Head-on collisions of electrostatic solitons in
nonthermal plasmas
Frank Verheest1,2 , Manfred A Hellberg2 and Willy A. Hereman3
1 Sterrenkundig Observatorium, Universiteit Gent, Belgium
School of Chemistry and Physics, University of KwaZulu-Natal, Durban, SA
Applied Mathematics and Statistics, Colorado School of Mines, Golden CO, USA
2
3
Phys. Rev. E 86, 036402 (2012)
Introduction
Basic formalism
Generic case
Critical case
Summary
Outline
1
Introduction
2
Basic formalism
3
Generic case
4
Critical case
5
Summary
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Motivation
Recently, many papers appeared in a plasma physics context about head-on collisions of two
electrostatic solitons, which intrinsically is a very interesting problem
Before going on, we remind ourselves that KdV equations possess exact solutions for
interactions between N solitons, with full nonlinearity during overtaking
t
x
Although rarely mentioned, this is valid only for solitons propagating in direction that
underlies basic derivation of parent equation
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Head-on collisions
In contrast, head-on collisions between two electrostatic solitons can only be dealt with by
approximate methods, which limit the range of validity but might offer interesting insight
Framework is based on Poincaré-Lighthill-Kuo formalism of strained coordinates (which
yields here KdV/mKdV families of equations plus phase/time shifts that occur during
interaction) and was adapted for solitary waves on shallow water
KdV equation is typical paradigm which fixes our intuitive ideas of what solitary waves look
like, having nonlinear steepening balanced by dispersion, on slow timescale compared to
linear propagation velocity
∂ϕ
∂ϕ
∂3ϕ
+ Aϕ
+B
=0
∂τ
∂ξ
∂ξ 3
However, when plasma composition is critical in sense that A = 0, another expansion and/or
stretching scheme is needed, leading to a modified KdV equation with cubic nonlinearity
∂ϕ
∂3ϕ
∂ϕ
+ C ϕ2
+D
=0
∂τ
∂ξ
∂ξ 3
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Remarks on models and procedures used in literature
Almost all papers in recent plasma literature are theoretical and use PLK formalism, without
worrying about or explaining inherent limitations or use of certain assumptions
Even though most plasma models investigated allow for criticality (A = 0), none of this is
discussed, so that only generic half of problem is treated
This is like deducing KdV equation by tacitly assuming that A 6= 0, and forgetting about
possible mKdV discussion
Original derivation of KdV equation was, of course, for water waves, which can only occur as
humps (so that A 6= 0 here), but plasma is much richer medium
Besides this plethora of theoretical papers, looking very similar and seldom giving any
figures, one experimental paper stands out: Harvey et al., Phys. Rev. E 81, 057401 (2010)
Harvey et al. discuss experimental observations of interaction of two counter-propagating
solitons of equal amplitude in a monolayer strongly coupled dusty plasma
Their results include propagation delays after interaction, compared to a single soliton, and
also that combined amplitudes during interaction are less than simple linear superposition
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Experimental results
Data from experiment [Harvey et al., Phys. Rev. E 81, 057401 (2010)]
t
6
4
2
-4
2
-2
4
x
t
-2
x
-4
Results schematically show delay in transmission after interaction, but do not indicate height,
which is found to be less than linear superposition
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Outline
1
Introduction
2
Basic formalism
3
Generic case
4
Critical case
5
Summary
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Model and basic equations
Although recent literature encompasses quite a variety of plasma compositions and
electrostatic waves, treatment is very similar and leads to KdV equations plus time delays,
only coefficients change
Hence, for clarity of exposition, consider plasma consisting of cold ions and nonthermal
(“Cairns") electrons, which is simplest two-species plasma allowing for criticality
Normalized continuity and momentum equations for ions are coupled to Poisson’s equation
∂n
∂
+
(nu) = 0
∂t
∂x
∂u
∂u
∂ϕ
+u
+
=0
∂t
∂x
∂x
∂2ϕ
+ n − (1 − βϕ + βϕ2 ) exp(ϕ) = 0
∂x2
where only parameter is β, measuring degree of nonthermality
For β = 0 Maxwellian electrons are recovered, and for β > 4/7 original phase space
distribution develops wings which might lead to bump-on-tail or two-beam unstable cases
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Stretching and expansions
Stretching of independent variables reflects propagation in opposite directions
ξ = ε(x − ca t) + ε2 P(ξ, η, τ ) + ...
η = ε(x + ca t) + ε2 Q(ξ, η, τ ) + ...
τ = ε3 t
Both space coordinates have to use same linear acoustic phase velocity in medium, here
1
ca = (1 − β)− 2 with proper choice of normalization
This is coupled to expansions for densities, velocities and electrostatic potential
n = 1 + εn1 + ε2 n2 + ε3 n3 + ε4 n4 + ...
u = εu1 + ε2 u2 + ε3 u3 + ε4 u4 + ...
ϕ = εϕ1 + ε2 ϕ2 + ε3 ϕ3 + ε4 ϕ4 + ...
Steps in expansion are left general, so as to deal with generic and critical compositions in
one coherent treatment
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Lower order results and bifurcation
To lowest order perturbations obey
∂n1
∂u1
∂u1
∂n1
1
−
+
+
=0
∂ξ
∂ξ
∂η
1 − β ∂η
1
∂u1
∂ϕ1
∂ϕ1
∂u1
√
−
+
+
=0
∂ξ
∂ξ
∂η
1 − β ∂η
√
n1 − (1 − β)ϕ1 = 0
and lead to separability at linear level
n1 = (1 − β) ϕ1ξ + ϕ1η
p
u1 =
1 − β ϕ1ξ − ϕ1η
ϕ1 = ϕ1ξ + ϕ1η
Next order leads to bifurcation
(2 − 6β + 3β 2 )ϕ21ξ = 0
&
(2 − 6β + 3β 2 )ϕ21η = 0
so that in generic case ϕ1ξ = ϕ1η = 0 or at critical parameters β = βc = (3 −
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
√
3)/3
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Outline
1
Introduction
2
Basic formalism
3
Generic case
4
Critical case
5
Summary
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Coupled Korteweg-de Vries equations and phase shifts
After much complicated/tedious algebra one arrives at KdV equations and phase shifts
∂ϕ2ξ
∂ϕ2ξ
∂ 3 ϕ2ξ
+ A ϕ2ξ
+B
=0
∂τ
∂ξ
∂ξ 3
&
∂P
= S ϕ2η
∂η
∂ϕ2η
∂ϕ2η
∂ 3 ϕ2η
− A ϕ2η
−B
=0
∂τ
∂η
∂η 3
&
∂Q
= S ϕ2ξ
∂ξ
where
A=
2 − 6β + 3β 2
2(1 − β)3/2
&
B=
1
2(1 − β)3/2
One-soliton solutions for each KdV equation are
3vξ
sech2 κξ (ξ − vξ τ )
A
3vη
=
sech2 [κη (η + vη τ )]
A
ϕ2ξ =
&
ϕ2η
&
where
κξ = (1 − β)3/4
r
vξ
2
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
6Sκη
{tanh [κη (η + vη τ )] + 1}
A(1 − β)3/2
6Sκξ
Q=
tanh κξ (ξ − vξ τ ) − 1
A(1 − β)3/2
P=
&
κη = (1 − β)3/4
r
vη
2
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Behaviour of KdV coefficients
Expressions for coefficients in KdV equations are
A=
2 − 6β + 3β 2
2(1 − β)3/2
&
B=
1
2(1 − β)3/2
Changes of A and B (full and dashed curves) as β is increased, showing how A goes from
positive to negative values at βc and that B remains positive and increases monotonically
from 1/2
A,B
2.0
1.5
1.0
0.5
Β
0.1
0.2
0.3
0.4
0.5
0.6
-0.5
-1.0
Case A = 1 and B = 1/2 is for Maxwellian electrons (β = 0)
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Illustrations for Maxwellian and Cairns electrons
Only same sign polarities (positive or negative) are possible for both waves
Left figure: values for Maxwellian (β = 0) electrons, so that A = 1 (potential and density
humps) and B = 1/2 (propagation delays due to interaction)
Right figure: values for Cairns (β = 0.5) electrons, so that A < 0 (potential and density dips)
but B > 1/2 > 0 (propagation delays due to interaction)
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Visualization of propagation delays
Comparison of analytical result with schematic corresponding to Harvey et al. experiments
t
6
4
2
-4
2
-2
4
x
-2
-4
Left figure: top view showing propagation delay for slower positive soliton at β = 0.25, where
stronger soliton has been omitted for graphical clarity
Right figure: repetition of earlier schematic of Harvey et al. results
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Comparison of amplitudes
Compare exact solution for overtaking solitons to PLK approach for head-on collisions
t
x
Overtaking soliton amplitudes are depressed during nonlinear interaction, whereas PLK
formalism leads to simple linear superposition
However, this linear superposition is also at variance with experiments by Harvey et al. and
points to weak point of PLK approach
Caution is needed here, but at present there seems to be no alternative theory to deal with
head-on collisions...
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Outline
1
Introduction
2
Basic formalism
3
Generic case
4
Critical case
5
Summary
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Modified Korteweg-de Vries equations and phase shifts
Algebra in critical case is even more involved but leads to mKdV equations plus phase shifts
∂ϕ1ξ
∂ϕ1ξ
∂ 3 ϕ1ξ
+ 0.353 ϕ21ξ
+ 1.140
=0
∂τ
∂ξ
∂ξ 3
&
∂P
= 0.449 ϕ21η
∂η
∂ϕ1η
∂ϕ1η
∂ 3 ϕ1η
− 0.353 ϕ21η
− 1.140
=0
∂τ
∂η
∂η 3
&
∂Q
= 0.449 ϕ21ξ
∂ξ
Solutions can now be of either sign and thus allow for collision between counterstreaming
negative and positive polarity solitons
√
√
ϕ1ξ = ± 4.125 vξ sech 0.937 vξ (ξ − vξ τ )
√
√
ϕ1η = ± 4.125 vη sech 0.937 vη (η + vη τ )
and
P = 8.162
Q = 8.162
√
√
√
tanh 0.937 vη (η + vη τ ) + 1
√
tanh 0.937 vξ (ξ − vξ τ ) − 1
vη
vξ
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Illustrations for combination of positive and negative solitons
Criticality can lead to two counter-propagating positive or negative solitons, but graphs are
similar to examples produced for generic case
Unusual is combination of a positive and a negative polarity soliton
Combining solitons of different polarity in two-soliton model for single KdV equation is
apparently not possible!
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Outline
1
Introduction
2
Basic formalism
3
Generic case
4
Critical case
5
Summary
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas
Introduction
Basic formalism
Generic case
Critical case
Summary
Summary
In generic case KdV equations govern collision of left- and right-propagating solitons, with
corresponding phase shifts
At critical plasma composition modified KdV equations are needed, with corresponding
phase shifts, a case not addressed before
For A ≷ 0 and B > 1/2 (no critical compositions): polarities are positive/negative, but phase
shifts are equivalent to delays compared to single-soliton trajectory
Criticality needs A = 0, but coefficient of nonlinear term in mKdV equations is positive,
avoiding hypothetical case of supercriticality which has occasionally surfaced in literature...
Comparison with recent experimental observations of two counter-propagating solitons of
equal amplitude in a monolayer strongly coupled dusty plasma indicates qualitative
agreement regarding delays occurring after interaction
However, amplitude of overlapping solitons during collision was less than sum of initial
soliton amplitudes, which cannot correctly be dealt with by available PLK formalism
Survey of literature indicates that analytical treatment of more complicated plasma models is
fully analogous, without qualitative changes
ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012
Head-on collisions of solitons in nonthermal plasmas