Supercritical solitons in two-electron temperature
plasmas
Frank Verheest1,2 , Carel P. Olivier3 & Willy A. Hereman4
1 Sterrenkundig
2 School
Observatorium, Universiteit Gent, Belgium
of Chemistry and Physics, University of KwaZulu-Natal, Durban, South Africa
3 Space
4 Applied
Science, South African National Space Agency, Hermanus, South Africa
Mathematics and Statistics, Colorado School of Mines, Golden CO, USA
Abstract
Solitary waves in a plasma with cold positive ions and a two-temperature electron population are
investigated with both a reductive perturbation approach and a Sagdeev pseudopotential method.
First, requiring that neither quadratic nor cubic nonlinearities are present in an evolution equation
of KdV-type fixes the set of compositional parameters, in what can be termed a supercritical
plasma composition. The result is a modified KdV (mKdV) equation with a quartic nonlinear term,
but conclusions about its one-soliton solution and integrability will also hold for more complicated
plasma compositions, with four or more constituents. Only three polynomial conservation laws can
be obtained and the new mKdV equation is not completely integrable, thus precluding the
existence of multi-soliton solutions.
Next, the full pseudopotential method has been investigated for this supercritical composition and
this allows for a detailed comparison with the mKdV-like results. The mKdV solitons are shown to
have larger amplitudes and widths than those obtained from the more complete Sagdeev solution.
Consequently, only slightly superacoustic mKdV solitons have acceptable amplitudes and widths,
in the light of the full solutions.
F. Verheest, C. P. Olivier & W. A. Hereman, J. Plasma Phys. 82 (2016) 905820208
2016 EPS Conf. Plasma Physics, Leuven, 4–8 July 2016
Supercritical solitons in two-electron temperature plasmas
1. Motivation and model
Since 1965 electrostatic solitons have been described by KdV equations, yielding superacoustic
phenomena of limited amplitudes (so as not to invalidate reductive perturbation analysis)
tending to zero as M → Ms , with typical relations between amplitude, width and speed,
“taller is faster and narrower"
Simultaneously, Sagdeev pseudopotential analysis described large amplitude solitons in frame
where structure is stationary, one at a time, without possibility of studying interactions or stability
KdV-type equations have balance between nonlinearities (steepening of soliton profile) and
dispersion (broadening of soliton profile), coupled to slow time variation with respect to
propagation at linear acoustic speed
Generic KdV equation has quadratic nonlinearity unless plasma composition is critical, annulling
coefficient of quadratic term and giving rise to cubic nonlinearity and modified KdV equation
Question is: can plasma configuration be supercritical, meaning that lowest-order nonlinearity is
quartic? If so, call resultant KdV equation supercritical
Not easily realized, but model with cold positive ions and two-temperature Boltzmann electron
distributions just achieves supercriticality
2016 EPS Conf. Plasma Physics, Leuven, 4–8 July 2016
Supercritical solitons in two-electron temperature plasmas
2. Basic formalism
Restricted to one-dimensional propagation in space and written in normalized variables, the
model equations for cold ions and two-temperature electrons reads
∂
∂n
+
(nu) = 0
∂t
∂x
∂u
∂u
∂ϕ
+u
+
=0
∂t
∂x
∂x
∂2ϕ
+ n − f exp[αc ϕ] − (1 − f ) exp[αh ϕ] = 0
∂x2
KdV equations have structure
∂ψ
∂3ψ
∂ψ
+ Bψ
+
=0
∂τ
∂ξ
∂ξ 3
∂ψ
∂ψ
∂3ψ
+ C ψ2
+
=0
∂τ
∂ξ
∂ξ 3
∂ψ
∂ψ
∂3ψ
+ D ψ3
+
=0
∂τ
∂ξ
∂ξ 3
Standard KdV
Modified KdV when B = 0
Supercritical KdV for B = C = 0
To obtain supercritical KdV equation in ψ = ϕ1 scaling is ∂/∂ξ ∼ ε3/2 , ∂/∂τ ∼ ε9/2 and leads
to stretched variables ξ = ε3/2 (x − t) & τ = ε9/2 t where expansion is taken as
n = 1 + εn1 + ε2 n2 + ...
u = εu1 + ε2 u2 + ...
2016 EPS Conf. Plasma Physics, Leuven, 4–8 July 2016
ϕ = εϕ1 + ε2 ϕ2 + ...
Supercritical solitons in two-electron temperature plasmas
3. Details of supercritical analysis
Combining ion continuity and momentum equations on one side and using Poisson’s equation
on other side yields (before slow time and Laplacian kick in)
n2 = ϕ2 + 32 ϕ21 = A1 ϕ2 + 12 A2 ϕ21
n1 = ϕ1 = A1 ϕ1
n3 = ϕ3 + 3ϕ1 ϕ2 + 52 ϕ31 = A1 ϕ3 + A2 ϕ1 ϕ2 + 16 A3 ϕ31
A` = f α`c + (1 − f )α`h
(` = 1, 2, 3, ...)
Having ϕ1 6= 0 demands that
⇒
A1 = 1 (linear dispersion law)
√
f = 61 (3 − 6) = 0.092
A2 = 3 (condition B = 0)
√
αc = 3 + 6 = 5.45
A3 = 15 (condition C = 0)
√
αh = 3 − 6 = 0.55
and leads to supercritical KdV equation
∂ϕ1
∂ϕ1
1 ∂ 3 ϕ1
+ 2ϕ31
+
=0
∂τ
∂ξ
2 ∂ξ 3
Common mistake is using B = C = 0 but treating afterwards D as free parameter!
2016 EPS Conf. Plasma Physics, Leuven, 4–8 July 2016
Supercritical solitons in two-electron temperature plasmas
4. Supercritical solitons and conserved densities
Use standard techniques to derive one-soliton solution in terms of excess W over acoustic speed
ϕ1 =
√
3
r
5Wsech2/3
3
!
W
(ξ − Wτ ) .
2
There are only three conservation laws:
"
#
∂ϕ21
1 ∂
∂ 2 ϕ1
∂ 4 5
∂ 2 ϕ1
1 ∂ϕ1 2
∂ϕ1
4
=0
+
ϕ1 +
=0
+
ϕ + ϕ1
−
∂τ
2 ∂ξ
∂ξ 2
∂τ
∂ξ 5 1
∂ξ 2
2
∂ξ
"
"
#
2
∂
∂ 5 8
5
5 ∂ϕ1 2
∂ϕ1 2
∂ 2 ϕ1
5 ∂ 2 ϕ1
+
+ ϕ41
ϕ51 −
ϕ1 − 10ϕ31
+
∂τ
2
∂ξ
∂ξ 4
∂ξ
2
∂ξ 2
4
∂ξ 2
5 ∂ϕ1 ∂ 3 ϕ1
=0
−
2 ∂ξ ∂ξ 3
In contrast to KdV and mKdV equations which have infinite string of conserved densities and are
completely integrable, supercritical KdV equation (and also higher orders) is not integrable and
looses typical soliton interactions and their famous stability
2016 EPS Conf. Plasma Physics, Leuven, 4–8 July 2016
Supercritical solitons in two-electron temperature plasmas
5. Comparison between KdV and Sagdeev solitons
This problem can also be treated by Sagdeev pseudopotential method, whereby basic
equations in co-moving frame can be integrated and combined into particle-like energy integral
1
2
dϕ
dχ
2
+ S(ϕ, M) = 0
"
S(ϕ, M) = M
2
2ϕ
1− 2
M
1−
with
1/2 #
+
h √
i
√
√
5
1
+ exp(3ϕ) 2 6 sinh( 6ϕ) − 5 cosh( 6ϕ)
3
3
j1
j1
SHjL
W=0.001
M=1.001
0.3
0.1
W=0.01
M=1.01
-0.2
-0.1
0.1
0.2
0.3
0.2
0.1
-100
-50
50
100
Ζ
-50
-25
25
50
Ζ
In left and middle panels dashed refers to KdV and solid to Sagdeev solitons, and M = 1 + W
2016 EPS Conf. Plasma Physics, Leuven, 4–8 July 2016
Supercritical solitons in two-electron temperature plasmas
j
6. Summary and remarks
Generic KdV equation has quadratic nonlinearities as lowest significant contribution, besides
slow time variation in frame moving at linear acoustic speed and typical dispersion
For specific plasma compositions lowest-order nonlinearities can change to cubic (modified
KdV) or more rarely to quartic (supercritical KdV), imposing restrictions on compositional
parameters which are more and more difficult to obey in combination
Plasma model with cold ions and two-temperature Boltzmann electrons allows supercriticality
and can also be treated by pseudopotential Sagdeev analysis
Supercritical KdV is shown to be nonintegrable, thereby loosing all beautiful interaction and
stability properties of real solitons, leaving only equivalent of one-soliton solution
Comparison between supercritical KdV and full Sagdeev soliton profiles at same excess over
linear speed shows that KdV solitons have larger amplitudes and widths than fully nonlinear
counterparts
Supercritical KdV solitons spring bounds on validity of reductive perturbation theory already at
very low excesses over acoustic speed, typically W ' 0.001 which is very slightly superacoustic
Hence, KdV theory should be used with great care, prsesumably at very small amplitudes
2016 EPS Conf. Plasma Physics, Leuven, 4–8 July 2016
Supercritical solitons in two-electron temperature plasmas