D’Angelo4 as dust acoustic wave Propagation characteristics and stability properties of dust acoustic and dust ion acoustic waves were experimentally studied by Merlino et al5 in University of Iowa Liang et al6 reported experimental observation of dust ion acoustic wave propagation down the steep density gradient in an inhomogeneous diffusive dusty plasma Nonlinear behavior of dust acoustic and dust ion acoustic waves were also investigated by experimental plasma physicists in last few years For example, excitation a Electronic mail: raicharanroykolkata@gmail.com b Electronic mail: samitpaul4@gmail.com c Electronic mail: susmita62@yahoo.co.in 2158-3226/2016/6(12)/125045/12 6, 125045-1 © Author(s) 2016 125045-2 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016) and propagation of low frequency finite amplitude solitary waves were investigated by Bandyopadhyay et al7 in Argon plasma impregnated with Kaolin dust particles Nonlinear propagation of dust acoustic shock was experimentally studied by Merlino et al8 in a direct current glow discharged dusty plasma But none of these experiments plasma involved suprathermal charge particles Theoretical investigation on linear dust acoustic and dust ion acoustic wave propagation were extensively studied by several authors9–12 in different physical situation even in presence of self gravitating dust grains.13–17 Extensive theoretical studies have also been reported on several cases of nonlinear propagation of dust acoustic and dust ion acoustic waves considering fixed as well as fluctuating gain charges, in case of Maxwellian plasmas.18–26 Space and astrophysical plasmas consist of charged dust particles whose existence was proved by space craft observation.27–31 Such plasmas consist of suprathermal charge particles whose velocities follow the generalized kappa distribution32 and is known as Lorentzian dusty plasma Characteristics of dust acoustic and dust ion acoustic wave propagation in Lorentzian plasmas differ from their characteristics in Maxwellian dusty plasmas Few such works have so far done on linear and non linear dust acoustic and dust ion acoustic wave propagation in Lorentzian dusty plasmas considering fixed charges on dust grains.33 In this paper we are interested to study nonlinear propagation of dust acoustic waves in a Lorentzian dusty plasma with effect of grain charge fluctuation Such grain charge fluctuation may be of adiabatic or nonadiabatic type For adiabatic dust charge variation dust charging frequency is very large compared to dust plasma frequency whereas for nonadiabatic dust charge variation dust charging frequency is comparatively small Thus in space or astrophysical plasmas, the model of adiabatic dust charge variation is appropriate if dust grains are charged in a very fast time scale(charging frequency is high) On the other hand the model of nonadiabatic dust charge variation is appropriate if dust grains are charged on slow time scale (charging frequency is low) The dust grains under our consideration are assumed to be cold and charged by plasma current As a consequence equilibrium dust charge is negative Since phase velocity of dust acoustic wave is less than electron and ion thermal velocities, both the electrons and ions are considered inertialess, only the dust inertia is taken into consideration Here electron and ion velocities are assumed to follow generalized Kappa distribution as plasma under our consideration is a Lorentzian dusty plasma Small amplitude structures are investigated using the reductive perturbation technique For adiabatic dust charge variation nonlinear propagation of dust acoustic wave is governed by KdV equation which has soliton solution To study the nature of this dust acoustic soliton, for simplicity of numerical calculation we have made ion-Kappa index tending to infinity in the analytical results Since ions are heavier than electrons, probability of existence of suprathermal electrons is much higher than of suprathermal ions Thus our assumption is physically consistent Our numerical study shows that in Lorentzian dusty plasma amplitude of dust acoustic soliton is negative and lower in magnitude than in Maxwellian dusty plasmas On the other hand width of the dust acoustic soliton is higher in Lorentzian than in Maxwellian dusty plasmas It is also observed that amplitude of this rarefied dust acoustic soliton decreases and width increases with decreasing kappa index, i.e with increasing number of suprathermal electrons For nonadiabatic dust charge variation nonlinear propagation of Dust Acoustic Wave is governed by KdV-Burger equation which possesses shock solution Numerical study in this case shows that in Lorentzian dusty plasma dust acoustic shock wave is dissipation dominated upto certain value of normalized grain charge number and dispersion dominated thereafter As value of the electron kappa index decreases this changeover is prominent Propagation of such dust acoustic shock wave causes dust density condensation and enhances the gravitational interaction This is an important phenomenon in astrophysical plasmas as it may be a variable process of star formation.34,35 In section II of this paper we have formulated the problem after a brief description of Kappa velocity distribution Section III deals with the nondimensionalization of basic equations and reductive perturbation analysis of the problem In the same section both adiabatic and non adiabatic dust charge variation have been considered separately Numerical estimation has been reported in section IV Results have been concluded in V 125045-3 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016) II MATHEMATICAL FORMULATION OF THE PROBLEM Presence of suprathermal charge particles were confirmed by several spacecraft measurements.36–41 Vasyliunas42 suggested that charge particles in natural space environment may be well described by the Kappa (κ) or generalized Lorentzian velocity distribution functions which fits both the thermal as well as the suprathermal parts of the observed energy spectra.42 The conventional isotropic, three dimensional form of the Kappa velocity distribution function for electrons in equilibrium may be written as,43   fe0 vx , vy , vz =  ne πθ e2 κ e  3/2 Γ (κ e + 1) Γ (κ e − 1/2) −(κ +1)  v + vy2 + vz2  e 1 + x    κ e θ e2 (1) g 1/2 f e is the thermal velocity, me , ne , Te are respectively the mass, number density, where θ e = (2κκee−3)T me temperature and ~v = (vx , vy , vz ) is the velocity of electrons, Γ(x) is the Gamma function The value of the index κ e determines the slope of the energy spectrum of the suprathermal electrons forming the tail of the velocity distribution functions In the limit κ e → ∞, this Kappa distribution reduces to Maxwellian distribution Kappa distributions with < κ e < have been found to fit the observations and satellite data in the solar wind, the terrestrial magnetosphere; the magnetosheath, the magnetosphere of other planets like Mercury, Jupiter, Saturn and Uranus as observed by Ulysses, Cassini and the Hubble Space Telescope.44 In this paper we consider a model of Lorentzian dusty plasma consisting of electrons, ions and negatively charged dust grains Charge neutrality at equilibrium reads as, ni0 = ne0 + zd0 nd0 (2) where ni0 , ne0 , nd0 are respectively the ion, electron and dust number densities in equilibrium, and zd0 is the unperturbed number of charges residing on the dust grain measured in the unit of electron charge For one-dimensional low-frequency dust acoustic wave motion cold dust grains satisfy the fluid equations ∂nd ∂ (nd ud ) = + ∂t ∂x (3) ∂ud qd ∂φ ∂ud + ud =− ∂t ∂x md ∂x (4) and the Poisson equation ∂2 φ = 4πe (zd nd + ne − ni ) (5) ∂x Here nd , ud , and md refer to the number density, fluid velocity and mass of the dust grains, qd = −ezd is the variable dust charge and zd is the variable charge number on dust grains Number densities of non inertial Kappa distributed electrons and ions are, ! −(ke − 21 ) −(ki − 21 ) 2eϕ 2eφ + and ni = ni0 *1 + (6) ne = ne0 − me ke θ e2 mi ki θ i2 , r r (ke − 23 )  2Te  (ki − 23 )  2Ti  where θ e = ke me , θ i = ki mi with temperatures Te , Ti are their thermal velocities, me , mi are masses and κ e , κ i are electron and ion kappa indices respectively The variable dust charge qd satisfies the grain charging equation, dqd = Ie + Ii dt where Ie = −πa ene 8Te πme ! 21 κe − ! 21 Γ (κ e − 1)   Γ κ e − 21 (7)   −(κe −1) 1 −  eΦd    κ e − 23 Te    (8) 125045-4 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016) ! 21 ! 21   eΦd (κ i − 1)   1 −    (9) Ii = πa eni κ i − 23 Ti    are electron and ion current flowing to the dust surface and Φd denotes the dust surface potential relative to plasma potential φ 8Ti πmi κi − Γ (κ i − 1)   Γ κ i − 21 III NONDIMENSIONALIZATION AND REDUCTIVE PERTURBATION In this section all the physical quantities are normalized as follows Electron, ion and dust number densities ne , ni , nd are normalized by corresponding unperturbed densities neo , nio and ndo whereas the dust charge number zd is normalized by its equilibrium value zd0 The space coordinate x, time t, electrostatic potential energy eϕ and dust velocity ud are normalized by the Debye length   21   21 T md −1 = λ Dd = 4πz effn e2 , the inverse of dust plasma frequency ωpd , the electron thermal 4πnd0 zd0 e2 d0 d0   21 z T energy Te (in eV) and the dust-acoustic speed cd = d0mdeff respectively Here (δ − 1)  (κe − 21 ) δ (κi − 21 ) + (κe − ) σ (κi − ) The set of dimensionless basic equations are then obtained in the form, ∂Nd ∂ (Nd Vd ) = + ∂T ∂x Teff = Te αd , with αd =  ∂Vd Q ∂Φ ∂Vd + Vd =− ∂T ∂X αd ∂X (10) (11) (12) αd ∂2Φ =− { (δ − 1) QNd − Ne + δNi } (δ − 1) ∂X ! ωpd ∂Q ∂Q (Ie + Ii ) +V = υd ∂T ∂X υd ezd0 (13) (14) with the dimensionless electron and ion number densities and current expressions,   Φ ne  = 1− Ne =  ne0 κe −  −    3 2 κe −  (15)   − κi −    ni  Φ  Ni = = + ni0  σ(κ i − 23 )     1 s −(κe −1)     κ e − 23 Γ (κ e − 1)   8T ZQ e    ne0 Ne  1−  Ie = −πr02 e  πme   Γ κ e − 21 κ i − 23     1 s      ZQ (κ i − 1)  8Ti κ i − Γ (κ i − 1)      Ii = πr0 e ni0 Ni  −   πmi σ  Γ κi − κi −    (16) (17) (18) where Z= e2 zd0 qd qd x t ud eφ nd Ti ,Q= , Φd = , X = , T = −1 , V = , Φ = , Nd = ,σ= , r0 Te ezd0 r0 λd cd Te nd0 Te ωpd r0 is the grain radius, νd is the grain charging frequency and κ e , κ i are respectively the electron and ion kappa indices Here Z is the normalized grain charge number, effect of whose variation on nonlinear wave propagation in Lorentzian dusty plasma is of prime interest of this paper 125045-5 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016) A Adiabatic dust charge variation For the adiabatic dust charge variation, dust charging time is very small and hence dust charging ω frequency is very large compared to dust plasma frequency, which implies υpdd ≈ Then normalized grain charging equation (14) reduces to, Ie + Ii = (19) which after substitution of the expressions of electron and ion currents from (17) and (18) becomes,    1 s − κi −         Φ 8Ti κ i − Γ (κ i − 1)    − ZQ (κ i − 1)     + ni0  πr02 e    πmi σ κi −    σ(κ i − )  Γ κi −       1 s −(κe −1) − κe −         Φ  8Te κ e − Γ (κ e − 1)   −  ZQ      − πr02 e − =0 ne0    πme   κ e − 23  κ e − 23  Γ κ e − 21     (20a) This can be further written in the form,     Φ  δ  + σ(κ − )  i     − κi −    Φ −A 1 − κ − e  −    3 2 κe −        − ZQ (κ i − 1)    σ κi −      −(κe −1)       ZQ 1 −   =0    κ e − 23    1    κ e − 23 Γ (κ e − 1) Γ κ i − 21 r mi ni0 where δ = and A =  1  σme  ne0 κ i − 23 Γ (κ i − 1) Γ κ e − 21 (20b) (20c) In equilibrium (Φ = 0, Q = −1) the ion-electron density ratio can therefore be expressed as,   −(κe −1) + κ Z− ni0 ( e 2)  (21) δ= =A  ne0 + σZ (κκ i−−1) ( i 2) Since dust grains are negatively charged the quasi neutrality condition (2) implies δ > Thus we have to choose the range of Z keeping δ greater than one Now for the study of small-amplitude dust acoustic waves in presence of self-consistent adiabatic dust-charge variation, we derive the KdV equation from equations (11–20b) by employing the reductive perturbation technique and using the stretched coordinates ξ = ε (X − λT ), and τ = ε T , where ε is a small parameter and λ is unknown normalized phase velocity of the linear dust acoustic wave The variables Nd , Vd , Φ, and Q are then expanded as, Nd = + εNd1 + ε Nd2 + ε Nd3 + · · · · · · Vd = εVd1 + ε Vd2 + ε Vd3 + · · · · · · Φ = εΦ1 + ε Φ2 + ε Φ3 + · · · · · · (22) Q = −1 + εQ1 + ε Q2 + ε Q3 + · · · · · · Substituting these expansions into equations (11)–(16), (20b) and comparing coefficient of ε from both sides we have the following relations, ! Φ1 Φ1 1 λNd1 = Vd1 , Vd1 = − , Nd1 = − , Q = − Φ , Q1 = − β d Φ (23) αd λ αd αd λ λ2 125045-6 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016)    (κi − 21 ) (κe − 21 ) Z (κi −1) 1 + + AB σ (κi − ) σ (κ e − 1) (κ e − 1) κ e (κi − ) (κe − 23 )    , B=1 −  Z +  where βd =  2 Z κe − 2 κ e − 23 − κκe−Z3 Z + σZ (κκ i−−1) A (κκ e−−1) 3 ( e 2) ( e 2) ( i 2)  (24) To the next higher order in, ε i.e ε we have the following set of equations, ∂Nd1 ∂Nd2 ∂Vd2 ∂ (Nd1 Vd1 ) −λ + + =0 ∂τ ∂ξ ∂ξ ∂ξ (25) ∂Vd2 ∂Vd1 ∂Φ2 ∂Φ1 ∂Vd1 −λ + Vd1 = + Q1 ∂τ ∂ξ ∂ξ αd ∂ξ ∂ξ (26) ∂ Φ1 = αd Nd2 − αd Q2 + Φ2 + EΦ21 ∂ξ (27) Q2 = − βd Φ2 + rd Φ21 (28) where  E= rd = − λ αd       !   1 1− −  2  λ          1 κi − κi + δ   σ2 κi − ( κe − 21 ) κe − +  −   1 κe − κe +   κe − δ (κi − ) σ κi −                    rd1 rd2        κ e − 21 (κ e − 1)  − 21 κ e + 21  (κ e − 1) κ e 2 1 −  κe  Z   Z βd rd1 = −AB  2 − A   Z βd + A 2      3 κ − e κe − 2 κe − κe − 2        1    (κ i − 1) κ i − Z βd κi − κi +   (κ i − 1) Z     + + −    2    σ2 σ2 κ i − 23 σ  κ i − 23 κ i − 23        (κ i − 1) Z (κ e − 1)  κe   +A     rd2 =  − Z   σ  κi − κe − κe −    Eliminating all second order terms from equations (25)–(28) we get the KdV equation,  ∂Φ1 ∂ Φ1 ∂Φ1 + aΦ1 +b =0 ∂τ ∂ξ ∂ξ   βd 2E  a = αd b 2rd − + − , λ =  α   α λ αd λ d (1 + αd βd ) 2 (1 + αd βd )  The solution of equation (29a) can be written as,   ϕ1 = ϕ1 sech2 (ξ − Mτ) /W where b = (29a) (29b) (30) r 3M b which represents soliton of amplitude ϕ1 = and width W = (31) a M Here M is the Mach number which is the ratio of the wave velocity and the velocity of sound Clearly amplitude and width of the soliton depend on electron and ion kappa indices and the grain charge number Z through the coefficient of nonlinearity a and the coefficient of dispersion b respectively 125045-7 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016) B Non adiabatic dust charge variation The nonadiabaticity of the dust charge variation provides an alternate physical mechanism for generating dissipative effect in dusty plasma In case of nonadiabatic dust charge variation dust charging time is large and hence dust charging frequency is small compared to the case of adiabatic dust charge variation This slow grain charging process captures charge fluctuation induced dissipation which cannot be captured in the fast adiabatic grain charging process In the slow charging process ratio ω of the dust plasma frequency ωpd to the grain charging frequency νd is small but finite, i.e., υpdd , With this assumption formation of dust acoustic and dust ion acoustic shock waves were already investigated in Maxwellian dusty plasmas.45,46 In collisionless plasma, however, particle velocity distributions can often depart from being Maxwellian For example, in naturally occurring plasma in the planetary magnetosphere and in solar wind, the particle velocity distribution is observed to have non-Maxwellian (power-law), high-energy tail.43 Such distribution may be accurately modelled by a generalized Lorentzian (kappa)distribution.42 Recently A Shah et al47 have studied the shock wave structure with kappa-distributed electrons and positrons They derived Korteweg-de Vries (KdV)– Burger equation and solved it analytically The effect of plasma parameters on the shock strength and steepness were also investigated Significant effect of nonplanar geometry in the formation of shock waves in adiabatic dusty plasma has also been recently reported.48 But in all these above mentioned studies in Lorentzian dusty plasma, effect of charge fluctuation were not taken into account In this paper we are interested to investigate the formation of dust acoustic shock acoustic waves in Lorentzian dusty plasma considering nonadibatic dust charge fluctuation √ ω = ν ε where ε is small and ν is of the order of unity Then To study this effect we assume νpd d the reductive perturbation technique gives from equation (14) the first and second order dust charge perturbation in the following form Q1 = − βd Φ1 and Q2 = − βd Φ2 + rd Φ21 − µ1 ∂Φ1 ∂ξ (32) where νλ βd µ1 =  C A (κκ e−−1) ( e 23 )  1− κe Z (κe − 23 )  z+ z (κi −1) σ (κi − )  and  C= " (κ i − 1) Z + κi −  σ ) ( (κe −1) σ ((κi − )+Z(κi)−1) (κi −1) (κe − 23 )+Z # i) In Lorentzian dusty plasma νd νd has been calculated from the expression, νd = − ∂(I∂qe +I in the form d (   )   (κ e − 1) σ κ i − + Z (κ i − 1)  r0 ωpi Γ (κ i )   (  ) νd = √  1   1 + (κ i − 1) 2π Vthi κ − Γ κ −  κ e − 23 + Z   i i 2 Eliminating all the 2nd order terms from (25)-(27) and (32) we get the KdV- Burger equation, ∂Φ1 ∂Φ1 ∂ Φ1 ∂ Φ1 + aΦ1 +b = µ ∂τ ∂ξ ∂ξ ∂ξ (33) where µ = µ1 λ2 αd is the coefficient of the Burger term In hydrodynamic fluid flow the dissipative Burger term arises if viscous effect is present in equation of motion In our problem no such viscous effect has considered The Burger term here is arising exclusively due to the nonadiabaticity of the dust charge variation which was absent in the adiabatic case This viscous like dissipative effect may be generated due to the relative velocity between the electron fluid and dust fluid layers which is coming into the picture in case of nonadiabatic dust charge variation as it is a slow charging process In case of rapid charging in the adiabatic process this effect is not captured 125045-8 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016) IV NUMERICAL ESTIMATION For numerical   estimation we have to first determine the range of the normalized grain charge zd0 e2 number Z = r0 Te satisfying the inequality  −(κe −1) + κ Z− ni0 ( e 2)  >1 δ= =A  ne0 + σZ (κκ i−−1) ( i 2)  The temperature ratio σ = TTei is less than one as Te >> Ti Figure (1) has been plotted to find the range of Z for κ e =2,4,6 These values of κ e have been considered as in most space and astrophysical plasmas43 kappa index ranges from to For simplicity of numerical calculation we have considered Lorentzian electrons and Maxwellian ions making κ i → ∞ in the analytical results Figure shows that range of z is different for different electron kappa indices.For lower kappa index it is small.This implies in presence of very high population of suprathermal electrons charge fluctuation is insignificant But for medium to low population grain charge fluctuation plays important role Our main objective of this paper is to study the effects of suprathermal particles i) on amplitude and width of dust acoustic soliton when grain charge fluctuation is adiabatic and ii) on the nature of dust acoustic shock wave when grain charge fluctuation is nonadiabatic Analytically it has been seen that in case of adiabatic dust charge variation nonlinear propagation of dust acoustic wave obeys KdV equation which has soliton solution Figures and have been respectively plotted to show the variation of amplitude and FIG Plot of δ versus Z considering Lorentzian electrons and Maxwellian ions for different values of κe FIG Plot of soliton amplitude ϕ1 versus Z for different κe when ions are Maxwellian 125045-9 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016) FIG Plot of soliton width W versus Z for different κe when ions are Maxwellian width of the dust acoustic soliton with grain charge number Z at κ e =2,4,6 Numerical data have been considered22 σ = 0.5 and M = 1.1 In both figures results approach to Maxwellian values at κ e → ∞ Figure (2) shows that amplitude of the dust acoustic soliton is negative and its magnitude decreases as κ e decreases On the otherhand Figure (3) shows that width of the dust acoustic soliton increases with decreasing electron kappa index Thus in case of adiabatic dust charge variation in Lorentzian dusty plasma, dust acoustic solitons are shorter and wider and hence move slower in presence of suprathermal electrons The dust charging time has been estimated in this case is of the order of 10−1 s In case of nonadiabatic dust charging, the estimated dust charging time is of the order of 104 s ≈ 2hr 47 It is large compared to the adiabatic dust charging time Since nonadiabatic dust charging process is slower than adiabatic dust charging process, larger time is required to set up equilibrium in nonadiabatic case For nonadiabatic dust charge variation, propagation of nonlinear dust acoustic wave is governed by KdV-Burger equation whose Burger coefficient µ is responsible for an alternative dissipation mechanism in the dusty plasma medium We have plotted the Burger coefficient µ in figure (4), the dispersion coefficient b in Figure (5) and the ratio µb in figure (6) respectively taking the ionelectron temperature ratio σ = 0.5 and the nonadiabaticity parameter ν = 0.5 These numerical data have been taken from Reference 22 Figure (4) shows that in presence of suprathermal electrons, µ first increases and then decreases with increasing Z values It is unlike Maxwellian electrons where µ only increases within this specified range of Z The figure (4) also shows that the maximum of µ is FIG Plot of the Burger coefficient µ versus Z considering Lorentzian electrons and Maxwellian ions for different values of κe at ν = 0.5 125045-10 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016) FIG Plot of the dispersion coefficient b versus Z considering Lorentzian electrons and Maxwellian ions for different values of κe FIG Plot of µ/b versus Z considering Lorentzian electrons and Maxwellian ions for different values of κe at ν = 0.5 achieved at lower value of Z for lower κ e index This implies the increasing number of suprathermal electrons increases the Burger dissipation quickly and then decreases with increasing Z Figure (5) shows that the dispersion coefficient b increases with Z for all electron kappa indices Moreover rate of its increase is faster for lower values of electron kappa index This implies presence of suprathermal electrons pronounces the dispersion effect Figure (6) shows that for ν = 0.5, the ratio µb is greater than one upto certain value of Z and less than one thereafter This implies in presence of nonadiabatic dust charge variation dust acoustic shock wave is dissipation dominated upto certain value of Z and thereafter it is dispersion dominated Since shock wave is monotonic if it is dissipation dominated and oscillatory if dispersion dominated, there will be a transformation of a monotonic shock to an oscillatory shock when it propagates in a Lorentzian dusty plasma Figure (6) also shows that this change over is faster and prominent for κ e =2 which is consistent with the graphs of Figure and Figure 5, as for κ e =2, µ increases first and then decreases but b increases for the whole permissible range of Z Thus in presence of large population of suprathermal electrons (κ e = 2), monotonic shock quickly transforms to an oscillatory shock V CONCLUSION Study of wave characteristics in Lorentzian dusty plasma is an important area of dusty plasma research In his paper using reductive perturbation technique, we have studied nonlinear characteristics of small amplitude dust acoustic wave propagation in a Lorentzian dusty plasma considering charge fluctuation on dust grains with negative equilibrium dust charge 125045-11 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016) Both adiabatic and nonadiabatic dust charge variation has considered Numerically for adiabatic dust charge variation we have obtained rarefied dust acoustic soliton whose amplitude decreases and width increases with decreasing electron kappa indices i.e., with the increasing number of suprathermal electrons Thus in presence of suprathermal electrons dust acoustic solitons move slower In this adiabatic dust charge variation charging time is very small i.e, charging frequency is very high compared to dust plasma frequency which makes the ratio of the dust plasma frequency to the dust charging frequency zero On the other hand for nonadiabatic dust charge variation dust charging frequency is comparatively smaller than that of the adiabatic case So the ratio of the dust plasma frequency and dust charging frequency in this case possesses a small but finite value This nonadiabaticity of dust charge variation induces a dissipative effect which generates dust acoustic shock waves in plasma In presence of Lorentzian electrons this dust acoustic shock wave is dissipation dominated upto certain value of the normalized grain charge number and dispersion dominated thereafter Thus in a Lorentzian dusty plasma monotonic dust acoustic shock transfers to an oscillatory shock after a certain grain charge number This transfer is quicker in presence of larger number of suprathermal electrons ACKNOWLEDGMENTS The authors wish to acknowledge the referee for his valuable comments on this paper F Verheest, Waves in Dusty Space Plasmas (Kluwer, Dordrecht, 2000) K Shukla and A A Mamun, Introduction to Dusty Plasma Physics (IOP, Bristol, 2002) J H Chu, J B Du, and I J Lin, J Phys D 27, 296 (1994) 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