Head-on collisions of electrostatic solitons in multi-ion plasmas
Frank Verheest
Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281, B–9000 Gent, Belgium &
School of Chemistry and Physics, University of KwaZulu-Natal, Durban 4000, South Africa
Manfred A. Hellberg
School of Chemistry and Physics, University of KwaZulu-Natal, Durban 4000, South Africa
Willy A. Hereman
Department of Applied Mathematics and Statistics,
Colorado School of Mines, Golden CO 80401–1887, U.S.A.
(Dated: September 15, 2012)
Head-on collisions between two electrostatic solitons are dealt with by the Poincaré-Lighthill-Kuo
method of strained coordinates, for a plasma composed of a number of cold (positive and negative)
ion species and Boltzmann electrons. The nonlinear evolution equations for both solitons and their
phase shift due to the collision, resulting in time delays, are established. A Korteweg-de Vries description is the generic conclusion, except when the plasma composition is special enough to replace
the quadratic by a cubic nonlinearity in the evolution equations, with concomitant repercussions
on the phase shifts. Applications include different two-ion plasmas, showing positive or negative
polarity solitons in the generic case. At critical composition, a combination of a positive and a
negative polarity soliton is possible.
PACS numbers: 52.27.Cm, 52.35.Fp, 52.35.Mw, 52.35.Sb
I.
INTRODUCTION
There has been a recent upsurge of interest in the investigating of head-on collisions between two acoustictype nonlinear waves in various plasma models.1–14 This
builds on earlier efforts by Su and Mirie,15 who dealt with
a similar problem for surface waves on shallow water, by
using an extension of the Poincaré-Lighthill-Kuo (PLK)
method of strained coordinates.16
Head-on collisions present a particular challenge, in
contrast to the overtaking interactions of several solitons
propagating in the same direction. Both the standard
and the modified Korteweg-de Vries (KdV and mKdV)
equations are integrable, possessing, i.a., an infinite chain
of conserved densities and exact solutions for interactions
between N solitons, for all natural numbers N .17,18 Such
N -soliton solutions correctly describe how taller (and
faster) solitons overtake the smaller (and slower), with
full nonlinearity during the interaction phase. Unfortunately, seeing that the typical solutions of the KdV equations are mildly superacoustic, all true KdV and mKdV
solitons propagate in the direction that underlies the basic derivation of the parent equation, when observed in
an inertial frame.
As far as we could ascertain, there is no such powerful
exact method to deal with head-on collisions, so that one
is forced to use an approximate description, as done indeed by relevant theoretical papers in the literature.1–14
In a recent paper, Harvey et al.19 investigated experimentally the interaction of two counter-propagating solitons of equal amplitude in a monolayer strongly coupled
dusty plasma. To quote their conclusions, it was found
that the solitons are delayed after the collision, with soli-
tons of higher amplitude experiencing longer delays. The
amplitude of the overlapping solitons during the collision
was less than the sum of the initial soliton amplitudes.
Harvey et al.19 discuss a comparison with the theoretical
analysis of Su and Mirie,15 which, however, like all papers
using the same methodology, predicts a superposition of
the amplitudes during the interaction. While the theoretical analyses give a qualitatively correct picture of the
soliton structures outside the head-on collision region,
plus the time delays,14 the PLK method is not really capable of accurately reproducing the amplitudes during
the collision.
In a previous paper14 we tried to give a detailed derivation of the relevant amplitude and time delay equations
for the head-on collisions in a plasma composed of cold
positive ions and hot Cairns distributed nonthermal electrons, with an emphasis on the fundamental, as well as
the often tacitly assumed underlying hypotheses. Theoretical papers in the literature1–12,15 derive KdV equations for the amplitudes, with the corresponding phase
shifts (time delay), and this for quite a variety of plasma
models. As we discovered,13,14 several plasma models
can also admit mKdV equations for the amplitudes, provided the model parameters can be chosen such that
the coefficient of the quadratic nonlinearity in the KdV
equations vanishes. Except for the papers by Demiray2
and Chatterjee et al.,5 whose plasma models do not give
rise to critical parameters, other models1,3,4,6–12 are sufficiently sophisticated to allow critical compositions, yet
these were not discussed.
In the present paper, we investigate a plasma model
composed of a number of cold (positive and negative) ion
species in the presence of Boltzmann electrons, a model
2
not studied before in this context. The particular interest is that such a multi-ion plasma can cover several
interesting situations, including some observed experimentally. In doing so, we will follow the outline of our
previous paper,14 but in an abbreviated way, without repeating some of our comments on the basic aspects of
the methodology, yet with sufficient detail to make this
paper coherent and self-contained.
The paper is organized as follows. In Section II the
basic formalism is given for the generic case, which leads
to KdV equations for the right- and the left-traveling
soliton, respectively, plus the phase changes due to the
head-on collision between the two solitons. Section III
is devoted to the special case when the derivation leads
to an mKdV equation, with corresponding changes in
phases. Section IV then briefly summarizes our conclusions.
So as not to hinder the flow of the main exposition,
some special topics have been relegated to two Appendices. Appendix A covers the proof that even plasmas
with only two cold ion species can exhibit critical parameters. Appendix B deals with some general properties of
the coefficients of the quadratic and cubic nonlinearities
in the KdV and mKdV equations.
II.
Basic Formalism
The normalized continuity and momentum equations
for the different ion species, with label i, are
∂ρi
∂
+
(ρi ui ) = 0,
∂t
∂x
∂ui
∂ui
∂ϕ
+ ui
+ zi
= 0.
∂t
∂x
∂x
All species are coupled through Poisson’s equation,
∂2ϕ X
+
ρi zi − exp(ϕ) = 0,
∂x2
i
(1)
(2)
The mass density ρi and fluid velocity ui have been normalized in terms of a reference mass density n0 M and
where exp(ϕ) represents the hot Boltzmann electron contribution. P
Charge neutrality in equilibrium requires the
condition
now adjust the referi ρi0 zi = 1,
Pand we
2
ence
i ρi0 zi = 1. In other words,
P 2mass M2 so that
i ωpi = n0 e /ε0 M defines M through the effective ion
plasma frequency of the system. This can be done without loss of generality and simplifies the notational details
of the algebra. One can also show that the normalization
is such that the linear acoustic phase velocity is c = 1.
ξ = ε(x − t) + ε2 P (ξ, η, τ ) + ...,
η = ε(x + t) + ε2 Q(ξ, η, τ ) + ...,
τ = ε3 t,
The series expansions of the dependent variables are
To lowest nonzero order (1), (2) and (3) lead to
(6)
(4)
referring in ξ and η to a right- and to a left-propagating
soliton, Sξ and Sη , respectively. The present description, to lowest order, treats the colliding waves as separate entities propagating in the same medium, and their
stretching involves the unique linear acoustic phase velocity c = 1. This substitution leads to
∂
∂
∂
∂P
∂P
∂
∂Q ∂Q ∂
3
=ε
+
+ε
+
+
+
+ ...,
∂x
∂ξ
∂η
∂ξ
∂η ∂ξ
∂ξ
∂η ∂η
∂
∂
∂
∂P
∂P
∂
∂Q ∂Q ∂
∂
=ε
−
+ ε3
−
+
−
+ ε3
+ ... .
∂t
∂η ∂ξ
∂η
∂ξ ∂ξ
∂η
∂ξ ∂η
∂τ
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 + ... .
(3)
As we are considering head-on collisions of two electrostatic solitons, and following methods in the cited literature, we introduce
BASIC FORMALISM AND GENERIC
COMPOSITION
A.
p
of an ion-acoustic velocity Via = Te /M . The chargeper-mass ratios are expressed through the parameters
zi = qi M/emi , so that only one combined parameter
per ion species is needed, rather than working with the
charges qi and masses mi separately. The electrostatic
potential ϕ is given in units of Te /e, where Te is the kinetic temperature of the hot electrons with equilibrium
density n0 .
∂ρi1
∂ρi1
∂ui1
∂ui1
−
+ ρi0
+
= 0,
∂η
∂ξ
∂ξ
∂η
∂ui1
∂ui1
∂ϕ1
∂ϕ1
−
+ zi
+
= 0,
∂η
∂ξ
∂ξ
∂η
(5)
3
X
ρi1 zi − ϕ1 = 0.
(7)
i
and τ , but not ξ,
ρi1 = ρi0 zi (ϕ1ξ + ϕ1η ) ,
ui1 = zi (ϕ1ξ − ϕ1η ) ,
ϕ1 = ϕ1ξ + ϕ1η ,
The linear perturbations will consist of a term depending
on ξ and τ , but not η, and another term depending on η
Compact notation denotes the dependence on the space
arguments ξ or η.
To the next higher order (1), (2) and (3) give
∂ρi2
∂ui2
∂ui2
∂
∂
∂ρi2
−
+ ρi0
+
+
+
(ρi1 ui1 ) = 0,
∂η
∂ξ
∂ξ
∂η
∂ξ
∂η
∂ui2
∂ui2
∂ui1
∂ui1
∂ϕ2
∂ϕ2
−
+ ui1
+
+ zi
+
= 0,
∂η
∂ξ
∂ξ
∂η
∂ξ
∂η
X
1
ρi2 zi − ϕ2 − ϕ21 = 0,
2
i
which yields
ρi2 = ρi0 zi (ϕ2ξ + ϕ2η ) +
ui2 = zi (ϕ2ξ − ϕ2η ) +
3
ρi0 zi2 (ϕ21ξ + ϕ21η ) + ρei2 ,
2
1 2 2
z (ϕ − ϕ21η ) + u
ei2 ,
2 i 1ξ
ϕ2 = ϕ2ξ + ϕ2η + ϕ
e2 .
Here ϕ
e2 has been defined as the part of ϕ2 which depends
on ξ and η together, besides τ , in a way which cannot
be disentangled, with similar notation for other variables
and orders.
The expressions in (10) are coupled through constraints that arise from Poisson’s equation in (9),
X
3
ρi0 zi3 − 1 ϕ21ξ = 0,
(11)
i
X
3
ρi0 zi3 − 1 ϕ21η = 0,
(12)
i
2
X
∂
∂
∂2
3
ϕ
e2 −
ρi0 zi
+
ϕ1ξ ϕ1η
4
∂ξ∂η
∂ξ
∂η
i
2
∂
∂
−
ϕ1ξ ϕ1η = 0.
−
∂ξ
∂η
(13)
Here, (13) comes from the elimination of ui2 between the
first two equations in (9) and using the third equation in
(9) to replace ρi2 , for the terms which depend on ξ and
η together. A similar procedure will be followed for the
higher-order equations.
The structure of (11) and (12) points to two possibilities:
either the ion composition is very special, in that
P
3
i ρi0 zi = 1/3, or ϕ1ξ = ϕ1η = 0. The
P former case corresponds to critical ion densities,Pas i ρi0 zi = 1 (charge
2
neutrality in equilibrium) and
i ρi0 zi = 1 (choice of
normalization), and a third condition on the densities
ρi0 is not generic. Examples of critical ion densities are
(9)
discussed in Appendix A, and the critical composition
itself will be investigated in detail in Section III.
B.
(10)
(8)
Generic case
Pursuing now the generic case, that is, setting ϕ1ξ =
ϕ1η = 0, one is, from (9)–(13), once again led to separability. The expressions for the second-order variables
then resemble (8) with the index 1 replaced by 2.
Next, we find that the third-order variables obey linear
equations akin to (7), and that these variables do not
appear in the equations for the fourth-order variables.
We thus simply set ρi3 = ui3 = ϕ3 = 0.
Finally, interesting new contributions involving the
fourth-order variables, as well as P and Q, will arise at
the last step we consider. Picking out those terms that
depend on τ , and on either ξ or η, but not both, we find
the typical KdV equations
∂ϕ2ξ
∂ϕ2ξ
1 ∂ 3 ϕ2ξ
+ Aϕ2ξ
+
= 0,
∂τ
∂ξ
2 ∂ξ 3
∂ϕ2η
∂ϕ2η
1 ∂ 3 ϕ2η
− Aϕ2η
−
= 0,
∂τ
∂η
2 ∂η 3
(14)
for the right- and left-propagating solitary wave, respectively. Here the coefficient of the quadratic nonlinearity,
1 X
A=
3
ρi0 zi3 − 1 ,
(15)
2
i
has already been encountered in a similar role in (11) and
(12), but for a numerical factor.
Further information is in the terms which contain both
ξ and η, besides τ , combining into
"
#
∂2
1 X
ϕ
e4 −
ρi0 zi3 − 1 ϕ2ξ ϕ2η
∂ξ∂η
2 i
4
∂P
∂ϕ2ξ
∂
− Bϕ2η
∂ξ
∂η
∂ξ
∂
∂Q
∂ϕ2η
+
− Bϕ2ξ
= 0,
∂η
∂ξ
∂η
+
(16)
with
B=
1 X
ρi0 zi3 + 1 .
4 i
(17)
The second and third terms in (16) would generate secular contributions at the next higher order, so they must
vanish, leading to equations for the phase shifts,
∂P
= B ϕ2η ,
∂η
∂Q
= B ϕ2ξ .
∂ξ
(18)
It is seen that ∂P/∂η cannot depend on ξ and thus P
itself might contain an additive part which would depend
on ξ and τ , but not on η. This would refer to changes in
the phase of the right-propagating soliton due to its own
propagation and can be omitted altogether. Analogous
arguments hold for the absence of η in Q. Note that B
might be positive or negative, an aspect that is further
addressed in Appendix B.
The one-soliton solutions of (14) are the well-known
“sech squared” solitons of KdV theory, here
r
3vξ
vξ
2
ϕ2ξ =
sech
(ξ − vξ τ ) ,
A
2
r
vη
3vη
sech2
(η + vη τ ) ,
(19)
ϕ2η =
A
2
expressed in terms of the velocities vξ and vη , respectively, for the right- and left-propagating soliton. This
requires that vξ > 0 and vη > 0 and also indicates that
the two interacting solitons have the same polarity, given
by the sign of A. For A > 0 the solitons will have positive
polarity, and for A < 0 negative polarity.
To obtain the phase shifts after the head-on collision
between the two solitons, we assume that Sξ and Sη are,
asymptotically, far from each other at the initial time
(t = −∞), i.e., Sξ is at ξ = 0, η = −∞ and Sη is at
η = 0, ξ = +∞, respectively, as done in several of the
cited papers, e.g., Ref. [12]. After the collision (t = +∞),
Sξ is far to the right of Sη , i.e., Sξ is at ξ = 0, η = +∞
and Sη is at η = 0, ξ = −∞.
The phase shifts expressed by P and Q are found from
substitution of (19) into (18) and integration, yielding
p
r
3B 2vη
vη
P =
tanh
(η + vη τ ) + 1 ,
A
2
p
r
3B 2vξ
vξ
Q=
tanh
(ξ − vξ τ ) − 1 . (20)
A
2
Note that, although the stretching (4) uses opposing velocities c = 1 of equal magnitude, this does not hold
FIG. 1. (Color online) Head-on collision for plasma composition with two positive ion species and Boltzmann electrons,
for z1 = 4, z2 = 1/2, vξ = 0.05 and vη = 0.1. The values for
z1 might refer to H+ and z2 to O+ . The solitons have positive
polarity and are compressive in the densities.
for the one-soliton solutions, which are superacoustic in
their direction of propagation but can have quite different
amplitudes. Increases in vξ entail increases in the amplitudes of Sξ as well as in the phase shift of Sη . Thus, the
larger of the two solitons travels faster than the smaller
one, but is less affected by the phase shift when emerging
from the collision region.
For the special case of a plasma with one species of
ions only, besides the electrons, as usually treated in the
literature,2 we adjust the normalization such that ρi0 = 1
and zi = 1 and find that A = 1 and B = 1/2, with
corresponding simplifications in (14) and (18)–(20).
We can illustrate the foregoing discussion with some
graphs, for the case when there are two ion species. In
Fig. 1 both ion species are positive, so that the solitons
have the positive polarity given by the sign of A. The
values for z1 might refer to hydrogen ions, H+ , and z2
to singly ionized oxygen, O+ . Furthermore, there is a
general proof in Appendix B that when all ions are positive, A > 1 and B > 1/2. The phase shifts P and Q
are then also positive, pointing to a delay in propagation
occurring during the interaction, as compared to a single
KdV soliton in the same physical system.
When the ion mixture contains both positive and negative ions, either of the two cases may be obtained: the
ratios z1 and z2 are then such that either both solitons
are positive, or they are both negative. This is illustrated in Fig. 2 for z1 = 0.1 and z2 = −0.1, which leads
to both solitons being negative. The charge-to-mass ratios have been taken equal in magnitude, which could
cover the case of a fullerene (pair-ion) plasma,20,21 with
an admixture of electrons. We note that these experiments were reported to involve only positive and negative
fullerene ions,20,21 without any contamination. However,
not all the observed wave dispersion could then be under-
5
∂ϕ1η
1 ∂ 3 ϕ1η
∂ϕ1η
= 0,
− C ϕ21η
−
∂τ
∂η
2 ∂η 3
(22)
with
C=
1 X
15
ρi0 zi4 − 1 .
4
i
(23)
The terms which contain both ξ and η, besides τ , allow
for the determination of the phase shifts through
∂P
= D ϕ21η ,
∂η
∂Q
= D ϕ21ξ ,
∂ξ
(24)
where
FIG. 2. (Color online) Head-on collision for plasma composition with positive and negative ion species, and Boltzmann
electrons, for z1 = 0.1, z2 = −0.1, vξ = 0.01 and vη = 0.02.
The underlying model is a fullerene (pair-ion) plasma with an
admixture of electrons. The solitons have negative polarity
and are rarefactive in the densities.
stood. Hence, the presence of some electron admixture
has been advocated in order to explain the wave dispersion diagrams.22
III.
SOLITONS AND PHASE SHIFTS AT
CRITICAL COMPOSITIONS
In this section we take the ion densities to be critical,
with A = 0, examples of which are discussed in Appendix
A. The quadratic nonlinearity in the KdV equations (14)
disappears, and in (10) we have to keep the contributions
in ϕ1 . Without loss of generality, we can set ϕξ2 = 0 and
ϕη2 = 0, but ϕ
e2 6= 0, as shown by the simplified version
of (13),
2
∂2ϕ
e2
1
∂
∂2
∂2
+
−
− 2 ϕ1ξ ϕ1η = 0.
(21)
∂ξ∂η 3 ∂ξ∂η ∂ξ 2
∂η
It is remarkable that, barring the coefficient in front of
the parentheses, this equation is identical to what one
obtains for quite different plasma models and waves, such
as for ion-acoustic solitons in a mixture of cold ions with
nonthermal electrons14 or for electron-acoustic solitary
waves in a two-electron plasma with hot nonextensive
and cool components, in the presence of a neutralizing
ion background.13
At critical densities, the new contributions involving
P and Q appear in the equations for the third order
variables. Combining again the parts of these equations
which contain terms that only depend on ξ or on η (besides τ ) yields the typical mKdV equations
∂ϕ1ξ
∂ϕ1ξ
1 ∂ 3 ϕ1ξ
+ C ϕ21ξ
+
= 0,
∂τ
∂ξ
2 ∂ξ 3
D=
X
1
1−
ρi0 zi4 .
8
i
(25)
Because mKdV equations like (22) are invariant for a sign
inversion of ϕ1ξ or ϕ1η , the one-soliton solutions of (22)
are
r
p
6vξ
ϕ1ξ = ±
sech
2vξ (ξ − vξ τ ) ,
C
r
p
6vη
ϕ1η = ±
sech
2vη (η + vη τ ) .
(26)
C
Note that the respective ± signs are not coupled. Substitution of (26) into (24) and integration yields
p
3D p
2vη tanh
2vη (η + vη τ ) + 1 ,
C
p
3D p
Q=
2vξ tanh
2vξ (ξ − vξ τ ) − 1 . (27)
C
P =
P and Q are defined in terms of ϕ21η and ϕ21ξ , respectively,
so that the polarity of the modes does not play a role in
this aspect of the problem. However, D could be negative
in certain parameter ranges.
For counter-propagating solitons of the same polarity
the graphs are qualitatively similar to those obtained in
the generic case, as illustrated in Figs. 1 and 2. The
characteristics are less steep, because one is now plotting “sech” rather than “sech squared” solutions. Interestingly, in the critical case the two mKdV equations
(22) admit a combination of positive and negative modes.
This is shown in Fig. 3 for a stronger right-propagating
negative soliton and a weaker left-propagating positive
soliton.
In order to bring out more clearly the interaction
around the time of the head-on collision, we present in
Fig. 4 three snapshots of the combined soliton profile ϕ1
as function of x, for the plasma parameters used to generate Fig. 3. The times are chosen before (t = −5, dotted curve), during (t = 0, solid curve) and after (t = 5,
dashed curve) the head-on collision. Although it is difficult to see in detail, the negative polarity contribution is
for t = −5 at x = −5.573 but for t = 5 only at x = 5.564,
6
IV.
FIG. 3. (Color online) Head-on collision between solitons of
opposite polarities, for critical plasma composition, for z1 =
0.739, z2 = −1.552, vξ = 0.1 (negative polarity) and vη = 0.05
(positive polarity). This mimics a plasma with Ar+ and F−
ions.
j
0.3
0.2
0.1
-20
10
-10
20
x
-0.1
-0.2
-0.3
CONCLUSIONS
The head-on collision between two electrostatic solitons in a multi-ion plasma with Boltzmann electrons has
been treated for a general model which can accommodate
a number of cold ion species. For the more specific applications, we have illustrated the essential characteristics
for different plasmas with two ion species, of the same or
of the opposite sign.
In the generic case, the solitons are governed by KdV
equations, with a phase shift occurring during the interaction, which usually leads to a time delay compared
to the trajectory of a single soliton in the same plasma.
Both left- and right-propagating solitons must have the
electrostatic polarity given by the sign of the coefficient,
A, of the nonlinear term in the KdV equation. For plasmas with two positive ion species, the solitons have positive polarity, as in, e.g., a hydrogen-oxygen model. To
obtain negative polarities, there should be at least one
negative ion species, besides one or more positive ones,
and this is shown for a fullerene (pair-ion) plasma with
an admixture of electrons.
When A vanishes, for critical plasma parameters, the
KdV equation disappears and the scaling leads to an
mKdV equation with cubic nonlinearity, a possibility
not addressed earlier in the literature, except in our
recent investigation of head-on collisions in nonthermal
plasmas.14 As the mKdV equation is invariant for inversion of the electrostatic polarity, it allows for combinations of solitons of different polarities. We have chosen to
discuss a negative right- and a positive left-propagating
soliton, for a plasma in which the dominant ions are argon and fluorine, experimentally studied by Nakamura et
al.23
-0.4
FIG. 4. (Color online) Graphs of ϕ1 as function of x for the
plasma parameters used in Fig. 3, before (for t = −5, dotted
curve), during (for t = 0, solid curve) and after (for t = 5,
dashed curve) the head-on collision.
indicating a delay in propagation time, compared to what
a single soliton would do. Similarly, the positive polarity
contribution is for t = −5 at x = 5.349 but for t = 5
only at x = −5.336, showing a similar delay. Moreover,
the amplitudes for t = 5 are smaller (in absolute values)
than their counterparts for t = −5.
Solitary waves under critical conditions were studied in
seminal experiments of Nakamura et al.23 They worked
with a negative ion plasma, having argon (Ar+ ) and fluorine (F− ) ions as the dominant ion species. From a mass
ratio of z1 /|z2 | = 0.476, they found that the coefficient
of the KdV equation vanishes at a critical density ratio
of 0.102. Our computations yield a critical density of
n20 /n10 = 0.1024, and thus confirm the value found by
Nakamura et al.
Appendix A: TWO-ION PLASMAS AND
CRITICAL DENSITIES
The simplest example of a plasma exhibiting a critical
composition is a two-ion model, for which the necessary
conditions that ρ10 and ρ20 must obey are
ρ10 z1 + ρ20 z2 = 1,
ρ10 z12 + ρ20 z22 = 1,
ρ10 z13 + ρ20 z23 = 31 .
(A1)
The first two equations determine ρ10 and ρ20 in terms
of z1 and z2 ,
ρ10 =
1 − z2
,
z1 (z1 − z2 )
ρ20 =
z1 − 1
.
z2 (z1 − z2 )
(A2)
Since at least one ion species needs to be positive, we
can assume, without loss of generality, that z1 > 0. In
the case of two positive species, we take z1 > z2 > 0,
and see that for the densities to be strictly positive, this
condition is sharpened to z1 > 1 > z2 . When there is
7
a negative ion species, then z2 < 0, and one has to take
z1 < 1, without any obvious restriction on z2 < 0.
Using (A2) one can rewrite the third condition (A1) as
(z1 − 1)(1 − z2 ) +
2
3
= 0.
(A3)
In the case of two positive ion species, for which z1 >
1 > z2 , this condition cannot hold, as the left-hand side
of (A3) is larger than 2/3. For such plasmas A > 0, the
solitons are always compressive, and hence there can be
no critical parameters. As shown in Appendix B, this
holds even for more than two ion species, provided all
are positive.
We thus return to the case of a positive and a negative
ion species and hence assume that z2 < 0. Solving (A3)
yields
z2 =
z1 − 1/3
< 0,
z1 − 1
(A4)
requiring that 1/3 < z1 < 1. This leads to
2
,
− 6z1 + 1)
9(z1 − 1)3
=
,
(3z1 − 1)(3z12 − 6z1 + 1)
ρ10 = −
ρ20
z1 (3z12
(A5)
and these densities are positive.
Hence, critical densities can occur in a two-ion plasma
only if there are both a positive and a negative ion
species. This condition is also required for the presence
of rarefactive solitons in the generic case, for A < 0.
Appendix B: COEFFICIENTS OF QUADRATIC
AND CUBIC NONLINEARITIES
First, we shall prove that, when all ion species in a
multi-ion plasma are positive (zi > 0), both the coefficient of the quadratic nonlinearity, A, and the sign of the
phase shifts, B, are also positive. The two solitons must
then have positive polarity and there can be no critical
compositions. In Appendix A it was already shown that
A > 0 for two ion species, but in fact it holds for any
number of ion species, provided all arePpositive.
i ρi0 zi = 1 and
PWhen2 all zi > 0, the conditions
ρ
z
=
1
are
used
to
deduce
a
chain
of
(in)equalities,
i0
i
i
!2
!2
X
X 1/2 1/2 1/2 3/2
1=
ρi0 zi2
=
ρi0 zi · ρi0 zi
i
i
!
6
X
i
ρi0 zi
!
X
i
ρi0 zi3
=
X
ρi0 zi3 ,
(B1)
i
with the help of the Cauchy-Schwarz inequality24 in the
penultimate step. The proof of the Cauchy-Schwarz
inequality is immediate for N -dimensional vectors, as
P 2
(a · b)2 6 ||a||2 ||b||2 underPthe norm ||a||2 =
i ai and
the scalar product a·b = i ai bi . We thus find that
!
X
1
3
A=
3
ρi0 zi − 1 > 1,
2
i
!
1
1 X
3
B=
ρi0 zi + 1 > ,
(B2)
4
2
i
for any number of cold positive ion species. The equality
sign in (B2) holds for a single ion species, and only then.
Next, we assume that the composition is critical, A =
0, for which a mix of positive and negative ion species is
needed. Now we prove that the coefficient of the cubic
nonlinearity in the mKdV equations
(22), C,P
cannot be
P
2
zero under
the
conditions
that
ρ
z
=
1,
i0
i
i
i ρi0 zi =
P
3
1 and i ρi0 zi = 1/3. From A = 0 we deduce another
chain of (in)equalities,
!2
!2
X
X 1/2
1
1/2 2
3
=
ρi0 zi
=
ρi0 zi · ρi0 zi
9
i
i
!
!
X
X
X
2
4
<
ρi0 zi
ρi0 zi =
ρi0 zi4 .
(B3)
i
i
i
Hence we arrive at
1
C=
4
!
15
X
i
ρi0 zi4 − 1
>
1
,
6
(B4)
for any number of cold ion species, regardless of their
charge signs. As in related investigations,25 there is
no need to resort to higher-order KdV equations, which
would then be nonintegrable.
On the other hand, we see that
!
X
1
1
D=
1−
ρi0 zi4 < ,
(B5)
8
9
i
indicating that D might be negative, depending on the
values for the model parameters.
ACKNOWLEDGMENTS
M.A.H. acknowledges the support of the NRF of South
Africa. The research was supported in part by the National Science Foundation (NSF) of the U.S.A. under
Grant No. CCF-0830783. Any opinions, findings and
conclusions or recommendations expressed in this material are those of the authors and therefore the NRF and
NSF do not accept any liability in regard thereto. Jacob Rezac is thanked for verifying the computations and
additional editing.
8
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