SOLITARY AND SHOCK WAVES SOLITARY AND SHOCK WAVES IN DUSTY PLASMASIN DUSTY PLASMAS

Dr. A A Dr. A A MamunMamunProfessor of PhysicsProfessor of PhysicsJahangirnagarJahangirnagar UniversityUniversityDhaka, BangladeshDhaka, Bangladesh

OUTLINEOUTLINEIntroductionIntroduction

Static Dust: DIA WavesStatic Dust: DIA WavesDIA Solitary WavesDIA Solitary WavesDIA Shock WavesDIA Shock Waves

Mobile Dust: DA WavesMobile Dust: DA WavesDA Solitary WavesDA Solitary WavesDA Shock WavesDA Shock Waves

ConclusionConclusion

IntroductionIntroductionWhat is What is dusty plasmadusty plasma??Dusty plasmaDusty plasma: plasma (electrons + ions + neutrals) : plasma (electrons + ions + neutrals) with with dust havingdust havingsize: size: µµmm -- mm: mm: not constant not constant in generalin general

mass: billions times heavier than ions: mass: billions times heavier than ions: not const. not const. in generalin general

charge: not neutral: charge: not neutral: -- or or ++ depending on charging processes: depending on charging processes:

1010--101044 electronic or electronic or protonicprotonic charge: charge: not const. not const. in general.in general.

The addition of such dust makes a plasma system very The addition of such dust makes a plasma system very complex. This is why, it is also known as a complex. This is why, it is also known as a complex plasma.complex plasma.

Ref: Review articles: Ref: Review articles: GoertzGoertz (1989); (1989); MendisMendis & Rosenberg (1994) & Rosenberg (1994) books: books: BouchouleBouchoule (1999); (1999); VerheestVerheest (2000); (2000); ShuklaShukla & & MamunMamun (2002)(2002)

What are exp. discoveries in dusty plasma physics?What are exp. discoveries in dusty plasma physics?

Γ>172 IKezi 1986Thomas et alChu & Lin IHayashi & Tachibana

DUST CRYSTAL 1994

DA WAVES ?? 1995 Barkan et al Rao, Shukla & Yu 1990

DL WAVES 1997 Morfill et alHomann et al Melandsø 1996

MACH CONES 1999 Samsonov et al Havnes et al 1996

DUST VOIDS 2001 Thomas et al Morfill et al 1999

DIA WAVES ?? 1996 Barkan et al Shukla & Silin 1992

What are main research areas in dusty plasma physics ?What are main research areas in dusty plasma physics ?

What are new waves in dusty plasmas?What are new waves in dusty plasmas?There are two new normal modes (waves) which are There are two new normal modes (waves) which are experimentally observed in experimentally observed in DPsDPs::

1. Static Dust:1. Static Dust: DIA WavesDIA Waves::

►► The restoring force comes from the electron thermal The restoring force comes from the electron thermal pressure and the inertia is provided by the ion masspressure and the inertia is provided by the ion mass. . The The dust charge maintains only the equilibrium charge neutrality.dust charge maintains only the equilibrium charge neutrality.

►► The phase speed of the DIA waves can be 10 The phase speed of the DIA waves can be 10 ------ 100 times 100 times higher than the ionhigher than the ion--acoustic speed acoustic speed CCii=(T=(Tee/m/mii))1/21/2 because of because of the factor nthe factor ni0i0/n/ne0e0 (=1+Z(=1+Zddnnd0d0/n/ne0e0) which can be 10) which can be 1022 ------ 101044 for for many space and laboratory dusty plasma situations.many space and laboratory dusty plasma situations.

►► DIA waves has been theoretically predicted by DIA waves has been theoretically predicted by ShuklaShukla & & SilinSilin(1992)(1992) and experimentally observed by and experimentally observed by DD’’AngeloAngelo et al (1996).et al (1996).

))((0

0

i

e

e

imT

nn

k =ω

2. Mobile Dust: DA Waves:2. Mobile Dust: DA Waves:

►► The restoring force mainly comes from ion thermal pressure The restoring force mainly comes from ion thermal pressure and the inertia is provided by dust mass: very low frequency and the inertia is provided by dust mass: very low frequency mode (V~9 cm/s, mode (V~9 cm/s, λλ~0.6 cm, f~15 Hz): visible with naked eyes: ~0.6 cm, f~15 Hz): visible with naked eyes:

►► DA waves has been theoretically predicted by DA waves has been theoretically predicted by RaoRao et al et al (1990)(1990) and experimentally observed by and experimentally observed by BarkanBarkan et al (1995).et al (1995).

))((0

0

d

id

i

dd

mTZ

nnZ

k≈

ω

What are solitary waves? How do they form? What are solitary waves? How do they form? How do they convert to shockHow do they convert to shock--like structures?like structures?

►► Solitary wavesSolitary waves are hump/dip shaped nonlinear waves of are hump/dip shaped nonlinear waves of permanent profile due to balance between nonlinearity and permanent profile due to balance between nonlinearity and dispersion (without dissipation). However, when dissipation dispersion (without dissipation). However, when dissipation is important, is important, shock wavesshock waves are formed. are formed.

0

x

y V(y)

y

x

y

A

A

00

m

m

The formation of arbitrary The formation of arbitrary amplitude stationary amplitude stationary solitarysolitaryand and shockshock waveswaves

•• Evolution equation for small amplitude solitary/shock Evolution equation for small amplitude solitary/shock waves: Kwaves: K--dVdV--Burgers equation (Burgers equation (KarpmanKarpman 1975):1975):

The term containing B The term containing B (C) is the dispersive (C) is the dispersive (dissipative) term(dissipative) term

When When thethe dispersive dispersive term is much more term is much more important than the important than the dissipative term: Kdissipative term: K--dVdV equation:equation:

When When thethe dissipative dissipative term is much more term is much more important than important than thethedispersive term dispersive term we we havehave: Burgers : Burgers equnequn::

Static Dust: DIA WavesStatic Dust: DIA WavesModel:Model:We consider an We consider an unmagnetizedunmagnetized DP with static dust [DP with static dust [ShuklaShukla & & SilinSilin 19921992].]. The dynamics of 1D DIA waves: The dynamics of 1D DIA waves:

where where nnii, , uuii, , , t, z are normalized by n, t, z are normalized by ni0i0, , CCii, T, Tee/e, 1//e, 1/ωωpipi, , μμ1/21/2λλdede, , respectively, and respectively, and μμ=n=ne0e0/n/ni0i0= 1= 1--ZZddnndd/n/ni0i0

..

DIA DIA SWsSWs::►► Small Amplitude:Small Amplitude:

RPM [RPM [WashimiWashimi & & TaniutiTaniuti 19661966]] reduces (1) reduces (1) -- (3) to a K(3) to a K--dVdV EqEq::

where where ζζ==εε1/21/2(z (z -- vv00t), t), ττ= = εε3/23/2tt , a, ass=(3=(3--1/1/μμ) ) μμ1/21/2/2, and /2, and bbss=(2=(2μμ))--3/23/2. .

The stationary SW solution of this KThe stationary SW solution of this K--dVdV EqEq::

where where mm=3u=3u00/a/ass and and ΔΔss=(4b=(4bss/u/u00))1/21/2. .

Since aSince ass is positive (negative) when is positive (negative) when μμ>(<)1/3, small amplitude >(<)1/3, small amplitude DIA DIA SWsSWs with with >(<)0 exist when >(<)0 exist when μμ>(<)1/3. >(<)1/3.

• Arbitrary Amplitude:SPA [SPA [SagdeevSagdeev 1966]1966] reduces (1) reduces (1) -- (3) to an energy integral:(3) to an energy integral:

where where ξξ=z =z -- MtMt and V(and V( ) reads [) reads [BharuthramBharuthram & & ShuklaShukla 1992]:1992]:

It is clear:It is clear: V(V( ) = ) = dV(dV( )/d)/d = 0 at = 0 at =0. So SW solution of =0. So SW solution of this energy integral exists if (dthis energy integral exists if (d22V/dV/d 22)) =0=0< 0 and < 0 and (d(d33V/dV/d 33

=0=0>(<) 0 for >(<) 0 for >(<)0. The vanishing of quadratic term: >(<)0. The vanishing of quadratic term: MMcc==μμ--1/21/2. At M=M. At M=Mcc the cubic term of V(the cubic term of V( ) becomes (3) becomes (3--1/1/μμ) ) μμ1/21/2/2= a/2= ass. .

This means that arbitrary amplitude DIA This means that arbitrary amplitude DIA SWsSWs with with >(<)0 >(<)0 exist when exist when μμ>(<)1/3. >(<)1/3.

• Effects of Non-Planar GeometryThe dynamics of DIA waves in planar (ν=0; r=z) or non-planar [cylindrical (ν=1) or spherical (ν=2)] geometry:

RPM [Maxon & Vieceli 1974] reduces these Eqs. to a mK-dV Eq:

where ζ=-ε1/2 (r + v0t), τ= ε3/2t. The 2nd term arises due to the effects of non-planar geometry. We have numerically solved, and studied the effects of cylindrical (ν=1) and spherical (ν=2) geometries on the

geometries on time-dependent DIA SWs. The numerical results are as follows [Mamun & Shukla 2002]:

ν=1red:τ=-9green:τ=-6blue:τ=-3

ν=2red:τ=-9green:τ=-6blue:τ=-3

μ=0.4

μ=0.2

μ=0.4

μ=0.2

• Effects of Ion-TemperatureTo examine the effects of ion-temperature we use Eqs. (1), (3) and

Pi is normalized by ni0Ti and σi=Ti/Te. The RPM reduces (1), (3)-(5) to

The numerical analysis of V( ,σi) or the EI: as we increase ion fluid temperature, amplitude of DIA SWs decreases, but their width increases. The SW solutions of the EI [Mamun 1997] for M=1.45:

σi=0

σi=0.1σi=0.2

σi=0

σi=0.1σi=0.2

μ=0.2

μ=0.4

DIA Shock WavesTheoretical Model:We consider an unmagnetized, dissipative DP [Nakamura et al 1999; Shukla 2000] which can be described by (1), (3) and

(6)

RPM and ηi=ε1/2η0 reduce (1), (3) and (6) to a K-dV-Bergers Eq:

(7)

where η=η0/2V0.

Eq. (7) has monotonic shock solution for η2>4U0bs (where U0=shock speed) and oscillatory shock solution for η2<4U0bs [Karpman 1975; Shukla & Mamun 2001]. For very small η, i.e. η2<<4U0bs, shock waves will have oscillatory profiles in which first few oscillations will be close to solitons moving with U0.

Experimental Observation• DIA shocks were experimentally observed by Nakamura et al (1999).

The plasma parameters used for this experiment: ne ≈108 cm-3, Te≈104 K, Ti ≈ 0.1Te, Zd ≈105 for nd<103 cm-3 and Zd≈ 102 for nd<105 cm-3.

• The oscillatory shock waves were excited in a plasma first without dust and then with dust: as nd is increased, oscillatory wave structures behind the shock decreases and completely disappears at nd=103 cm-3.

A source of Dissipation: Dust Charge Fluctuation

To find an alternate mechanism for the formation of DIA shock waves, we consider charge fluctuating dust grains. Then the dynamics of DIA waves can be described by (1), (2) and

(8)

Zd (normalized by Zd0) is not constant, but varies according to

(9)

where η=[αme(1-μ)/2mi]1/2, β=(rd/ad)1/2, βi=β(πme/mi)1/2, ad=nd0-1/3 and

α=Zd0e2/Terd. At equilibrium we have

where u0 (normalized by Ci) is the streaming speed of ions.

RPM, ηc=ε1/2ηc0 reduce (1), (2), (8), (9) to K-dV-Bergers Eq.

(10)

where [Mamun & Shukla 2001]

Two situations for analytic solution of Eq. (10):

• C2>4U0B: This corresponds to monotonic shock solution of Eq. (10). For C2>>4U0B its monotonic shock solution [Karpman 1975] is

• C2<4U0B: This corresponds to oscillatory shock solution of Eq. (10). For C2<<4U0B its oscillatory shock solution [Karpman 1975] is

where 0= (ζ=0) and K is a constant. This solution means that for very small C, shock waves will have oscillatory profiles in which first few oscillations will be close to solitons moving with U0.

• Therefore, dust charge fluctuation can act as a source of dissipation which may responsible for the formation of monotonic or oscillatory DIA shock profiles depending on the plasma parameters.

Mobile Dust: DA WavesModel:We consider an unmagnetized DP with mobile dust [Rao et al 1990]. The dynamics of 1D DA waves:

(11)

(12)

(13)

where nd, ud, , t, z are normalized by nd0, Cd, Ti/e, 1/ωpd, Cd/ωpd, respectively, μe=1/(δ-1), μi=δ/(δ-1), δ=ni0/ne0, σi=Ti/Te.

DA SWs:Small Amplitude:RPM [Washimi & Taniuti 1966] reduces (11)- (13) to a K-dV Eq.

and . The stationary SW solution of this K-dV Eq.

where m=3u0/as and Δs=(4bs/u0)1/2.

Since δ>1 and σi>0, i.e. as is always negative, small amplitude DA SWs with <0 can only exist [Mamun 1999].

• Arbitrary Amplitude:SPA [Sagdeev 1966] reduces (11) - (13) to an energy integral:

where V( ) reads [Mamun1999]:

where ξ=z - Mt and V( ) reads [Mamun 1992]:

V( ) = dV( )/d = 0 at =0. So SW solution of this energy integral exists if (d2V/d 2) =0< 0 and (d3V/d 3

=0>(<) 0 for >(<)0. The vanishing of quadratic term: Mc=[(δ-1)/(δ+σi)]1/2. At M=Mc the cubic term of V( ):

• Effects of Trapped Ion DistributionThe trapped ion distribution of Schamel reduces the ion number density ni for | |<<1 to a simple form [Schamel 1975] :

(14)

where |σit|=Tit/Ti. Eq. (14) is valid for both σit<0 and σit>0.

Now, using RPM and Eq. (11) - (13) with the replacement of Boltzmann ion distribution [exp(- )] by the trapped ion distribution [Eq. (14)] we obtain

This is a modified KdV equation with stronger nonlinearity: larger amplitude, smaller width and larger propagation speed.

• Effects of Nonthermal Ion DistributionThe nonthermal ion distribution of Cairns et al reduces the ion number density ni to a form [Cairns et al 1995)] :

(15)

where α determines the fraction of fast particles present.

Now, using SPA and Eq. (11) - (13) with the replacement of Boltzmann ion distribution [exp(- )] by the nonthermal ion distribution [Eq. (15)] we obtain:

where V( ) is [Mamun et al 1996]:

and α0=4α/(1+3α). The analysis of V( ) predicts that the presence of nonthermal (fast) ions [with α>0.155] supports coexistence of DA SWs with

<0 and >0.

These results are clearly observed from the following plots:

δ=10σi=0.1

δ=10σi=0.1

δ=10σi=0.1

δ=10σi=0.1

• Effects of Positive Dust ComponentModel:We consider an unmagnetized DP with dust of opposite polarity [Sayed & Mamun 2007]. The dynamics of 1D DA waves:

where subscript 1 (2) corresponds to - (+) dust. N1 (N2) are normalized by n10 (n02), U1 and U2 are normalized by C1=(Z1kBTi/m1)1/2,

ψ is normalized by kBTi/e, α=Z2/Z1, β=m1/m2, µe=ne0/Z1n10, µi=ni0/Z1n10, and σ=Ti/Te.

RPM [Washimi & Taniuti 1966] reduces (1) - (5) to a K-dV Eq:

(6)

where A and B are given by

(7)

The steady state solution of K-dV Eq. (6) is

(8)

where the amplitude ψm and the width Δ are given by

(9)

Eqs. (7) – (9) implies that the solitary potential profile is + (-) if A> (<) 0. So, we have numerically analyzed A and obtain A=0 curves displayed in figure below:

0.2 0.3 0.4 0.5 0.6 0.7Μi

40

60

80

100

120

140

160

Β + (-) solitary profiles exist for the parameters whose values lie above (below) these A=0 curves, where σ=0.5, α=0.01, µe=0.1 (solid curve), µe=0.15 (dotted curve) and µe=0.2 (dash curve).

We have graphically shown how amplitude of + (corresponding to parameters whose values lie above A=0 curves) and –(corresponding to the parameters whose values lie below A=0 curves) solitary potential profiles vary with µi.

0.45 0.5 0.55 0.6 0.65 0.7Μi

0.275

0.3

0.325

0.35

0.375

0.4

0.425

0.45�m

0.45 0.5 0.55 0.6 0.65 0.7Μi

-0.7

-0.6

-0.5

-0.4

�m

α=0.01, σ=0.5, µe=0.2, β=150 (solidcurve), β=160 (dotted curve), and β=170 (dash curve).

α=0.01, σ=0.5, µe=0.2, β=5 (solidcurve), β=10 (dotted curve), and β=15 (dash curve).

We have graphically shown how width of + (corresponding to parameters whose values lie above A=0 curves) and –(corresponding to the parameters whose values lie below A=0 curves) solitary potential profiles vary with µi.

0.45 0.5 0.55 0.6 0.65 0.7Μi

7

8

9

�

α=0.01, σ=0.5, µe=0.2, β=150 (solidcurve), β=160 (dotted curve), and β=170 (dash curve).

α=0.01, σ=0.5, µe=0.2, β=5 (solidcurve), β=10 (dotted curve), and β=15 (dash curve).

0.45 0.5 0.55 0.6 0.65 0.7Μi

5.5

6

6.5

7

7.5

8

�

DA Shock WavesWe consider an unmagnetized strongly coupled DP described by GH equations [Kaw & Sen 1998]: Eqs. (11), (13) and

(16)

where

is the normalized longitudinal viscosity coefficient, τm is the normalized visco-elastic relaxation time, ηb and ζb are shear and bulk viscosity coefficients, νdn is the normalized dust-neutral collision frequency. All transport coefficients are well defined by Kaw & Sen (1999), Mamun et al (2000), Shukla & Mamun (2001), etc.

Using RPM, ηd=ε1/2η0 and Eqs. (11), (13) & (16) we have:

(17)

where

It is obvious that for a collision-less limit (νdn=0) A<0, B>0, and C>0, but for a highly collision limit (νnτm>2, since aδσ≈2 for δ=10 and σi=1) A>0, B>0 and C>0.

There are two situations for analytic solutions of Eq. (17):

• C2>4U0B: This corresponds to monotonic shock solution of Eq. (17). For C2>>4U0B its monotonic shock solution [Karpman 1975] is

• C2<4U0B: This corresponds to oscillatory shock solution of Eq. (17). For C2<<4U0B its oscillatory shock solution [Karpman 1975] is

where 0= (ζ=0) and K is a constant. This solution means that for very small ηd shock waves will have oscillatory profiles in which first few oscillations will be close to solitons moving with U0.

• Therefore, strong correlation of dust can act as a source of dissipation which may responsible for the formation of monotonic or oscillatory DA shock profiles depending on the plasma parameters.

ConclusionConclusionThe dust particle does not only modify the existing plasma The dust particle does not only modify the existing plasma waves, but also introduces a number of new waves, but also introduces a number of new eigeneigen modes, modes, e.g. DIA, DA, DL, etc., which in nonlinear regime form e.g. DIA, DA, DL, etc., which in nonlinear regime form different types of interesting coherent structures. different types of interesting coherent structures.

The basic features of DIA & DA The basic features of DIA & DA SWsSWs [[MamunMamun et al 1996; et al 1996; MamunMamun & & ShuklaShukla 2001, 2002]2001, 2002] in comparison with IA in comparison with IA SWsSWs::

IA IA SWsSWs((VVTeTe>V>VPP>>VVTiTi))

DIA DIA SWsSWs((VVTeTe>V>VPP>>VVTiTi))

DA DA SWsSWs((VVTiTi>V>VPP>>VVTdTd))

>0 (only)>0 (only) >0 when >0 when μμ>1/3>1/3<0 when <0 when μμ<1/3<1/3

<0 (only)<0 (only)

nnee>0>0nnii>0>0

nnee>(<)0 >(<)0 when when μμ>(<)1/3>(<)1/3nnii>(<) 0 >(<) 0 when when

μμ>(<)1/3>(<)1/3

nnee<0; <0; nnii>0>0nndd<0<0

V>V>CCii

CCii=(T=(Tee/m/mii))1/21/2V>V>CCii μμ--1/21/2

μμ=n=ne0e0/n/ni0i0

V>V>CCdd

CCdd=(Z=(ZddTTii/m/mdd))1/21/2

Effect of ion (dust) fluid temperature on DIA and DA Effect of ion (dust) fluid temperature on DIA and DA SWsSWs: As : As ion (dust) fluid temperature increases, amplitude of DIA (DA) ion (dust) fluid temperature increases, amplitude of DIA (DA) SWsSWs decreases, but their width increases decreases, but their width increases [[MamunMamun 1997; 1997; MamunMamun & & ShuklaShukla 2000].2000].

Effect of a nonEffect of a non--planer geometry on DIA and DA planer geometry on DIA and DA SWsSWs: It : It reduces to a modified Kreduces to a modified K--dVdV equation containing an extraequation containing an extra--term term ((νν/2T)/2T) withwith νν=1 (2) is for cylindrical (spherical) geometry =1 (2) is for cylindrical (spherical) geometry [[MamunMamun & & ShuklaShukla 2001, 2002].2001, 2002].

Effect of dust charge fluctuation on DIA Effect of dust charge fluctuation on DIA SWsSWs: The dust grain : The dust grain charge fluctuations do not only change the amplitude and charge fluctuations do not only change the amplitude and width of DIA width of DIA SWsSWs, but also provide a source of dissipation, , but also provide a source of dissipation, and may be responsible for the formation of DIA shock waves and may be responsible for the formation of DIA shock waves [[MamunMamun & & ShuklaShukla 2002].2002].

Effect of fast ions on DA Effect of fast ions on DA SWsSWs: The presence of fast ions may : The presence of fast ions may allow compressive and allow compressive and rarefactiverarefactive SWsSWs to coexist: to coexist: electrostatic electrostatic SWsSWs observed by observed by FrejaFreja and Viking spacecrafts and Viking spacecrafts [[MamunMamun et al 1996].et al 1996].

Effects of trapped ion distribution on DA Effects of trapped ion distribution on DA SWsSWs: it gives rise to : it gives rise to a modified Ka modified K--dVdV equation exhibiting stronger nonlinearity: equation exhibiting stronger nonlinearity: smaller width & larger propagation speed smaller width & larger propagation speed [[MamunMamun et al 1996; et al 1996; MamunMamun 1997]. 1997].

Effects of positive dust component on DA Effects of positive dust component on DA SWsSWs: The : The consideration of positive dust gives rise to the existence of consideration of positive dust gives rise to the existence of positive positive solitonssolitons for some parametric regimes as well as it for some parametric regimes as well as it significantly modify the basic properties (amplitude, width & significantly modify the basic properties (amplitude, width & Mach number) of DA solitary waves Mach number) of DA solitary waves [[MamunMamun and and ShuklaShukla 2002; 2002; SayedSayed & & MamunMamun 2007].2007].

Effect of strong dust correlation on DA Effect of strong dust correlation on DA SWsSWs: It provides a : It provides a source of dissipation, and is responsible for the formation of source of dissipation, and is responsible for the formation of DA shock waves DA shock waves [[ShuklaShukla & & MamunMamun 2001].2001]. The combined The combined effects of strong dust correlation and trapped ion distribution effects of strong dust correlation and trapped ion distribution reduce to a modified Kreduce to a modified K--dVdV--Burgers equation with some new Burgers equation with some new features features [[MamunMamun et al 2004]et al 2004]

Because of time limit, I confined my talk to an Because of time limit, I confined my talk to an unmagnetizedunmagnetizeddusty plasma. However, a number of investigations dusty plasma. However, a number of investigations [[MamunMamun1998a; 1998a; MamunMamun et al 1998, et al 1998, MamunMamun & & HassanHassan 1999] 1999] on DA on DA SWsSWsin a magnetized dusty plasma have been made: (i) external in a magnetized dusty plasma have been made: (i) external magnetic field makes ESmagnetic field makes ES--SWsSWs more spiky and (ii) ESmore spiky and (ii) ES--SWsSWsbecomes unstable: multibecomes unstable: multi--dimensional instability dimensional instability [[MamunMamun1998b, 1998c].1998b, 1998c].

Also because of time limit, I confined my talk to ES Also because of time limit, I confined my talk to ES SWsSWs. . However, a number of investigations However, a number of investigations [[MamunMamun 1999; 1999; MamamunMamamun & & GebreelGebreel 2000; 2000; MamunMamun & & ShuklaShukla 2003]2003] on dust on dust EM EM SWsSWs in a magnetized dusty plasma have been made.in a magnetized dusty plasma have been made.

The physics of nonlinear waves that we have The physics of nonlinear waves that we have discussed must play a significant role in discussed must play a significant role in understanding the properties of localized understanding the properties of localized electrostatic as well as electromagnetic solitary electrostatic as well as electromagnetic solitary structures in space & laboratory dusty plasmas.structures in space & laboratory dusty plasmas.

Last but, of course, not the least: