modulation - Theoretische Physik IV
modulation - Theoretische Physik IV
modulation - Theoretische Physik IV
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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
2ω
∂ 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:
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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).
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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.
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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.
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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 !)
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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, ...
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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.)
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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.
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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.
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