here - Ruhr-UniversitÃ¤t Bochum

Nonlinear Excitations in Dusty Plasmas

and Debye Crystals

A survey of theoretical results

Ioannis Kourakis

Centre for Plasma Physics, School of Mathematics and Physics

Queen’s University of Belfast, BT7 1NN, Belfast, Northern Ireland, UK

In collaboration with:

V Koukouloyannis, Aristotle Univ. Thessaloniki, Greece

P Kevrekidis & K Law, Amherst, Massachussets USA

D Frantzeskakis, U. Athens, Greece

B Farokhi, Arak University, Iran

P K Shukla, Ruhr Universität Bochum, Germany

2008 GRTR Conference on Statistical Physics

Rhodes & Marmaris, 11-17 September 2008

www.tp4.rub.de/~ioannis/conf/200809-GRTR-oral.pdf

I. Kourakis, Nonlinear excitations in dusty plasma crystals 1

Dusty Plasmas (or Complex Plasmas): prerequisites

Plasma textbook model:

→ electrons e − (charge q e = −e, mass m e ),

→ ions i + (charge q i = +Z i e, mass m i ≫ m e ); q i /m i ≪ q e /m e

www.tp4.rub.de/∼ioannis/conf/200809-GRTR-oral.pdf GRTR 2008, Rhodes 12 July 2008


Dusty Plasmas (or Complex Plasmas): prerequisites

Dusty Plasmas (DP):

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

→ ions i + (charge +Z i e, mass m i ≫ m e ); q i /m i ≪ q e /m e

→ charged particulates ≡ dust grains d ± (most often d − ):

charge Q = sZ d e ∼ ±(10 3 − 10 4 ) e, (s = ±1)

mass M ∼ 10 9 m p ∼ 10 13 m e ,

radius r ∼ 10 −2 µm up to 10 2 µm.



Ultra-strongly coupled Dusty Plasmas (DPs)

• Low Q/M ratio (ω p,d ≈ 10 − 100 Hz)

•

Γ eff = < E Q

potential >

2

< E kinetic > ≈ r

e−r/λ D

k B T

• Contrary to “ordinary” (weakly-coupled) e-i plasmas (Γ ≪ 1),

DPs may be strongly coupled: “liquid”, “crystalline” states

• → Dusty Plasma (Debye/Yukawa) Crystals!!! (DPCs)

• Theoretical prediction: 1986 [H. Ikezi, Phys. Fluids 29, 1764 (1986)]

• Experimental realization: 1994

[H. Thomas, A. Melzer et al. PRL 73, 652 (1994); Chu & Lin I J. Phys. D 27 296 (1994),

Hayashi & Tachibana, Jap. J. Appl. Phys. 33 L804 (1994)]



Dust Crystal experiments

[Source: H. Thomas, A. Melzer et al., PRL 1994].





Overview of current activity on NL crystals: theory vs

experiment and community vs community

• Experimental DP Research “Community A”:

G E Morfill (MPIeP Garching, Germany), A Piel (Kiel, Germany), A Melzer(Greifswald,

Germany), D. Samsonov (Manchester, UK), J Goree (Iowa, US), V Fortov (Moscow,

Russia), Lin I (Taiwan), A Samarian (Sydney, AUS), S Takamura (Nagoya, Japan), ...

• Focus on: statistical mechanics, kinetic phenomena,

linear dust-lattice waves (DLWs), nonlinear (NL) DLWs, ...

• Theoretical NL DPC Research “Community B”:

F. Melandsø(1996); M Amin, P K Shukla et al (1998+); B Farokhi et al (1999+); S.

Zhdanov et al. (2002); K. Avinash et al. (2003); V. Fortov (2004); I Kourakis et al (2004+).

• The Nonlinear Science “Community C”

lattice theories, nonlinear optics, photonic crystals, granular

“phononic” crystals, periodic metamaterials, complex systems



Outline – Menu

1. Dusty plasmas (DP) & DP crystals (DPCs): Prerequisites

Nonlinearity in 1D DP crystals: Origin and modelling

2. Transverse dust-lattice (TDL) excitations:

amplitude modulation, transverse envelope structures

3. Longitudinal dust-lattice (LDL) excitations:

amplitude modulation, envelope structures, solitons

4. 1D Discrete Breathers (DBs)

5. 2D DBs in hexagonal crystals

6. Conclusions



Preliminaries – 1D DP crystals (undamped): Model Hamiltonian

H = ∑ n

( ) 2

1

2 M drn

+ ∑ U int (r nm ) + Φ ext (r n )

dt

m≠n

Terms include:

– Kinetic 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 .

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

Q.: Nonlinearity: Origin: where from ?

Effect: which consequence(s) ?



Nonlinearity in DPCs: Origin & modelling (1)

→ 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 experimentally

Source: Sorasio et al. (2002).




→ Interactions between grains: anharmonic, screened ES:

U Debye (r) = q2

r e(−r/λ D)

Expanding U int (r nm ) = U int ( √ (∆x nm ) 2 + (∆z nm ) 2 ) near:

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

one obtains:

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

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

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

Details in: I Kourakis and P. K. Shukla, Int. J. Bif. Chaos 16, 1711 (2006).




→ Coupling among degrees of freedom induces nonlinearity:

anisotropic motion, not confined along principal axes (∼ ˆx, ẑ).

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



Discrete coupled equations of motion

d 2 (δx n )

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

−a 20

[

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

+ a 30

[

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

+ a 02

[

(δz n+1 − δz n ) 2 − (δz n − δz n−1 ) 2 ]

−a 12

[

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

,

d 2 (δz n )

dt 2

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

− K 1 (δz n ) 2 − K 2 (δz n ) 3 + a [

]

02

(δz n+1 − δz n ) 3 − (δz n − δz n−1 ) 3

r 0

]

+ 2 a 02

[(δx n+1 − δx n )(δz n+1 − δz n ) − (δx n − δx n−1 )(δz n − δz n−1 )

]

− a 12

[(δx n+1 − δx n ) 2 (δz n+1 − δz n ) − (δx n − δx n−1 ) 2 (δz n − δz n−1 ) .



Continuum coupled equations of motion

ü − c 2 L u xx − c2 L

12 r2 0 u xxxx =

− 2 a 20 r 3 0 u x u xx + 3 a 30 r 4 0 (u x ) 2 u xx

− a 12 r 4 0 [(w x ) 2 u xx + 2w x w xx u x ] + 2 a 02 r 3 0 w x w xx ,

ẅ + c 2 T w xx + c2 T

12 r2 0 w xxxx + ω 2 g w =

− K 1 w 2 − K 2 w 3 + 3 a 02 r 3 0 (w x ) 2 w xx

+ 2 a 02 r 3 0 (u x w xx + w x u xx ) − a 12 r 4 0 [(u x ) 2 w xx + 2u x u xx w x ] ,



Outline – Menu







4. 1D Discrete Breather excitations (intrinsic localized modes)

5. 2D Discrete Breather excitations in hexagonal crystals

6. Conclusions



2. Transverse DP-lattice excitations

The 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

* ω T,0 = [ −qU ′ (r 0 )/(Mr 0 ) ] 1/2

= ω

2

DL exp(−κ) (1 + κ)/κ 3 (†)

* ω DL = [q 2 /(Mλ 3 D )]1/2 ; λ D is the Debye length;

* Optical dispersion relation

(backward wave, v g < 0):

ω 2 = ω 2 g − 4ω 2 T,0 sin2( kr 0 /2 )





d 2 (δz n )


* ω T,0 = [ −qU ′ (r 0 )/(Mr 0 ) ] 1/2

= ω

2

DL exp(−κ) (1 + κ)/κ 3 (†)



(backward wave, v g < 0) † :

ω 2 = ω 2 g − 4ω 2 T,0 sin2( kr 0 /2 )

† Cf. experiment:

T. Misawa et al., PRL 86, 1219 (2001)

(Nagoya, Japan).





d 2 (δz n )


* ω T,0 = [ −qU ′ (r 0 )/(Mr 0 ) ] 1/2

= ω

2

DL exp(−κ) (1 + κ)/κ 3 (†)



(backward wave, v g < 0) † :

ω 2 = ω 2 g − 4ω 2 T,0 sin2( kr 0 /2 )

† Cf. experiment:

B. Liu et al., PRL 91, 255003 (2003)

(Iowa, USA).



What if nonlinearity is taken into account?

d 2 δz n

dt 2

+ 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 .

* Intermezzo: NL wave amplitude modulation → localisation!

→

→



Large amplitude oscillations - envelope structures

A reductive multiple scales technique, yields

[ (

)]

δz n ≈ ɛ (A e iφ n

+ c.c.) + ɛ 2 α + e 2iφ n

+ c.c. + ...

− 2|A|2

ω 2 g

A 2

3ω 2 g

• Harmonic Generation (multiples of φ n = nkr 0 − ωt)

• The harmonic amplitude A(X, T ):

– depends on the slow variables {X, T } = {ɛ(x − v g t), ɛ 2 t} ;

– obeys the nonlinear Schrödinger equation (NLSE):

i ∂A

∂T + P ∂2 A

∂X 2 + Q |A|2 A = 0 , (1)

– Dispersion coefficient: P = ω ′′ (k)/2 → see DR;

– Nonlinearity coefficient: Q = [ 10α 2 /(3ω 2 g) − 3 β ] /2ω.

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



Modulational stability analysis & envelope structures

• P Q > 0: Modulational instability, 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)].



Experimental confirmation (2005)

(Qualitative study; modulational instability not investigated.)



Outline – Menu









6. Conclusions



3a. (Nonlinear) longitudinal excitations.

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]

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

– Cf. Fermi-Pasta-Ulam (FPU) problem:

anharmonic spring chain model.



In general:

Coefficients (for Debye interactions)

a 20 = − 1

2M U ′′′ (r 0 ) , a 30 = 1

6M U ′′′′ (r 0 ) ,

ω 2 0,L = U ′′ (r 0 )/M ,

For a Debye (Yukawa) potential U D (r) = Q 2 e −r/λ D/r:

ω 2 0,L = 2Q2

Mλ 3 D

q 0 ≡ 3a 30 κ 4 λ 4 D =

e −κ 1 + κ + κ2 /2

κ 3 ≡ c 2 L/(κ 2 λ 2 D) ,

(

p 0 ≡ 2a 20 κ 3 λ 3 D = 6Q2 1

e −κ

Mλ D κ + 1 + κ )

2 + κ2

,

6

(

Q2

κ 4 + 4κ 3 + 12κ 2 + 24κ + 24

2Mλ D

e −κ 1 κ

)

.



Longitudinal Dust-Lattice wave (LDLW) modulation

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. K. Avinash PoP 2004].

where the amplitudes now obey the coupled equations:

i ∂u(1) 1

∂T

+ P L

∂ 2 u (1)

0

∂ 2 u (1)

1

∂X 2 + Q 0 |u (1)

1 |2 u (1)

1 + p 0k 2

∂X 2 = − p 0k 2

vg,L 2 − c2 L

( )

– Q 0 = − k2

2ω

q 0 k 2 + 2p2 0

; P

c 2 L = ω

L r2 L ′′(k)/2 ;

0

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

u (1)

1

2ω L

∂u (1)

0

∂X = 0 ,

∂

∂X |u(1) 1 |2 ≡ R(k) ∂

∂X |u(1) 1 |2

– p 0 = −r 3 0U ′′′ (r 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 (sound velocity).



Asymmetric longitudinal envelope structures.

– The system of Eqs. for u (1)

1 , u(1) 0 may be solved exactly:

→ asymmetric envelope solutions.

– P = P L = ω L ′′ (k)/2 < 0;

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

(at high k)

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

(at low k)



A link to soliton theories:

3b. Longitudinal soliton formalism.

– 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.

– 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 (for ν = 0) the Korteweg-deVries Equation:

w τ − a w w ζ + b w ζζζ = 0

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



The KdV description: The Melandsø (1996) theory

The Korteweg-deVries (KdV) Equation


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 .

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



The Korteweg-deVries (KdV) Equation


yields compressive (only, since a > 0) solutions, in the form:

]

w 1 (ζ, τ) = − 3v a

[(v/4b) sech2 1/2 (ζ − vτ − ζ 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 † .

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



Characteristics of the KdV theory

The Korteweg - deVries theory:

– provides a qualitative description of compressive excitations;

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

integrability, multi-soliton solutions, conservation laws, ... ;

but possesses certain drawbacks:

– approximate derivation:

– propagation velocity v near (longitudinal) sound velocity c L ,

– time evolution terms omitted ‘by hand’,

– higher order nonlinear contributions omitted;

– only accounts for compressive solitary excitations (for Debye

interactions).



Experimental observation of rarefactive LDL solitons

[Samsonov et al., PRL 2002].

• Only compressive solitons predicted by KdV theory;

• Only compressive solitons experimentally anticipated and,

hence, reported in 2002;

• What about rarefactive longitudinal solitons?



Extended longitudinal soliton formalism

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 ∼ −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!)



Extended 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 has

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

(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.



Characteristics of the KdV/mKdV/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 .



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;



– has been widely studied in literature;

and, ...



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;



and, ...

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

[I Kourakis & P K Shukla, European Phys. J. D, 29, 247 (2004)] .



Added “in proof”:

Experimental observation of rarefactive LDL solitons (2008)

Poster by Ralf Heidemann et al, ICPDP5 conf, Acores May 2008



Outline – Menu







4. 1D Discrete Breather excitations (intrinsic localized

modes)


6. Conclusions



4. Transverse Discrete Breathers (DBs)



4. Transverse Discrete Breathers (DBs)



4. Transverse Discrete Breathers (DBs) (cont.)

• Eq. of motion in the transverse direction:

d 2 u n

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

• Damping may be neglected (for low plasma density and/or

pressure): ν/ω g ≃ 0.00154 ([Misawa et al., PRL 2001]).

• 1D DPCs are highly discrete lattice configurations:

ɛ = ω 2 0/ω 2 g ≃ 0.016 ([Misawa et al., 2001]); ɛ ≃ 0.181 ([Liu et al., 2003]).

• One may seek discrete breather solutions (localized modes):

u n (t) = ∑ m

A n (m) exp(imω B t)

where only few (m ≃ 1 − 5) sites are excited.



4. 1D Transverse Discrete Breathers (DBs) (cont.)

* Non - resonance condition: mω B ≠ ω T (k) ∀k, m = 1, 2, ...

* In-phase and out-of-phase DBs may occur in DPCs:



Stable vs. unstable Transverse DBs in DPCs

Ab-initio study, adopting values from the Kiel experiment

[Zafiu et al., PRE 63 066403 (2001).]



Outline – Menu









6. Conclusions



5. 2D Transverse Discrete Breathers (DBs)



2D T-DBs: the anticontinuum limit method

Ab-initio study, adopting values from the Kiel experiment

[Zafiu et al., PRE 63 066403 (2001).]; see Videos

V. Koukouloyannis & I. Kourakis, in preparation.



5. 2D T-DBs: the DNLS method

K Law, P Kevrekidis, V Koukouloyannis, I Kourakis, D Frantzeskakis & A R Bishop, sub PRE.



Conclusions - State of the Art

– Dust crystals provide an excellent test-bed for NL theories

– Observations are possible at the kinetic level: unique

possibility for physical data processing & real-time analysis

– Link between Plasma Phys., Solid State Physics, Stat. Mech.

– Theory (1D): Envelope solitons, (non-topological) solitons,

Discrete Breathers: predicted

– Theory (2D): Discrete Breathers, vortices, ...: predicted

– Experiment: Harmonic generation, density solitons, NL TDL

oscillations, backward wave: observed (Urge for more :-) )



Overview of existing results

1. 1D: Transverse dust-lattice (TDL) motion (∼ NL KG, inv. disp.):

→ Envelope (NLS) solitons [IK & P K Shukla, Phys. Plasmas 11, 1384 (2004)]

→ DBs (ILMs) [V. Koukouloyannis & IK, PRE 76, 016402 (2007)]

2. 1D: Longitudinal dust-lattice (LDL) motion) (∼ FPU):

→ Asymmetric envelope structures (coupled 0th/1st harmonics)

[IK & P K Shukla, Phys. Plasmas 11, 3665 (2004)]

→ KdV vs. eKdV / Bq solitons [IK & PKS, Eur. Phys. J. D 29, 247 (2004)]

Rem.: experimentally observed (compressive case only)

3. 2D: In-plane (“LDL”) motion in hexagonal DP crystals:

→ Envelope structures [Farokhi, IK & PKS, Phys. Plasmas 13, 122304 (2006)]

4. 2D: Out-of-plane (TDL) motion in hexagonal DP crystals:

→ Envelope structures, DBs, vortices.



Future considerations & perspectives

1. LDL-DBs ? (∼ FPU);

2. Damping (dissipative system), ion drag, wake potentials, ...

3. Mixed T-L Mode: coupled FPU-NLKG Eqs. (ongoing work);

4. 2D hexagonal dust lattices: vortices ? (seen experimentally);

5. Experimental feedback!?:

- establish & pursue contacts,

- seek confirmation of results, motivate experiments ...

→ A lot still to be done!



Thank You !

Ioannis Kourakis

Acknowledgments: P K Shukla (RUB, Bochum, Germany),

V Koukouloyannis (AUTh, Greece), P. Kevrekidis (Amherst, USA),

K Law (Amherst, USA), D Franzeskakis (Athens, Greece),

B Farokhi (Arak, Iran), V. Basios (ULB, Brussels, Belgium), ...

Material from:

I Kourakis & P K Shukla, Phys. Plasmas, 11, 2322 (2004); ibid, 11, 3665 (2004)

idem, Phys. Plasmas, 11, 1384 (2004); idem, European Phys. J. D, 29, 247 (2004)

V Koukouloyannis & IK, PRE 76, 016402 (2007)

F Farokhi, I Kourakis & P K Shukla, PoP 13 (12), 122304 (2006)

K Law, P Kevrekidis, V Koukouloyannis, I Kourakis, D Frantzeskakis & A R Bishop, sub PRE.

Slides available at:

www.kourakis.eu


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