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TP4 Seminar, 16 November 2005

Fluid theory for electrostatic

wave packet amplitude self-modulation

in Space and laboratory plasmas

Focus on electrostatic modes in dusty plasmas

and dust crystals (?)

Ioannis Kourakis

Ruhr-Universität Bochum

Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie

D–44780 Bochum, Germany

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf


I. Kourakis, Reductive perturbation method ... 1

Outline

❏ Introduction

– Amplitude Modulation: a rapid overview of notions and ideas;

– Relevance with space and laboratory observations;

– Intermezzo: Dusty Plasmas (DP) (or Complex Plasmas) & dust crystals.

❏ Case study 1 (Part A): Fluid model for ES waves in weakly-coupled DP.

– A pedagogical paradigm: Ion–acoustic plasma waves;

– Other examples: EAWs, DAWs, ...

❏ The reductive perturbation (multiple scales) formalism.

❏ Modulational Instability (MI) & envelope excitations.

❏ Case study 2 (Part B): Solitary excitations in dust (Debye) lattices.

❏ Conclusions.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 2

1. Intro. The mechanism of wave amplitude modulation

The amplitude of a harmonic wave may vary in space and time:


www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 3

1. Intro. The mechanism of wave amplitude modulation

The amplitude of a harmonic wave may vary in space and time:


This modulation (due to nonlinearity) may be strong enough to lead to wave

collapse (modulational instability) or ...

→ ?

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 4

1. Intro. The mechanism of wave amplitude modulation

The amplitude of a harmonic wave may vary in space and time:


This modulation (due to nonlinearity) may be strong enough to lead to wave

collapse (modulational instability) or to the formation of envelope solitons:

→ ?

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 5

Modulated structures occur widely in Nature,

e.g. in oceans (freak waves, or rogue waves) ...

(from: [Kharif & Pelinovsky, Eur. Journal of Mechanics B/Fluids 22, 603 (2003)])

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 6

... during freak wave reconstitution in water basins, ...

(from: [Klauss, Applied Ocean Research 24, 147 (2002)])

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 7

..., in EM field measurements in the magnetosphere, ...

(from: [Ya. Alpert, Phys. Reports 339, 323 (2001)])

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 8

..., in satellite (e.g. CLUSTER, FAST, ...) observations:

(*) From: O. Santolik et al., JGR 108, 1278 (2003); R. Pottelette et al., GRL 26 2629 (1999).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 9

Modulational instability (MI) was observed in simulations,

e.g. early (1972) numerical experiments of EM cyclotron waves:

[from: A. Hasegawa, PRA 1, 1746 (1970); Phys. Fluids 15, 870 (1972)].

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 10

Spontaneous MI has been observed in experiments,:

e.g. on ion acoustic waves

[from: Bailung and Nakamura, J. Plasma Phys. 50 (2), 231 (1993)].

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 11

Questions to be addressed in this brief presentation:

❏ How can one describe the (slow) evolution (modulation) of a

wave amplitude in space and time?

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 12

Questions to be addressed in this brief presentation:

❏ How can one describe the (slow) evolution (modulation) of a

wave amplitude in space and time?

❏ Can Modulational Instability (MI) of plasma “fluid” modes be

predicted by a simple, tractable analytical model?

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 13

Questions to be addressed in this brief presentation:

❏ How can one describe the (slow) evolution (modulation) of a

wave amplitude in space and time?

❏ Can Modulational Instability (MI) of plasma “fluid” modes be

predicted by a simple, tractable analytical model?

❏ Can envelope modulated localized structures (such as those

observed in space and laboratory plasmas) be modeled by an

exact theory?

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 14

Questions to be addressed in this brief presentation:

❏ How can one describe the (slow) evolution (modulation) of a

wave amplitude in space and time?

❏ Can Modulational Instability (MI) of plasma “fluid” modes be

predicted by a simple, tractable analytical model?

❏ Can envelope modulated localized structures (such as those

observed in space and laboratory plasmas) be modeled by an

exact theory?

❏ Focus: electrostatic waves; e.g. dust-ion acoustic waves

(DIAW); electron acoustic (EA), dust acoustic (DA) waves, ...

also (part B): dust-lattice waves in Debye crystals.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 15

Intermezzo: DP – Dusty Plasmas (or Complex Plasmas):

definition and characteristics of a focus issue

❏ Ingredients:

– electrons e − (charge −e, mass m e ),

– ions i + (charge +Z i e, mass m i ), and

– charged micro-particles ≡ dust grains d (most often d − ):

charge Q = ±Z d e ∼ ±(10 3 − 10 4 ) e,

mass M ∼ 10 9 m p ∼ 10 13 m e ,

radius r ∼ 10 −2 µm up to 10 2 µm.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 16

Origin: Where does the dust come from?

❏ Space: cosmic debris (silicates, graphite, amorphous carbon), comet dust,

man-made pollution (Shuttle exhaust, satellite remnants), ...

❏ Atmosphere: extraterrestrial dust (meteorites): ≥ 2 · 10 4 tons a year (!)(*),

atmospheric pollution, chemical aerosols, ...

❏ Fusion reactors: plasma-surface interaction, carbonaceous particulates

resulting from wall erosion-created debris (graphite, CFCs: Carbon Fiber

Composites, ...)

❏ Laboratory: (man-injected) melamine–formaldehyde particulates (**)

injected in rf or dc discharges; 3d (= multiple 2d layers) or 1d (by

appropriate experimental setting) crystals.

Sources: [P. K. Shukla & A. Mamun 2002], (*) [DeAngelis 1992], (**) [G. E. Morfill et al. 1998]

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 17

Some unique features of the Physics of Dusty Plasmas:

❏ Complex plasmas are overall charge neutral; most (sometimes all!) of the

negative charge resides on the microparticles;

❏ The microparticles can be dynamically dominant: mass density ≈ 10 2 times

higher than the neutral gas density and ≈ 10 6 times higher than the ion

density !

❏ Studies in slow motion are possible due to high M i.e. low Q/M ratio (e.g.

dust plasma frequency: ω p,d ≈ 10 − 100 Hz);

❏ The (large) microparticles can be visualised individually and studied at the

kinetic level (with a digital camera!) → video;

❏ Dust charge (Q ≠ const.) is now a dynamical variable, associated to a new

collisionless damping mechanism;

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 18

(...continued) More “heretical” features are:

❏ Important gravitational (compared to the electrostatic) interaction effects;

gravito-plasma physics; gravito-electrodynamics; Jeans-type (gravitational)

plasma instabilities etc. [Verheest PPCF 41 A445, 1999]

❏ Complex plasmas can be strongly coupled and exist in “liquid”

(1 < Γ < 170) and “crystalline” (Γ > 170 [IKEZI 1986]) states, depending on the

value of the effective coupling (plasma) parameter Γ;

Γ eff = < E potential >

< E kinetic > ∼ Q2

r T e−r/λ D

(r: inter-particle distance, T : temperature, λ D : Debye length).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 19

Part 2. Fluid model for electrostatic waves in weakly-coupled DP.

The standard recipe involves the following ingredients:

– A dynamical constituent (particle species) α ;

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 20

Part 2. Fluid model for electrostatic waves in weakly-coupled DP.

The standard recipe involves the following ingredients:

– A dynamical constituent (particle species) α ;

– a neutralizing background of (one or several) species α ′

(in a – presumably – known state).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 21

Part 2. Fluid model for electrostatic waves in weakly-coupled DP.

The standard recipe involves the following ingredients:

– A dynamical constituent (particle species) α ;

– a neutralizing background of (one or several) species α ′

(in a – presumably – known state).

Typical paradigm (cf. textbooks) to focus upon:

– Ion acoustic waves (IAW): ions (α = i) in a background of thermalized

electrons (α ′ = e): n e = n e,0 e eΦ/K BT e

.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 22

Part 2. Fluid model for electrostatic waves in weakly-coupled DP.

The standard recipe involves the following ingredients:

– A dynamical constituent (particle species) α ;

– a neutralizing background of (one or several) species α ′

(in a – presumably – known state).

Typical paradigm (cf. textbooks) to focus upon:

– Dust-ion acoustic waves (DIAW): ions (α = i) in a background of thermalized

electrons (α ′ = e): n e = n e,0 e eΦ/K BT e

and immobile dust grains: n d = const.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 23

Part 2. Fluid model for electrostatic waves in weakly-coupled DP.

The standard recipe involves the following ingredients:

– A dynamical constituent (particle species) α ;

– a neutralizing background of (one or several) species α ′

(in a – presumably – known state).

Typical paradigm (cf. textbooks) to focus upon:

– Ion acoustic waves (IAW): ions (α = i) in a background of thermalized

electrons (α ′ = e): n e = n e,0 e eΦ/K BT e

.

The theory applies to a variety of other modes, including e.g.

– Electron acoustic waves (EAW): electrons (α = e) in a background of

stationary ions (α ′ = i): n i = cst.;

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 24

Part 2. Fluid model for electrostatic waves in weakly-coupled DP.

The standard recipe involves the following ingredients:

– A dynamical constituent (particle species) α ;

– a neutralizing background of (one or several) species α ′

(in a – presumably – known state).

Typical paradigm (cf. textbooks) to focus upon:

– Ion acoustic waves (IAW): ions (α = i) in a background of thermalized

electrons (α ′ = e): n e = n e,0 e eΦ/K BT e

.

The theory applies to a variety of other modes, including e.g.

– Electron acoustic waves (EAW): electrons (α = e) in a background of

stationary ions (α ′ = i): n i = cst.;

– DAW: dust grains (α = d) against thermalized electrons and ions (α ′ = e, i):

n e = n e,0 e eΦ/K BT e

, n i = n i,0 e −Z ieΦ/K B T i .

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 25

Density n α (continuity) equation:

Fluid moment equations:

∂n α

∂t + ∇ · (n α u α ) = 0

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 26

Density n α (continuity) equation:

Mean velocity u α equation:

Fluid moment equations:

∂n α

∂t + ∇ · (n α u α ) = 0

[(*) Cold fluid model]

∂u α

∂t + u α · ∇u α = − q α

m α

∇ Φ

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 27

Density n α (continuity) equation:

Mean velocity u α equation:

[(*) Cold vs. Warm fluid model]

Fluid moment equations:

∂n α

∂t + ∇ · (n α u α ) = 0

∂u α

∂t + u α · ∇u α = − q α

m α

∇ Φ− 1

m α n α

∇p α

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 28

Density n α (continuity) equation:

Mean velocity u α equation:

Fluid moment equations:

∂n α

∂t + ∇ · (n α u α ) = 0

∂u α

∂t + u α · ∇u α = − q α

m α

∇ Φ− 1

m α n α

∇p α

Pressure p α equation:

[(*) Cold vs. Warm fluid model]

∂p α

∂t + u α · ∇p α = −γ p α ∇ · u α

[γ = (f + 2)/f = c P /c V : ratio of specific heats e.g. γ = 3 for 1d, γ = 2 for 2d, etc.].

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 29

Fluid moment equations:

Density n α (continuity) equation:

∂n α

∂t + ∇ · (n α u α ) = 0

Mean velocity u α equation:

∂u α

∂t + u α · ∇u α = − q α

m α

∇ Φ− 1

m α n α

∇p α

Pressure p α equation:

∂p α

∂t + u α · ∇p α = −γ p α ∇ · u α

The potential Φ obeys Poisson’s eq.:


∇ 2 Φ = −4π q α ′′ n α ′′ = 4π e (n e − Z i n i + ...)

α ′′ =α,{α ′ }

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 30

Fluid moment equations:

∂n α

∂t + ∇ · (n α u α ) = 0

∂u α

∂t + u α · ∇u α = − q α

∇ Φ− 1 ∇p α

m α m α n α

∂p α

∂t + u α · ∇p α = −γ p α ∇ · u α


∇ 2 Φ = −4π q α ′′ n α ′′ = 4π e (n e − Z i n i + ...)

α ′′ =α,{α ′ }

Hypothesis: Overall charge neutrality at equilibrium:

q α n α,0 = − ∑ {α ′ }

q α ′ n α ′ ,0 ,

e.g. for DIAW: n e,0 − Z i n i,0 − s Z d n d,0 = 0 (s = q d /|q d | = ±1).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 31

Reduced moment evolution equations:

Defining appropriate scales (see next slide) one obtains:

also,

∂n

+ ∇ · (n u) = 0 ,

∂t

∂u

∂t + u · ∇u = −s ∇φ − σ n ∇p ,

∂p

+ u · ∇p = −γ p ∇ · u ;

∂t

∇ 2 φ = φ − α φ 2 + α ′ φ 3 − s β (n − 1) ; (1)

i.e. Poisson’s Eq. close to equilibrium: φ ≪ 1; s = sgnq α = ±1.

- The dimensionless parameters α, α ′ and β must be determined exactly for

any specific problem. They incorporate all the essential dependence on the

plasma parameters.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 32

We have defined the reduced (dimensionless) quantities:

- particle density: n = n α /n α,0 ;

- mean (fluid) velocity: u = [m α /(k B T ∗ )] 1/2 u α ≡ u α /c ∗ ;

where c ∗ = (k B T ∗ /m α ) 1/2 is a characteristic “sound” velocity (∗) ;

- dust pressure: p = p α /p 0 = p α /(n α,0 k B T ∗ );

- electric potential: φ = Z α eΦ/(k B T ∗ ) = |q α |Φ/(k B T ∗ );

Also, time and space are scaled over:

- a characteristic time scale t 0 , e.g. the inverse DP plasma frequency

ω −1

p,α = (4πn α,0 q 2 α/m α ) −1/2

- a characteristic length scale r 0 = c ∗ t 0 , e.g. an effective Debye length

λ D,eff = (k B T ∗ /4πn α,0 qα) 2 1/2 .

Finally, σ = T α /T ∗ is the temperature (ratio)

(∗) e.g. T∗ = T e for DIAWs.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 33

3. Reductive Perturbation Technique

– 1st step. Define multiple scales (fast and slow) i.e. (in 2d)

X 0 = x , X 1 = ɛ x , X 2 = ɛ 2 x , ...

Y 0 = y , Y 1 = ɛ y , Y 2 = ɛ 2 y , ...

T 0 = t , T 1 = ɛ t , T 2 = ɛ 2 t , ...

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 34

3. Reductive Perturbation Technique

– 1st step. Define multiple scales (fast and slow) i.e. (in 2d)

and modify operators appropriately:

X 0 = x , X 1 = ɛ x , X 2 = ɛ 2 x , ...

Y 0 = y , Y 1 = ɛ y , Y 2 = ɛ 2 y , ...

T 0 = t , T 1 = ɛ t , T 2 = ɛ 2 t , ...


∂x →


∂y →


∂t →

∂ + ɛ ∂ + ɛ 2

∂X 0 ∂X 1

∂ + ɛ ∂ + ɛ 2

∂Y 0 ∂Y 1

∂ + ɛ ∂ + ɛ 2

∂T 0 ∂T 1


+

∂X 2

...

∂ +

∂Y 2

...


∂T 2

+ ...

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 35

3. Reductive Perturbation Technique

– 1st step. Define multiple scales (fast and slow) i.e. (in 2d)

X 0 = x , X 1 = ɛ x , X 2 = ɛ 2 x , ...

Y 0 = y , Y 1 = ɛ y , Y 2 = ɛ 2 y , ...

T 0 = t , T 1 = ɛ t , T 2 = ɛ 2 t , ...

– 2nd step. Expand near equilibrium:

n α ≈ n α,0 + ɛ n α,1 + ɛ 2 n α,2 + ...

u α ≈ 0 + ɛ u α,1 + ɛ 2 u α,2 + ...

p α ≈ p α,0 + ɛ p α,1 + ɛ 2 p α,2 + ...

φ ≈ 0 + ɛ φ 1 + ɛ 2 φ 2 + ...

(p α,0 = n α,0 k B T α ; ɛ ≪ 1 is a smallness parameter).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 36

Reductive perturbation technique (continued)

– 3rd step. Project on Fourier space, i.e. consider ∀m = 1, 2, ...

S m =

m∑

l=−m

Ŝ (m)

l

e il(k·r−ωt) = Ŝ(m) 0 + 2

m∑

l=1

Ŝ (m)

l

cos l(k · r − ωt)

for S m = (n m , {u x,m , u y,m }, p m , φ m ),

i.e. essentially:

n 1 = n (1)

0 + ñ (1)

1 cos θ , n 2 = n (2)

0 + ñ (2)

1 cos θ + ñ (2)

2 cos 2θ, etc.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 37

Reductive perturbation technique (continued)

– 3rd step. Project on Fourier space, i.e. consider ∀m = 1, 2, ...

S m =

m∑

l=−m

Ŝ (m)

l

e il(k·r−ωt) = Ŝ(m) 0 + 2

m∑

l=1

Ŝ (m)

l

cos l(k · r − ωt)

for S m = (n m , {u x,m , u y,m }, p m , φ m ),

i.e. essentially:

n 1 = n (1)

0 + ñ (1)

1 cos θ , n 2 = n (2)

0 + ñ (2)

1 cos θ + ñ (2)

2 cos 2θ, etc.

– 4rth step. (for multi-dimensional propagation) Modulation obliqueness:

(m)

the slow amplitudes ˆφ

l

, etc. vary only along the x-axis:

Ŝ (m)

l

= Ŝ(m) l

(X j , T j ) , j = 1, 2, ...

while the fast carrier phase θ = k · r − ωt is now:

k x x + k y y − ωt = k r cos α − ωt .

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 38

First-order solution (∼ ɛ 1 )

Substituting in the model Eqs., and isolating terms in m = 1, we obtain:

❏ The dispersion relation ω = ω(k):

ω 2 =

βk2

k 2 + 1 + γ σ k2 (2)

e.g. for DIAWs

ω ≈

( ) 1/2 ( ) 1/2 ( ) 1/2 ( ) 1/2 ni,0 kB T e

n d,0 kB T e

k = 1−sZ d k

n e,0 m i n e,0 m i

❏ The solution(s) for the 1st–harmonic amplitudes (e.g. ∝ φ (1)

1 ):

n (1)

1 = s 1 + k2

φ (1)

1 = 1 β γ p(1) 1 =

k

ω cos θ u(1) 1,x = k

ω sin θ u(1)

1,y (3)

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 39

Second-order solution (∼ ɛ 2 )

❏ From m = 2, l = 1, we obtain the relation:

where

– ψ = φ (1)

1 is the potential correction (∼ ɛ 1 );

– v g = ∂ω(k)

∂k x

is the group velocity along ˆx;

∂ψ ∂ψ

+ v g = 0 (4)

∂T 1 ∂X 1

– the wave’s envelope satisfies: ψ = ψ(ɛ(x − v g t)) ≡ ψ(ζ).

❏ The solution, up to ∼ ɛ 2 , is of the form:

φ ≈ ɛ ψ cos θ + ɛ 2 [ φ (2)

0 + φ (2)

1 cos θ + φ (2)

2 cos 2θ ] + O(ɛ 3 ) ,

etc. (+ similar expressions for n d , u x , u y , p d ): → Fourier harmonics!.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 40

Third-order solution (∼ ɛ 3 )

❏ Compatibility equation (from m = 3, l = 1), in the form of:

i ∂ψ

∂τ + P ∂2 ψ

∂ζ 2 + Q |ψ|2 ψ = 0 . (5)

i.e. a Nonlinear Schrödinger–type Equation (NLSE) .

❏ Variables: ζ = ɛ(x − v g t) and τ = ɛ 2 t;

❏ Dispersion coefficient P :

P = 1 2

∂ 2 ω

∂k 2 x

= 1 2

[

]

ω ′′ (k) cos 2 α + ω ′ (k) sin2 α

; (6)

k

❏ Nonlinearity coefficient Q: ...

A (lengthy!) function of k, angle α and T e , T i , ... → (omitted).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 41

4a. Modulational (in)stability analysis

❏ The NLSE admits the harmonic wave solution:

ψ = ˆψ e iQ| ˆψ| 2τ + c.c.

❏ Perturb the amplitude by setting: ˆψ = ˆψ 0 + ɛ ˆψ 1,0 cos (˜kζ − ˜ωτ)

❏ We obtain the (perturbation) dispersion relation:

˜ω 2 = P 2 ˜k2

(˜k 2 − 2 Q )

P | ˆψ 1,0 | 2 . (7)

❏ If P Q < 0: the amplitude ψ is stable to external perturbations;

❏ If P Q > 0: the amplitude ψ is unstable for ˜k <


2 Q P |ψ 1,0|.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 42

Stability profile (IAW): Angle α versus wavenumber k

Typical values: Z i = +1 (hydrogen plasma), γ = 2.

– Ion-acoustic waves; cold (σ = 0) vs. warm (σ ≠ 0) fluid:

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 43

Stability profile (IAW): Angle α versus wavenumber k

Typical values: Z i = +1 (hydrogen plasma), γ = 2.

– Ion-acoustic waves; cold (σ = 0) vs. warm (σ ≠ 0) fluid:

– Dust-ion acoustic waves, i.e. in the presence of negative dust (n d,0 /n i,0 = 0.5):

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 44

Stability profile (IAW): Angle α versus wavenumber k

Typical values: Z i = +1 (hydrogen plasma), γ = 2.

– Ion-acoustic waves; cold (σ = 0) vs. warm (σ ≠ 0) fluid:

– Dust-ion acoustic waves, i.e. in the presence of positive dust (n d,0 /n i,0 = 0.5):

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 45

Stability profile (IAW/EAW): Angle α versus wavenumber k

Typical values: Z i = +1 (hydrogen plasma), γ = 2.

– Ion acoustic waves, in the presence of 2 electron populations:

– Electron acoustic waves (+ cold electrons):

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 46

Stability profile (DAW): Angle α versus wavenumber k

Typical values: Z d /Z i ≈ 10 3 , T e /T i ≈ 10, n d,0 /n i,0 ≈ 10 −3 , γ = 2.

– Negative dust: s = −1; cold (σ = 0) vs. warm (σ ≠ 0) fluid:

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 47

Stability profile (DAW): Angle α versus wavenumber k

Typical values: Z d /Z i ≈ 10 3 , T e /T i ≈ 10, n d,0 /n i,0 ≈ 10 −3 , γ = 2.

– Negative dust: s = −1; cold (σ = 0) vs. warm (σ ≠ 0) fluid:

– The same plot for positive dust (s = +1):

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 48

4b. Localized envelope excitations (solitons)

❏ The NLSE:

i ∂ψ

∂τ + P ∂2 ψ

∂ζ 2 + Q |ψ|2 ψ = 0

accepts various solutions in the form: ψ = ρ e iΘ ;

the total electric potential is then: φ ≈ ɛ ρ cos(kr − ωt + Θ) where the

amplitude ρ and phase correction Θ depend on ζ, τ.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 49

4b. Localized envelope excitations (solitons)

❏ The NLSE accepts various solutions in the form: ψ = ρ e iΘ ;

the total electric potential is then: φ ≈ ɛ ρ cos(kr − ωt + Θ) where the

amplitude ρ and phase correction Θ depend on ζ, τ.

❏ Bright–type envelope soliton (pulse):

( ) ζ − v τ

ρ = ρ 0 sech , Θ = 1

L

2P

[

v ζ − (Ω +

1

2 v2 )τ ] . (8)

L =


2P

Q

1

ρ 0

This is a

propagating

(and oscillating)

localized pulse:

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 50

Propagation of a bright envelope soliton (pulse)

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 51

Propagation of a bright envelope soliton (pulse)

Cf. electrostatic plasma wave data from satellite observations:

(from: [Ya. Alpert, Phys. Reports 339, 323 (2001)] )

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 52

Propagation of a bright envelope soliton (continued...)

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 53

Propagation of a bright envelope soliton (continued...)

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 54

Propagation of a bright envelope soliton (continued...)

Rem.: Time-dependent phase → breathing effect (at rest frame):

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 55

Localized envelope excitations

❏ Dark–type envelope solution (hole soliton):

This is a

propagating

localized hole

(zero density void):

ρ = ±ρ 1

[

1 − sech 2 ( ζ − vτ

L ′

)] 1/2 ( ) ζ − v τ

= ±ρ 1 tanh

L ′

Θ = 1 [ ( ) ]

1

v ζ −

2P 2 v2 − 2P Qρ 2 1 τ


L ′ = 2 ∣ P ∣ 1 (9)

Q ρ 1

,

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 56

Localized envelope excitations

❏ Grey–type envelope solution (void soliton):

This is a

propagating

(non zero-density)

void:

ρ = ±ρ 2

[

1 − a 2 sech 2 ( ζ − v τ

L ′′ )] 1/2

Θ = ...


L ′′ = 2 ∣ P ∣ 1

(10)

Q aρ 2

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 57

5. (Part B): Focusing on 1d DP crystals:

[Figure from: S. Takamura et al., Phys. Plasmas 8, 1886 (2001).]

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 58

Focusing on 1d DP crystals: known linear modes.

❏ Longitudinal Dust Lattice (LDL) mode:

☛ Horizontal oscillations (∼ ˆx): cf. phonons in atomic chains;

☛ Acoustic mode: ω(k = 0) = 0;

☛ Restoring force provided by electrostatic interactions.

❏ Transverse Dust Lattice (TDL) mode:

☛ Vertical oscillations (∼ ẑ);

☛ Optical mode:

ω(k = 0) = ω g ≠ 0

(center of mass motion);

☛ Single grain vibrations (propagating ∼ ˆx for k ≠ 0): Restoring force

provided by the sheath electric potential (and interactions).

❏ Transverse (∼ ŷ, in-plane, optical) d.o.f. suppressed.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 59

Model Hamiltonian:

H = ∑ n

( ) 2

1

2 M drn

+ ∑ U int (r nm ) + Φ ext (r n )

dt

m≠n

where:

– Kinetic Energy (1st term);

– U int (r nm ) is the (binary) interaction potential energy;

– Φ ext (r n ) accounts for ‘external’ force fields:

may account for confinement potentials and/or sheath electric forces, i.e.

F sheath (z) = −∂Φ/∂z .

Q.: Nonlinearity: Origin: where from ? Consequence(s) ?

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 60

Nonlinearity: Where does it come from?

❏ (i) Interactions between grains: Intrinsically anharmonic!

☛ Electrostatic (e.g. Debye), long-range, screened (r 0 /λ D ≈ 1); typically:

U Debye (r) = q2

r exp (−r/λ D) .

☛ Expanding U pot (r nm ) near equilibrium:

one obtains:

∆x n = x n − x n−m = mr 0 , ∆z n = z n − z n−m = 0

U nm (r) ≈ 1 2 Mω2 L,0(∆x n ) 2 + 1 2 Mω2 T,0(∆z n ) 2

+ 1 3 u 30(∆x n ) 3 + 1 4 u 40(∆x n ) 4 + ... + 1 4 u 04(∆z n ) 4 + ...

+ 1 2 u 12(∆x n )(∆z n ) 2 + 1 4 u 22(∆x n ) 2 (∆z n ) 2 + ...

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 61

Nonlinearity: Where from? (continued ...)

❏ (ii) Mode coupling also induces non linearity:

anisotropic motion, not confined along one of the main axes (∼ ˆx, ẑ).

[cf. A. Ivlev et al., PRE 68, 066402 (2003); I. Kourakis & P. K. Shukla, Phys. Scr. (2004)]

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 62

Nonlinearity: Where from? (continued ...)

❏ (iii) Sheath environment: anharmonic vertical potential:

Φ(z) ≈ Φ(z 0 ) + 1 2 Mω2 g(δz n ) 2 + 1 3 Mα (δz n) 3 + 1 4 Mβ (δz n) 4 + ...

cf. experiments [Ivlev et al., PRL 85, 4060 (2000); Zafiu et al., PRE 63 066403 (2001)];

δz n = z n − z (0) ; α, β, ω g are defined via E(z), [B(z)] † and Q(z);

(in fact, functions of n and P ) [ † V. Yaroshenko et al., NJP 2003; PRE 2004]

Source: Sorasio et al. (2002).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 63

Overview: Localized excitations in 1d dust (Debye) crystals

❏ B1. Transverse degree of freedom (∼ ẑ): envelope solitons, breathers

(continuum theory)

❏ B2. Longitudinal d.o.f. (∼ ˆx): asymmetric envelope solitons, ...

❏ B3. Longitudinal solitons: Korteweg-deVries (KdV) vs. Boussinesq

theories, ...

→ Appendix

❏ B4. Discrete Breathers (Intrinsic Localized Modes).

→ Appendix

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 64

B1. Transverse oscillations

The (linear) vertical n−th grain displacement δz n = z n − z (0) obeys (*):

d 2 (δz n )

dt 2 +ν d(δz n)

dt

+ ω 2 T,0 ( δz n+1 + δz n−1 − 2 δz n ) + ω 2 g δz n = 0 (11)

❏ TDL eigenfrequency:

ω T,0 = [ −qU ′ (r 0 )/(Mr 0 ) ] 1/2

= ω

2

DL exp(−κ) (1 + κ)/κ 3

(for Debye interactions); κ = r 0 /λ D is the lattice parameter;

❏ ω DL = [q 2 /(Mλ 3 D )]1/2

is the characteristic DL frequency scale;

❏ λ D is the Debye length.

(*) [Vladimirov, Shevchenko and Cramer, PRE 1997]

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 65

Transverse oscillations (linear)

The (linear) vertical n−th grain displacement δz n = z n − z (0) obeys

d 2 (δz n )

dt 2 +ν d(δz n)

dt

+ ω 2 T,0 ( δz n+1 + δz n−1 − 2 δz n ) + ω 2 g δz n = 0 (12)

❏ Neglect dissipation, i.e. set ν = 0 in the following;

❏ Continuum analogue: δz n (t) → u(x, t), where

∂ 2 u

∂t 2 + ∂ 2 u

c2 T

∂x 2 + ω2 g u = 0

where c T = ω T,0 r 0 is the transverse “sound” velocity.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 66

Transverse oscillations (linear, “undamped”)

The (linear) vertical n−th grain displacement δz n = z n − z (0) obeys

d 2 (δz n )

dt 2 + ω 2 T,0 ( δz n+1 + δz n−1 − 2 δz n ) + ω 2 g δz n = 0 (13)

Optical dispersion relation

(backward wave, v g < 0) † :

ω 2 = ω 2 g−4ω 2 T,0 sin2( kr 0 /2 )

† Cf. experiments: T. Misawa et al., PRL 86, 1219 (2001); B. Liu et al., PRL 91, 255003 (2003).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 67

What if nonlinearity is taken into account?

d 2 δz n

dt 2 +ν d(δz n)

dt

+ ω 2 T,0 ( δz n+1 + δz n−1 − 2 δz n ) + ω 2 g δz n

+α (δz n ) 2 + β (δz n ) 3 = 0 . (14)

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 68

What if nonlinearity is taken into account?

d 2 δz n

dt 2 +ν d(δz n)

dt

+ ω 2 T,0 ( δz n+1 + δz n−1 − 2 δz n ) + ω 2 g δz n

+α (δz n ) 2 + β (δz n ) 3 = 0 . (15)

* Intermezzo: The mechanism of wave amplitude modulation:

The amplitude of a harmonic wave may vary in space and time:


www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 69

What if nonlinearity is taken into account?

d 2 δz n

dt 2 +ν d(δz n)

dt

+ ω 2 T,0 ( δz n+1 + δz n−1 − 2 δz n ) + ω 2 g δz n

+α (δz n ) 2 + β (δz n ) 3 = 0 . (16)

* Intermezzo: The mechanism of wave amplitude modulation:

The amplitude of a harmonic wave may vary in space and time:


This modulation (due to nonlinearity) may be strong enough to lead to wave

collapse or formation of envelope solitons:

→ ?

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 70

Large amplitude oscillations - envelope structures

A reductive perturbation (multiple scale) technique, viz.

t → {t 0 , t 1 = ɛt, t 2 = ɛ 2 t, ...}, x → {x 0 , x 1 = ɛx, x 2 = ɛ 2 x, ...}

yields (ɛ ≪ 1; damping omitted):

δz n ≈ ɛ (A e iφ n

+ c.c.) + ɛ 2 α

[− 2|A|2

ωg

2

+

( A

2

3ω 2 g

)]

e 2iφ n

+ c.c.

+ ...

Here,

❏ φ n = nkr 0 − ωt is the (fast) TDLW carrier phase;

❏ the amplitude A(X, T ) depends on the (slow) variables

{X, T } = {ɛ(x − v g t), ɛ 2 t} .

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 71

Transverse oscillations - the envelope evolution equation

The amplitude A(X, T ) obeys the nonlinear Schrödinger equation (NLSE):

i ∂A

∂T + P ∂2 A

∂X 2 + Q |A|2 A = 0 , (17)

where

❏ The dispersion coefficient (→ see dispersion relation)

P = 1 2

d 2 ω T (k)

dk 2 = ...

is negative/positive for low/high values of k.

❏ The nonlinearity coefficient is Q = [ 10α 2 /(3ω 2 g) − 3 β ] /2ω.

❏ Cf.: known properties of the NLS Eq. (cf. previous part).

[I. Kourakis & P. K. Shukla, Phys. Plasmas, 11, 2322 (2004); also PoP, 11, 3665 (2004).]

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 72

Modulational stability analysis & envelope structures

❏ P Q > 0: Modulational instability of the carrier, bright solitons:

→ TDLW s: possible for short wavelengths i.e. k cr < k < π/r 0 .

Rem.: Q > 0 for all known experimental values of α, β.

[Ivlev et al., PRL 85, 4060 (2000); Zafiu et al., PRE 63 066403 (2001)]

Source: G. Sorasio et al. (2002).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 73

Modulational stability analysis & envelope structures

❏ P Q > 0: Modulational instability of the carrier, bright solitons:

→ TDLW s: possible for short wavelengths i.e. k cr < k < π/r 0 .

❏ P Q < 0: Carrier wave is stable, dark/grey solitons:

→ TDLW s: possible for long wavelengths i.e. k < k cr .

Rem.: Q > 0 for all known experimental values of α, β

[Ivlev et al., PRL 85, 4060 (2000); Zafiu et al., PRE 63 066403 (2001)] (end of TDL).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 74

B2. Longitudinal excitations

The (linearized) equation of longitudinal (∼ ˆx) motion reads (*):

d 2 (δx n )

dt 2 +ν d(δx n)

dt

= ω 2 0,L (δx n+1 + δx n−1 − 2δx n )

– δx n = x n − nr 0 : longitudinal dust grain displacements

– Acoustic dispersion relation:

ω 2 = 4ω 2 L,0 sin 2( kr 0 /2 ) ≡ ω 2 L(k)

– LDL eigenfrequency: ω 2 0,L = U ′′ (r 0 )/M = 2 ω 2 DL exp(−κ) (1 + κ + κ2 /2)/κ 3

(∗)

(∗) for Debye interactions; Rem.: ω DL = [q 2 /(Mλ 3 D )]1/2 .

– Neglect damping in the following, viz. ν → 0 .

(*) [Melandsø PoP 1996, Farokhi et al, PLA 1999]

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 75

Longitudinal excitations (linear, “undamped”)

The (linearized) equation of longitudinal motion reads:

or, in the continuum approximation:

d 2 (δx n )

dt 2 = ω 2 0,L (δx n+1 + δx n−1 − 2δx n )

∂ 2 (δx n )

∂t 2

− c 2 L

∂ 2 (δx n )

∂t 2 = 0

(c L = ω 0,L r 0 )

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 76

Longitudinal excitations (nonlinear).

The nonlinear equation of longitudinal motion reads:

d 2 (δx n )

dt 2 = ω 2 0,L (δx n+1 + δx n−1 − 2δx n )

−a 20

[

(δxn+1 − δx n ) 2 − (δx n − δx n−1 ) 2]

+ a 30

[

(δxn+1 − δx n ) 3 − (δx n − δx n−1 ) 3] (18)

– δx n = x n − nr 0 : longitudinal dust grain displacements

– Cf. Fermi-Pasta-Ulam (FPU) problem: anharmonic spring chain model:

U int (r) ≈ 1 2 Mω2 0,Lr 2 − 1 3 Ma 20r 3 + 1 4 Ma 30r 4 .

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 77

Longitudinal envelope structures.

The reductive perturbation technique (cf. above) now yields:

δx n ≈ ɛ [ u (1)

0 + (u (1)

1 e iφ n

+ c.c.) ] + ɛ 2 (u (2)

2 e 2iφ n

+ c.c.) + ... ,

[Harmonic generation; Cf. experiments: K. Avinash PoP 2004].

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 78

Longitudinal envelope structures.

The reductive perturbation technique (cf. above) now yields:

δx n ≈ ɛ [ u (1)

0 + (u (1)

1 e iφ n

+ c.c.) ] + ɛ 2 (u (2)

2 e 2iφ n

+ c.c.) + ... ,

where the amplitudes obey the coupled equations:

i ∂u(1) 1

∂T

+ P L

∂ 2 u (1)

1

∂X 2 + Q 0 |u (1)

1 |2 u (1)

1 + p 0k 2

u (1)

1

2ω L

∂u (1)

0

∂X = 0 ,

– Q 0 = − k2


∂ 2 u (1)

0

∂X 2 = − p 0k 2

vg,L 2 − c2 L

( )

q 0 k 2 + 2p2 0

c 2 L r2 0


∂X |u(1) 1 |2 ≡ R(k) ∂

∂X |u(1) 1 |2

; v g,L = ω L ′ (k); {X, T }: slow variables;

– p 0 = −U ′′′ (r 0 )r 3 0/M ≡ 2a 20 r 3 0 , q 0 = U ′′′′ (r 0 )r 4 0/(2M) ≡ 3a 30 r 4 0.

– R(k) > 0, since ∀ k v g,L < ω L,0 r 0 ≡ c L (subsonic LDLW envelopes).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 79

Asymmetric longitudinal envelope structures.

– The system of Eqs. for u (1)

1 , u(1) 0 may be combined into a closed (NLSE)

equation (for A = u (1)

1 , here);

i ∂A

∂T + P ∂2 A

∂X 2 + Q |A|2 A = 0

– P = P L = ω L ′′ (k)/2 < 0;

– Q > 0 (< 0) prescribes stability (instability) at low (high) k.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 80

Asymmetric longitudinal envelope structures.

– The system of Eqs. for u (1)

1 , u(1) 0 may be combined into a closed (NLSE)

equation (for A = u (1)

1 , here);

– P = P L = ω L ′′ (k)/2 < 0;

i ∂A

∂T + P ∂2 A

∂X 2 + Q |A|2 A = 0

– Q > 0 (< 0) prescribes stability (instability) at low (high) k.

– Envelope excitations are now asymmetric:

(at high k, i.e. P Q > 0).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 81

Asymmetric longitudinal envelope structures.

(at high k)

(at low k)

[I. Kourakis & P. K. Shukla, Phys. Plasmas, 11, 1384 (2004).]

(end of L-Part).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 82

6. Conclusions

❏ Amplitude Modulation (due to carrier self-interaction) is an inherent feature

of oscillatory mode dynamics in dynamical systems;

❏ modulated waves may undergo spontaneous modulational instability; this is

an intrinsic feature of nonlinear dynamics, which ...

❏ ... may lead to the formation of envelope localized structures (envelope

solitons), in account of energy localization phenomena observed in Nature;

❏ Modulated electrostatic (ES) plasma wave packets observed in Space and

in the lab, in addition to dust-lattice excitations, may be modelled this way.

❏ The RP analytical framework permits modelling of these mechanisms in

terms of intrinsic physical (plasma) parameters.

→ an efficient model for a weakly nonlinear dynamical system in {x, t}.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 83

Ioannis Kourakis

Thank You !

Acknowledgments:

Padma Kant Shukla and co-workers at RUB (Germany)

Material from:

I. Kourakis & P. K. Shukla, Phys. Plasmas, 10 (9), 3459 (2003);

idem, PRE, 69 (3), 036411 (2003).

idem, J. Phys. A , 36 (47), 11901 (2003).

idem, European Phys. J. D, 28, 109 (2004).

Available at:

ioannis@tp4.rub.de

www.tp4.rub.de/∼ioannis

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 84

Appendix: DP – Dusty Plasmas (or Complex Plasmas):

definition and characteristics

❏ Ingredients:

– electrons e − (charge −e, mass m e ),

– ions i + (charge +Z i e, mass m i ), and

– charged micro-particles ≡ dust grains d (most often d − ):

charge Q = ±Z d e ∼ ±(10 3 − 10 4 ) e,

mass M ∼ 10 9 m p ∼ 10 13 m e ,

radius r ∼ 10 −2 µm up to 10 2 µm.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 85

Dust laboratory experiments on Earth:

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 86

Earth experiments are

subject to gravity:

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 87

Earth experiments are

subject to gravity:

thus ...: Dust experiments in ISS (International Space Station)

(Online data from: Max Planck Institüt - CIPS).

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 88

B3. Longitudinal soliton formalism.

Q.: A link to soliton theories: the Korteweg-deVries Equation.

– Continuum approximation, viz. δx n (t) → u(x, t).

– “Standard” description: keeping lowest order nonlinearity,

ü +ν ˙u − c 2 L u xx − c2 L

12

r 2 0 u xxxx = − p 0 u x u xx

c L = ω L,0 r 0 ; ω L,0 and p 0 were defined above.

– Let us neglect damping (ν → 0), once more.

– For near-sonic propagation (i.e. v ≈ c L ), slow profile evolution in time τ and

defining the relative displacement w = u ζ , one obtains the KdV equation:

w τ − a w w ζ + b w ζζζ = 0

(for ν = 0); ζ = x − vt; a = p 0 /(2c L ) > 0; b = c L r 2 0/24 > 0.

– This KdV Equation yields soliton solutions, ... (→ next page)

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 89

The KdV description

The Korteweg-deVries (KdV) Equation

w τ − a w w ζ + b w ζζζ = 0

yields compressive (only, here) solutions, in the form (here):

[

]

w 1 (ζ, τ) = − w 1,m sech 2 (ζ − vτ − ζ 0 )/L 0

– This solution is a negative pulse for w = u x ,

describing a compressive excitation for the displacement δx = u,

i.e. a localized increase of density n ∼ −u x .

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 90

The KdV description

The Korteweg-deVries (KdV) Equation

w τ − a w w ζ + b w ζζζ = 0

yields compressive (only, here) solutions, in the form (here):

[

]

w 1 (ζ, τ) = − w 1,m sech 2 (ζ − vτ − ζ 0 )/L 0

– Pulse amplitude: w 1,m = 3v/a = 6vv 0 /|p 0 |;

– Pulse width: L 0 = (4b/v) 1/2 = [2v 2 1r 2 0/(vv 0 )] 1/2 ;

– Note that: w 1,m L 2 0 = constant (cf. experiments) † .

– This solution is a negative pulse for w = u x ,

describing a compressive excitation for the displacement δx = u,

i.e. a localized increase of density n ∼ −u x .

– This is the standard treatment of dust-lattice solitons today ... †

† F. Melandsø 1996; S. Zhdanov et al. 2002; K. Avinash et al. 2003; V. Fortov et al. 2004.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 91

Characteristics of the KdV theory

The Korteweg - deVries theory, as applied in DP crystals:

– provides a correct qualitative description of compressive excitations

observed in experiments;

– benefits from the KdV “artillery” of analytical know-how obtained throughout

the years: integrability, multi-soliton solutions, conservation laws, ... ;

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 92

Characteristics of the KdV theory

The Korteweg - deVries theory, as applied in DP crystals:

– provides a correct qualitative description of compressive excitations

observed in experiments;

– benefits from the KdV “artillery” of analytical know-how obtained throughout

the years: integrability, multi-soliton solutions, conservation laws, ... ;

but possesses a few drawbacks:

– approximate derivation: (i) propagation velocity v near (longitudinal) sound

velocity c L , (ii) time evolution terms omitted ‘by hand’, (iii) higher order

nonlinear contributions omitted;

– only accounts for compressive solitary excitations (for Debye interactions);

nevertheless, the existence of rarefactive dust lattice excitations is, in

principle, not excluded.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 93

Longitudinal soliton formalism (continued)

Q.: What if we also kept the next order in nonlinearity ?

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 94

Longitudinal soliton formalism (continued)

Q.: What if we also kept the next order in nonlinearity ?

– “Extended” description: :

ü − c 2 L u xx − c2 L

12 r2 0 u xxxx = − p 0 u x u xx + q 0 (u x ) 2 u xx

c L = ω L,0 r 0 ;

ω L,0 , p 0 ∼ −U ′′′ (r) and q 0 ∼ U ′′′′ (r) (cf. above).

Rq.: q 0 is not negligible, compared to p 0 ! (instead, q 0 ≈ 2p 0 practically, for r 0 ≈ λ D !)

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 95

Longitudinal soliton formalism (continued)

Q.: What if we also kept the next order in nonlinearity ?

– “Extended” description: :

ü + ν ˙u − c 2 L u xx − c2 L

12 r2 0 u xxxx = − p 0 u x u xx + q 0 (u x ) 2 u xx

c L = ω L,0 r 0 ; ω L,0 , p 0 and q 0 were defined above.

– For near-sonic propagation (i.e. v ≈ c L ), and defining the relative

displacement w = u ζ , one obtains the E-KdV equation:

w τ − a w w ζ + â w 2 w ζ + b w ζζζ = 0 (19)

(for ν = 0); ζ = x − vt; a = p 0 /(2c L ) > 0; b = c L r 2 0/24 > 0;

â = q 0 /(2c L ) > 0.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 96

Characteristics of the EKdV theory

The extended Korteweg - deVries Equation:

– accounts for both compressive and rarefactive excitations;

(horizontal grain displacement u(x, t))

– reproduces the correct qualitative character of the KdV solutions (amplitude

- velocity dependence, ... );

– is previously widely studied, in literature;

Still, ...

– It was derived under the assumption: v ≈ c L .

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 97

One more alternative: the Boussinesq theory

The Generalized Boussinesq (Bq) Equation (for w = u x ):

ẅ − c 2 L w xx = c2 L r2 0

12 w xxxx − p 0

2

(w 2 ) xx + q 0

2

(w 3 ) xx

– predicts both compressive and rarefactive excitations;

– reproduces the correct qualitative character of the KdV solutions (amplitude

- velocity dependence, ... );

– has been widely studied in literature;

and, ...

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 98

One more alternative: the Boussinesq theory

The Generalized Boussinesq (Bq) Equation (for w = u x ):

ẅ − c 2 L w xx = c2 L r2 0

12 w xxxx − p 0

2

(w 2 ) xx + q 0

2

(w 3 ) xx

– predicts both compressive and rarefactive excitations;

– reproduces the correct qualitative character of the KdV solutions (amplitude

- velocity dependence, ... );

– has been widely studied in literature;

and, ...

– relaxes the velocity assumption, i.e. is valid ∀ v > c L .

(end of L-Part 4.)

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 99

B4. Transverse Discrete Breathers (DB)

❏ DBs are highly discrete oscillations (Intrinsic Localized Modes, ILMs);

❏ Looking for DB solutions in the transverse direction, viz.

d 2 u n

dt 2 + ω2 T,0 ( u n+1 + u n−1 − 2 u n ) + ω 2 g δz n + α u 2 + β u 3 = 0

one obtains the bright-type DB solutions (localized pulses):

as well as the dark-type excitations (holes; Kivshar dark modes):

❏ Similar modes may be sought in the longitudinal direction.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005


I. Kourakis, Reductive perturbation method ... 100

Transverse Discrete Breathers (DB)

❏ Existence and stability criteria still need to be examined.

❏ It seems established that DBs exist if the non-resonance criterion:

n ω B ≠ ω k

∀n ∈ N

is fulfilled, where:

– ω B is the breather frequency;

– ω k is the linear (“phonon”) frequency (cf. dispersion relation).

❏ If ω B (or its harmonics) enter(s) into resonance with the linear spectrum ω k ,

discrete oscillations will decay into a “sea” of linear lattice waves.

❏ The DB existence condition is satisfied in all known lattice wave

experiments.

www.tp4.rub.de/∼ioannis/conf/200511-TP4-oral.pdf TP4, Bochum, 16 Nov. 2005

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