Head-on collisions of electrostatic solitons in multispecies 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
1. Motivation and model
KdV equations possess exact solutions for interactions between N solitons, with full nonlinearity
during overtaking, but only for solitons propagating in the direction that underlies the basic
derivation of the parent equation
In contrast, head-on collisions between two electrostatic solitons can only be dealt with by
approximate methods, which limit the range of validity but offer valuable insight
Framework is based on Poincaré-Lighthill-Kuo formalism of strained coordinates, which yields
here (m)KdV families of equations plus phase (time) shifts that occur in the interaction
Plasma consists of number of cold (positive and negative) ion species and Boltzmann electrons
Continuity and momentum equations for different ion species are coupled to Poisson’s equation
∂ρi
∂
+
(ρi ui ) = 0
∂t
∂x
∂ui
∂ui
∂ϕ
+ ui
+ zi
=0
∂t
∂x
∂x
∂2ϕ X
+
ρi zi − exp(ϕ) = 0
∂x2
i
2012 EPS/ICPP Stockholm, 2–6 July 2012
Head-on collisions of solitons in multispecies plasmas
2. Basic formalism
Stretching of independent variables reflects propagation in opposite directions
ξ = ε(x − t) + ε2 P(ξ, η, τ ) + ...
η = ε(x + t) + ε2 Q(ξ, η, τ ) + ...
τ = ε3 t
Both space coordinates have to use same linear acoustic phase velocity in medium, here ca = 1
with proper choice of normalization
This is coupled to expansions for densities, velocities and electrostatic potential
ρi = ρi0 + ερi1 + ε2 ρi2 + ε3 ρi3 + ε4 ρi4 + ...
ui = εui1 + ε2 ui2 + ε3 ui3 + ε4 ui4 + ...
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
2012 EPS/ICPP Stockholm, 2–6 July 2012
Head-on collisions of solitons in multispecies plasmas
3. Lower order results and bifurcation
To lowest order perturbations obey
∂ρi1
∂ρi1
∂ui1
∂ui1
−
+ ρi0
+
=0
∂η
∂ξ
∂ξ
∂η
∂ui1
∂ui1
∂ϕ1
∂ϕ1
−
ui1 + zi
+
=0
∂η
∂ξ
∂ξ
∂η
X
ρi1 zi − ϕ1 = 0
i
and lead to separability at linear level
ρi1 = ρi0 zi ϕ1ξ + ϕ1η
ui1 = zi ϕ1ξ − ϕ1η
Next order leads to bifurcation
X
3
ρi0 z3i − 1 ϕ21ξ = 0
i
ϕ1 = ϕ1ξ + ϕ1η
X
3
ρi0 z3i − 1 ϕ21η = 0
i
so that in generic case ϕ1ξ = ϕ1η = 0 or else, at critical parameters
X
ρi0 z3i =
1
3
i
2012 EPS/ICPP Stockholm, 2–6 July 2012
Head-on collisions of solitons in multispecies plasmas
4. Generic case: Korteweg-de Vries equations and phase shifts
After much complicated algebra one arrives at KdV equations and phase shifts
∂ϕ2ξ
∂ϕ2ξ
1 ∂ 3 ϕ2ξ
+ Aϕ2ξ
+
=0
∂τ
∂ξ
2 ∂ξ 3
&
∂P
= B ϕ2η
∂η
∂ϕ2η
∂ϕ2η
1 ∂ 3 ϕ2η
− Aϕ2η
−
=0
∂τ
∂η
2 ∂η 3
&
∂Q
= B ϕ2ξ
∂ξ
where
A=
1 X
3
ρi0 z3i − 1 ≷ 0
2
i
B=
1X
ρi0 z3i + 1
4
i
One-soliton solutions for each KdV equation are
vξ
(ξ − vξ τ )
2
r
3vη
vη
2
sech
(η + vη τ )
=
A
2
ϕ2ξ =
ϕ2η
3vξ
sech2
A
r
&
&
p
r
3B 2vη
vη
tanh
(η + vη τ ) + 1
A
2
p
r
3B 2vξ
vξ
(ξ − vξ τ ) − 1
Q=
tanh
A
2
P=
Only same sign polarities (positive or negative) are possible for both waves
2012 EPS/ICPP Stockholm, 2–6 July 2012
Head-on collisions of solitons in multispecies plasmas
5. Illustrations for two ion species: head-on collisions in generic case
Normalization yields for two ion species that
ρ10 =
1 − z2
z1 (z1 − z2 )
and
ρ20 =
z1 − 1
z2 (z1 − z2 )
Left figure: values z1 = 4 and z2 = 1/4, typical for H+ and O+ , so that A > 0 (potential and
density humps) and B > 0 (propagation delays due to interaction)
Right figure: values z1 = 0.1 and z2 = −0.1, typical for a fullerene plasma with electron
contamination, so that A < 0 (potential and density dips) but B > 0 (propagation delays due to
interaction)
2012 EPS/ICPP Stockholm, 2–6 July 2012
Head-on collisions of solitons in multispecies plasmas
6. Critical case: 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
1 ∂ 3 ϕ1ξ
+ C ϕ21ξ
+
=0
∂τ
∂ξ
2 ∂ξ 3
&
∂P
= D ϕ21η
∂η
∂ϕ1η
∂ϕ1η
1 ∂ 3 ϕ1η
− C ϕ21η
−
=0
∂τ
∂η
2 ∂η 3
&
∂Q
= D ϕ21ξ
∂ξ
where
C=
1 X
1
15
ρi0 z4i − 1 >
4
6
i
D=
X
1
1
1−
ρi0 z4i <
8
9
i
Solutions can now be of either sign and thus allow for collision between counterstreaming
negative and positive polarity solitons
r
hp
i
6vξ
ϕ1ξ = ±
sech
2vξ (ξ − vξ τ )
C
r
hp
i
6vη
ϕ1η = ±
sech
2vη (η + vη τ )
C
&
&
2012 EPS/ICPP Stockholm, 2–6 July 2012
n
hp
i
o
3D p
2vη tanh
2vη (η + vη τ ) + 1
C
n
hp
i
o
3D p
Q=
2vξ tanh
2vξ (ξ − vξ τ ) − 1
C
P=
Head-on collisions of solitons in multispecies plasmas
7. Illustrations for two ion species: head-on collisions in critical case
At criticality, one also requires
z2 =
z1 − 1/3
<0
z1 − 1
Values z1 = 0.739 and z2 = −1.552, typical for Ar+ and F− plasma experiments at critical
densities
Here C > 0 and D > 0, but negative polarity soliton propagates to the right and positive polarity
soliton to the left, with positive phase shifts (propagation delays)
2012 EPS/ICPP Stockholm, 2–6 July 2012
Head-on collisions of solitons in multispecies plasmas
8. Summary and remarks
In generic case KdV equations govern collisions between 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
When all ion species are positive, A > 1 and B > 1/2 (no critical compositions): polarities and
phase shifts are positive (equivalent to delays compared to single-soliton trajectory)
Negative polarities in general require at least one negative ion species, in addition to necessary
positive ions, hence A < 0
Criticality also needs at least one negative species to make A = 0, but then always has C > 0
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, and general behaviour
However, amplitude of overlapping solitons during collision was less than sum of initial soliton
amplitudes, which cannot correctly be dealt with by available Poincaré-Lighthill-Kuo formalism
Analytical treatment of more complicated plasma models is qualitatively analogous
2012 EPS/ICPP Stockholm, 2–6 July 2012
Head-on collisions of solitons in multispecies plasmas