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1. Introduction
Nonlinear solitary acoustic waves in plasmas have been studied over the last half-century through reductive perturbation theory (Washimi & Taniuti Reference Washimi and Taniuti1966), if the nonlinearities are sufficiently weak, or through a pseudopotential Sagdeev analysis (Sagdeev Reference Sagdeev and Leontovich1966), for larger nonlinearities. In the former case, one deduces a Korteweg–de Vries (KdV) equation (Korteweg & de Vries Reference Korteweg and de Vries1895) or extended versions of it, such as the modified KdV (mKdV) equation, having a cubic rather than the usual quadratic nonlinearity. In the latter case, one can a posteriori expand the Sagdeev pseudopotential, if the nonlinearities are deemed small. If that is the case, there is an interesting link with results obtained through reductive perturbation theory, seeing that this method employs expansions of the dependent variables from the start (Baluku, Hellberg & Verheest Reference Baluku, Hellberg and Verheest2010).
Denoting the Sagdeev pseudopotential by
$S({\it\varphi})$
, where
${\it\varphi}$
is the electrostatic potential of the solitary wave, and derivatives of
$S({\it\varphi})$
with respect to
${\it\varphi}$
by primes,
$S^{\prime \prime \prime }(0)$
is related to the coefficient
$B$
of the quadratic nonlinearity in the KdV equation (Baluku et al.
Reference Baluku, Hellberg and Verheest2010),
$S^{\prime \prime \prime \prime }(0)$
to the coefficient
$C$
of the cubic nonlinearity in the mKdV equation, and so on, if there is need to go to higher orders in the KdV hierarchy. Expressions for
$B$
and
$C$
will be determined below. Phrased differently,
$S(0)=S^{\prime }(0)=0$
ensures that the origin or undisturbed condition
${\it\varphi}=0$
is a double root and one obtains a solitary wave. Equating in addition
$B$
or
$S^{\prime \prime \prime }(0)$
to zero amounts to having a triple root at the origin, and annulling on top of that also
$C$
or
$S^{\prime \prime \prime \prime }(0)$
means a quadruple root. It is clear at this stage that one needs a sufficient flexibility in the number of adjustable parameters to make that possible.
All this is relevant for the polarity of the acoustic modes in terms of the sign of
${\it\varphi}$
. When the plasma model allows changes in polarities for suitable values of the compositional parameters (Das Reference Das1975; Das & Tagare Reference Das and Tagare1975; Tagare Reference Tagare1986), other fundamental questions arise. One of these is whether the plasma admits modes of both polarities for the same set of compositional parameters (Nishihara & Tajiri Reference Nishihara and Tajiri1981). This cannot be described by reductive perturbation analysis, but requires a pseudopotential Sagdeev approach, but even then one can only realize one polarity at a time, given the initial infinitesimal perturbations from
${\it\varphi}=0$
.
Nevertheless, aspects of such transitions in polarity can be obtained from the reductive perturbation analysis, which has the advantage of being very algorithmic and can usually tackle more complicated plasma compositions than the Sagdeev pseudopotential method, but at the price of allowing only weaker nonlinearities. In particular, changes in polarity are possible when the coefficient of the nonlinearities in the evolution equation such as KdV or mKdV vanish for appropriate compositional parameters, and one has to rearrange the reductive perturbation analysis (Das Reference Das1975; Das & Tagare Reference Das and Tagare1975; Watanabe Reference Watanabe1984; Tagare Reference Tagare1986).
This has been documented in vastly more papers than can possibly be cited here, so that the references quoted are limited to selected ones, without any hope of being exhaustive. It is not the aim of the present paper to add yet another plasma model where such changes can occur, but to try to narrow down the field by showing that one cannot easily find supercritical densities, by establishing results for a class of plasma models. By ‘supercritical’ I mean that the nonlinear terms in both the KdV and mKdV equations vanish simultaneously for the same set of compositional parameters (Tagare Reference Tagare1986; Verheest Reference Verheest1988).
Up to now, as far as can be ascertained, having a quadruple root in the origin
${\it\varphi}=0$
has only been advanced for hypothetical plasma compositions, without working that out for a specific model and indicating that it can indeed occur, or by forgetting that one needs to find parameters such that
$B=C=0$
or equivalently
$S(0)=S^{\prime }(0)=S^{\prime \prime }(0)=S^{\prime \prime \prime }(0)=S^{\prime \prime \prime \prime }(0)=0$
. A common mistake here is to assume that
$C=0$
, without checking that
$B=0$
for those parameters, as happened in Das & Sarma (Reference Das and Sarma1998). Very recently, however, it was shown that
$B=C=0$
can occur in a four-component plasma with either four fluid adiabatic species, or with two fluid adiabatic cooler species and two-temperature Boltzmann electrons, but only for the slow ion-acoustic modes (Olivier, Maharaj & Bharuthram Reference Olivier, Maharaj and Bharuthram2015). This has prompted revisiting an older proof that quadruple roots of
$S({\it\varphi})$
in
${\it\varphi}=0$
(or the equivalent reductive perturbation requirement that
$B=C=0$
) could not occur in plasmas with a number of adiabatic fluid ion species in the presence of Boltzmann electrons (Verheest Reference Verheest1988). This general proof was valid for the fast ion-acoustic mode, where all the fluid ion thermal velocities are smaller than the acoustic speed.
In the present paper I will therefore generalize this proof to include other polytropic pressure–density relations for the cooler ions, which even may include cool fluid electrons. At the same time, it admits for the hot electrons more recent generalizations of the Boltzmann distributions, with the inclusion of non-thermal effects, including the Cairns (Cairns et al. Reference Cairns, Mamun, Bingham, Boström, Dendy, Nairn and Shukla1995), kappa (Vasyliunas Reference Vasyliunas1968; Hellberg et al. Reference Hellberg, Mace, Baluku, Kourakis and Saini2009) or Tsallis (Tsallis Reference Tsallis1988; Lima, Silva & Santos Reference Lima, Silva and Santos2000) distributions. As shown in the next section, while the generalization of the older proof is fairly straightforward, it is only for the fast acoustic mode that definite results can be obtained. Unfortunately, there seems to be no equivalent statement for the slow modes, so that these have to be studied on a case-by-case basis, which for a model with more than three species is far from straightforward, given the complicated parameter ranges to be considered.
To clarify the use of ‘fast mode’ in this paper, I recall that in plasmas with
$N$
different constituents, there are at most
$N-1$
acoustic regimes, one in each of the intervals between the thermal speeds when all different, less if some intervals are collapsed, as in the case of two strictly cold species or of two inertialess hot species (equivalent to having infinite thermal velocities). The largest of these acoustic velocities is called here the fast mode, for simplicity.
In particular, one may have two acoustic branches for a plasma with three components, so that the fast mode is easily found upon comparing the two acoustic speeds. Suppose now that there are two ion species at different temperatures in the presence of hot electrons, then the slow mode has a velocity between the two ion thermal speeds, whereas the fast mode has a velocity between the warmer ion and the electron thermal speeds. Both are traditionally referred to as ion-acoustic in nature. To the contrary, in a model with one kind of fluid ions and two-temperature electrons, the slow mode has a velocity between the ion and the cooler electron thermal speeds, whereas the fast mode has a velocity between the two electron thermal speeds. In this case, the usual parlance is different, the slow mode in such plasmas is often called the ion-acoustic mode, whereas the fast mode is the electron-acoustic one (Watanabe & Taniuti Reference Watanabe and Taniuti1977; Buti Reference Buti1980; Nishihara & Tajiri Reference Nishihara and Tajiri1981; Verheest et al. Reference Verheest, Cattaert, Hellberg and Mace2006).
2. Reductive perturbation formalism and results
2.1. Model equations and basic formalism
For clarity, I will follow the line of thought of an earlier paper (Verheest Reference Verheest1988), which was restricted to adiabatic multi-ion plasmas in the presence of Boltzmann electrons. Here, general polytropic ion pressures are allowed, with indices
${\it\gamma}_{{\it\alpha}}$
that can vary from one ion species to the next, including in particular the traditional isothermal (
${\it\gamma}_{{\it\alpha}}=1$
, but inertial) and adiabatic pressure–density (
${\it\gamma}_{{\it\alpha}}=3$
) relations. Not all ions have to be positive; there can also be one or more negative ion species. In addition, the hot electrons are described by a non-thermal Cairns distribution (Cairns et al.
Reference Cairns, Mamun, Bingham, Boström, Dendy, Nairn and Shukla1995), having a normalized density
$$\begin{eqnarray}n_{e}=(1-{\it\beta}{\it\varphi}+{\it\beta}{\it\varphi}^{2})\exp [{\it\varphi}].\end{eqnarray}$$
This includes the Boltzmann case when the non-thermality parameter
${\it\beta}=0$
. Other non-thermal electron models will be discussed below.
Thus, the normalized basic equations are
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial {\it\rho}_{{\it\alpha}}}{\partial t}+\frac{\partial }{\partial x}({\it\rho}_{{\it\alpha}}u_{{\it\alpha}})=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial u_{{\it\alpha}}}{\partial t}+u_{{\it\alpha}}\frac{\partial u_{{\it\alpha}}}{\partial x}+\frac{1}{{\it\rho}_{{\it\alpha}}}\,\frac{\partial p_{{\it\alpha}}}{\partial x}+z_{{\it\alpha}}\frac{\partial {\it\varphi}}{\partial x}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial p_{{\it\alpha}}}{\partial t}+u_{{\it\alpha}}\frac{\partial p_{{\it\alpha}}}{\partial x}+{\it\gamma}_{{\it\alpha}}p_{{\it\alpha}}\,\frac{\partial u_{{\it\alpha}}}{\partial x}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial ^{2}{\it\varphi}}{\partial x^{2}}+\mathop{\sum }_{{\it\alpha}}z_{{\it\alpha}}{\it\rho}_{{\it\alpha}}-(1-{\it\beta}{\it\varphi}+{\it\beta}{\it\varphi}^{2})\exp [{\it\varphi}]=0. & \displaystyle\end{eqnarray}$$
${\it\alpha}$
. The expansion of the dependent variables is slightly different from Verheest (Reference Verheest1988): 
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle {\it\rho}_{{\it\alpha}}={\it\rho}_{{\it\alpha}0}+{\it\varepsilon}{\it\rho}_{{\it\alpha}1}+{\it\varepsilon}^{2}{\it\rho}_{{\it\alpha}2}+{\it\varepsilon}^{3}{\it\rho}_{{\it\alpha}3}+\cdots \,,\\ \displaystyle u_{{\it\alpha}}={\it\varepsilon}u_{{\it\alpha}1}+{\it\varepsilon}^{2}u_{{\it\alpha}2}+{\it\varepsilon}^{3}u_{{\it\alpha}3}+\cdots \,,\\ \displaystyle p_{{\it\alpha}}=p_{{\it\alpha}0}+{\it\varepsilon}p_{{\it\alpha}1}+{\it\varepsilon}^{2}p_{{\it\alpha}2}+{\it\varepsilon}^{3}p_{{\it\alpha}3}+\cdots \,,\\ {\it\varphi}={\it\varepsilon}{\it\varphi}_{1}+{\it\varepsilon}^{2}{\it\varphi}_{2}+{\it\varepsilon}^{3}{\it\varphi}_{3}+\cdots \,.\end{array}\right\} & & \displaystyle\end{eqnarray}$$
The stretching, however, will be taken as
$$\begin{eqnarray}{\it\xi}={\it\varepsilon}(x-Vt),\quad {\it\tau}={\it\varepsilon}^{3}t,\end{eqnarray}$$
where
$V$
is the linear acoustic phase velocity. This has the advantage of having one stretching throughout, instead of starting from the more usual
${\it\xi}={\it\varepsilon}^{1/2}(x-Vt)$
and
${\it\tau}={\it\varepsilon}^{3/2}t$
for the standard case and then changing at critical ion densities to (2.7) (Das Reference Das1975; Das & Tagare Reference Das and Tagare1975; Watanabe Reference Watanabe1984; Tagare Reference Tagare1986). The present stretching implies, as will be seen, that in the generic case all terms in (2.6) with odd subscripts will be zero, leaving only the even subscripted variables. This is mathematically completely equivalent to the common approach to derivations of KdV-like evolution equations.
Applying (2.6) and (2.7) to (2.2)–(2.4) yields the linear contributions, after integration with respect to
${\it\xi}$
and zero boundary conditions for
${\it\xi}\rightarrow \pm \infty$
, as befitting solitary structures:
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle {\it\rho}_{{\it\alpha}1}=\frac{z_{{\it\alpha}}{\it\rho}_{{\it\alpha}0}}{V^{2}-{\it\sigma}_{{\it\alpha}}}\,{\it\varphi}_{1},\\ \displaystyle u_{{\it\alpha}1}=\frac{z_{{\it\alpha}}V}{V^{2}-{\it\sigma}_{{\it\alpha}}}\,{\it\varphi}_{1},\\ \displaystyle p_{{\it\alpha}1}=\frac{z_{{\it\alpha}}{\it\sigma}_{{\it\alpha}}{\it\rho}_{{\it\alpha}0}}{V^{2}-{\it\sigma}_{{\it\alpha}}}\,{\it\varphi}_{1},\end{array}\right\} & & \displaystyle\end{eqnarray}$$
where
${\it\sigma}_{{\it\alpha}}={\it\gamma}_{{\it\alpha}}p_{{\it\alpha}0}/{\it\rho}_{{\it\alpha}0}$
has been introduced as a measure of the thermal speed squared of the cooler species.
Poisson’s equation (2.5) yields to zero order the equilibrium charge neutrality,
$$\begin{eqnarray}\mathop{\sum }_{{\it\alpha}}z_{{\it\alpha}}{\it\rho}_{{\it\alpha}0}-1=0,\end{eqnarray}$$
and, after substitution of (2.8), to first order the linear dispersion law,
$$\begin{eqnarray}D:=\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{2}{\it\rho}_{{\it\alpha}0}}{V^{2}-{\it\sigma}_{{\it\alpha}}}-(1-{\it\beta})=0.\end{eqnarray}$$
The solutions
$V$
of
$D=0$
determine the linear acoustic modes in the plasma composition under consideration.
When there are only fluid ions, besides the inertialess electrons, we will restrict ourselves, in what follows, to the fast (ion-acoustic) mode, characterized by
$V^{2}>{\it\sigma}_{{\it\alpha}}$
for all ion species. When there are also cool fluid electrons, the condition
$V^{2}>{\it\sigma}_{{\it\alpha}}$
selects the electron-acoustic mode (Watanabe & Taniuti Reference Watanabe and Taniuti1977; Buti Reference Buti1980; Nishihara & Tajiri Reference Nishihara and Tajiri1981; Verheest et al.
Reference Verheest, Cattaert, Hellberg and Mace2006), as mentioned already at the end of the Introduction, where
$V$
lies between the cool and the hot electron thermal velocities.
There are, of course, one or more slow modes included in the formalism, as soon as there are three or more plasma constituents, but these are not the focus of the present analysis. Slow modes require that
$V$
lies between some of the fluid thermal velocities (Verheest et al.
Reference Verheest, Cattaert, Lakhina and Singh2004).
Leaving out lengthy intermediate expressions (Verheest Reference Verheest2000), one obtains to the next higher order from (2.5) that
$$\begin{eqnarray}D{\it\varphi}_{2}+B{\it\varphi}_{1}^{2}=0,\end{eqnarray}$$
where
$$\begin{eqnarray}B:=\frac{1}{2}\left\{\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{3}{\it\rho}_{{\it\alpha}0}[3V^{2}+({\it\gamma}_{{\it\alpha}}-2){\it\sigma}_{{\it\alpha}}]}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{3}}-1\right\}.\end{eqnarray}$$
Assuming that
$V$
annuls
$D$
, we have reached a bifurcation: the plasma composition is either generic (
$B\not =0$
) or critical (
$B=0$
). In the generic case, the only possibility is that
${\it\varphi}_{1}=0$
, which leads again to
$D=0$
, now for
${\it\varphi}_{2}\not =0$
, and to lowest non-zero linear level we obtain (2.8), but expressed with subscripts 2.
2.2. Generic plasma composition
Going through the motions for the generic case when
$B\not =0$
, and collecting terms to orders four and five, one notices that the odd-numbered variables in the expansion decouple from the even-numbered ones, so that without loss of generality one can put
${\it\varphi}_{3}={\it\varphi}_{5}=\cdots =0$
. To the relevant higher order, the well-known KdV equation is derived, here written as
$$\begin{eqnarray}A\frac{\partial {\it\varphi}_{2}}{\partial {\it\tau}}+B{\it\varphi}_{2}\,\frac{\partial {\it\varphi}_{2}}{\partial {\it\xi}}+\frac{1}{2}\,\frac{\partial ^{3}{\it\varphi}_{2}}{\partial x^{3}}=0,\end{eqnarray}$$
with
$$\begin{eqnarray}A:=\mathop{\sum }_{{\it\alpha}}\frac{Vz_{{\it\alpha}}^{2}{\it\rho}_{{\it\alpha}0}}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{2}}>0.\end{eqnarray}$$
Clearly
$A$
is positive, but
$B$
might vanish, given appropriate relations between the compositional parameters.
2.3. Critical plasma composition
Now it is assumed that the plasma composition is special enough to have
$B=0$
, which means that one keeps
${\it\varphi}_{1}$
non-zero. Thus, the algebra leads to the next higher order to a modified KdV (mKdV) equation, having a cubic nonlinearity,
$$\begin{eqnarray}A\frac{\partial {\it\varphi}_{1}}{\partial {\it\tau}}+C{\it\varphi}_{1}^{2}\,\frac{\partial {\it\varphi}_{1}}{\partial {\it\xi}}+\frac{1}{2}\,\frac{\partial ^{3}{\it\varphi}_{1}}{\partial x^{3}}=0,\end{eqnarray}$$
with
$$\begin{eqnarray}C:=\frac{1}{4}\left\{\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{4}{\it\rho}_{{\it\alpha}0}P_{{\it\alpha}}}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{5}}-(1+3{\it\beta})\right\}.\end{eqnarray}$$
For notational brevity I have defined a bi-quadratic polynomial in
$V$
,
$$\begin{eqnarray}P_{{\it\alpha}}:=15V^{4}+V^{2}{\it\sigma}_{{\it\alpha}}({\it\gamma}_{{\it\alpha}}^{2}+13{\it\gamma}_{{\it\alpha}}-18)+{\it\sigma}_{{\it\alpha}}^{2}({\it\gamma}_{{\it\alpha}}-2)(2{\it\gamma}_{{\it\alpha}}-3).\end{eqnarray}$$
An interesting question is whether
$C$
could vanish, under the combined assumptions that there is charge neutrality in equilibrium (2.9),
$V$
annuls the dispersion law (2.10) and
$B=0$
. If that were indeed the case, one would have to go to yet another stretching, leading to a higher-order KdV equation, with a quartic nonlinearity. However, all such evolution equations are no longer completely integrable, meaning, inter alia, that there is no infinite chain of conserved densities and the characteristic interaction properties of true solitons is lost. Moreover, it requires a lot of flexibility in the choice of parameters, which is rarely available (Verheest, Hereman & Malfliet Reference Verheest, Hereman and Malfliet1999).
However, if one can find compositional parameters and soliton velocities such that
$C=0$
, but under conditions that
$B\not =0$
, then clearly the KdV equation (2.13) holds and higher-order corrections can only be small and physically uninteresting, lest the iterative nature of reductive perturbation theory and the convergence of its concomitant expansions is violated.
2.4. Higher-order criticalities?
I now adapt an earlier proof (Verheest Reference Verheest1988) that for the fast mode (meaning
$V^{2}>{\it\sigma}_{{\it\alpha}}$
for all
${\it\alpha}$
)
$C$
cannot vanish under (2.9), (2.10) and
$B=0$
together, based on a Cauchy–Schwartz inequality (Gradshteyn & Ryzhik Reference Gradshteyn and Ryzhik2000). Starting from
$B=0$
, and with the help of (2.10), it is deduced that
$$\begin{eqnarray}\displaystyle 1 & = & \displaystyle \left(\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{3}{\it\rho}_{{\it\alpha}0}[3V^{2}+({\it\gamma}_{{\it\alpha}}-2){\it\sigma}_{{\it\alpha}}]}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{3}}\right)^{2}\nonumber\\ \displaystyle & = & \displaystyle \left(\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{2}{\it\rho}_{{\it\alpha}0}^{1/2}[3V^{2}+({\it\gamma}_{{\it\alpha}}-2){\it\sigma}_{{\it\alpha}}]}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{5/2}}\frac{z_{{\it\alpha}}{\it\rho}_{{\it\alpha}0}^{1/2}}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{1/2}}\right)^{2}\nonumber\\ \displaystyle & {<} & \displaystyle \mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{4}{\it\rho}_{{\it\alpha}0}[3V^{2}+({\it\gamma}_{{\it\alpha}}-2){\it\sigma}_{{\it\alpha}}]^{2}}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{5}}\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{2}{\it\rho}_{{\it\alpha}0}}{V^{2}-{\it\sigma}_{{\it\alpha}}}\nonumber\\ \displaystyle & = & \displaystyle (1-{\it\beta})\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{4}{\it\rho}_{{\it\alpha}0}[3V^{2}+({\it\gamma}_{{\it\alpha}}-2){\it\sigma}_{{\it\alpha}}]^{2}}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{5}}.\end{eqnarray}$$
The proof of the Cauchy–Schwarz inequality is immediate for
$N$
-dimensional vectors, as
$(\boldsymbol{a}\boldsymbol{\cdot }\boldsymbol{b})^{2}\leqslant \Vert \boldsymbol{a}\Vert ^{2}\Vert \boldsymbol{b}\Vert ^{2}$
under the norm
$\Vert \boldsymbol{a}\Vert ^{2}=\sum _{i}a_{i}^{2}$
and the scalar product
$\boldsymbol{a}\boldsymbol{\cdot }\boldsymbol{b}=\sum _{i}a_{i}b_{i}$
. The inequality in the third step of (2.18) is strict, since, in the parlance of
$N$
-dimensional vectors,
$\boldsymbol{a}$
and
$\boldsymbol{b}$
are here not parallel, as the expressions in (2.18) clearly indicate for two or more fluid species.
Next, one easily proves, again for
$V^{2}>{\it\sigma}_{{\it\alpha}}$
, that
$$\begin{eqnarray}P_{{\it\alpha}}-{\textstyle \frac{5}{3}}[3V^{2}+({\it\gamma}_{{\it\alpha}}-2){\it\sigma}_{{\it\alpha}}]^{2}\geqslant {\textstyle \frac{4}{3}}\,{\it\sigma}_{{\it\alpha}}^{2}({\it\gamma}_{{\it\alpha}}+1)^{2}\geqslant 0.\end{eqnarray}$$
Consequently, with the help of (2.18),
$$\begin{eqnarray}\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{4}{\it\rho}_{{\it\alpha}0}P_{{\it\alpha}}}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{5}}\geqslant \frac{5}{3}\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{4}{\it\rho}_{{\it\alpha}0}[3V^{2}+({\it\gamma}_{{\it\alpha}}-2){\it\sigma}_{{\it\alpha}}]^{2}}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{5}}>\frac{5}{3}\frac{1}{1-{\it\beta}}.\end{eqnarray}$$
The equality signs in (2.19) and (2.20) only hold when all fluid species are strictly cold (
${\it\sigma}_{{\it\alpha}}=0$
). We conclude from this that
$$\begin{eqnarray}C>\frac{(3{\it\beta}-1)^{2}+1}{12(1-{\it\beta})}>0.\end{eqnarray}$$
For Boltzmann hot electrons (
${\it\beta}=0$
) this reduces to
$C>1/6$
, which is rather remarkable, because such a lower bound was already obtained by Verheest (Reference Verheest1988), but then for fluid species that were adiabatic (
${\it\gamma}_{{\it\alpha}}=3$
), whereas now all
${\it\gamma}_{{\it\alpha}}$
can, in principle, be arbitrary, at least from a mathematical point of view. The minimum bound for
$C$
might be increased by using the intermediate expression in (2.19) for those species having
${\it\sigma}_{{\it\alpha}}\not =0$
, but that is difficult to quantify, given the wide range of
$V$
,
$z_{{\it\alpha}}$
and
${\it\rho}_{{\it\alpha}0}$
that might exist.
The minimum of
$C$
starts at
${\it\beta}=0$
from
$1/6$
, then drops to 0.118 at
${\it\beta}=0.255$
to increase monotonically for larger
${\it\beta}$
, reaching
$1/6$
again for
${\it\beta}=4/9$
, up to 0.294 for
${\it\beta}=4/7$
. The latter
${\it\beta}$
limit is where the Cairns distribution in phase space, from which (2.1) is derived, starts to behave non-monotonically and might become beam-unstable (Verheest & Pillay Reference Verheest and Pillay2008).
2.5. Kappa electrons
We now look at an alternative electron description, where the common Boltzmann or Cairns description is replaced by a kappa non-thermal distribution (Vasyliunas Reference Vasyliunas1968; Hellberg et al. Reference Hellberg, Mace, Baluku, Kourakis and Saini2009), with
$$\begin{eqnarray}n_{e}=\left(1-\frac{{\it\varphi}}{{\it\kappa}-3/2}\right)^{-{\it\kappa}+1/2},\end{eqnarray}$$
requiring
${\it\kappa}>3/2$
. This has repercussions on some of the expressions computed above, in that
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle D:=\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{2}{\it\rho}_{{\it\alpha}0}}{V^{2}-{\it\sigma}_{{\it\alpha}}}-\frac{2{\it\kappa}-1}{2{\it\kappa}-3}=0,\\ \displaystyle B:=\frac{1}{2}\left\{\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{3}{\it\rho}_{{\it\alpha}0}[3V^{2}+({\it\gamma}_{{\it\alpha}}-2){\it\sigma}_{{\it\alpha}}]}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{3}}-\frac{4{\it\kappa}^{2}-1}{(2{\it\kappa}-3)^{2}}\right\},\\ \displaystyle C:=\frac{1}{4}\left\{\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{4}{\it\rho}_{{\it\alpha}0}P_{{\it\alpha}}}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{5}}-\frac{(4{\it\kappa}^{2}-1)(2{\it\kappa}+3)}{(2{\it\kappa}-3)^{3}}\right\}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$
The analysis developed in (2.18) rests essentially on the ion contributions, but the kappa electrons modify this to
$$\begin{eqnarray}\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{4}{\it\rho}_{{\it\alpha}0}[3V^{2}+({\it\gamma}_{{\it\alpha}}-2){\it\sigma}_{{\it\alpha}}]^{2}}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{5}}>\frac{(4{\it\kappa}^{2}-1)(2{\it\kappa}+1)}{(2{\it\kappa}-3)^{3}},\end{eqnarray}$$
which changes the last step in (2.20) to yield
$$\begin{eqnarray}C>\frac{(4{\it\kappa}^{2}-1)({\it\kappa}-1)}{3(2{\it\kappa}-3)^{3}}\geqslant \frac{1}{6}.\end{eqnarray}$$
The equality sign is obtained in the Maxwellian limit
${\it\kappa}\rightarrow +\infty$
. Small
${\it\kappa}$
raises the lower limit for
$C$
above
$1/6$
so that
$C$
remains positive.
2.6. Tsallis electrons
Closely related to the kappa distribution is the non-extensive Tsallis distribution (Tsallis Reference Tsallis1988; Lima et al. Reference Lima, Silva and Santos2000), which leads to the electron density
$$\begin{eqnarray}n_{e}=[1+(q-1){\it\varphi}]^{(q+1)/2(q-1)}.\end{eqnarray}$$
In line with looking for suitable descriptions of superthermal effects, the superextensive part of the underlying phase-space distribution is needed, which requires that
$1/3<q\leqslant 1$
(Lima et al.
Reference Lima, Silva and Santos2000; Saberian & Esfandyari-Kalejahi Reference Saberian and Esfandyari-Kalejahi2013; Verheest Reference Verheest2013). The lower limit,
$1/3<q$
, ensures that the energy computed from the original phase space distribution does not diverge and thus that temperatures can be defined (here, used in the normalization), although this is often overlooked in applications.
Now the modifications are
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle D:=\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{2}{\it\rho}_{{\it\alpha}0}}{V^{2}-{\it\sigma}_{{\it\alpha}}}-\frac{1}{2}(q+1)=0,\\ \displaystyle B:=\frac{1}{2}\left\{\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{3}{\it\rho}_{{\it\alpha}0}[3V^{2}+({\it\gamma}_{{\it\alpha}}-2){\it\sigma}_{{\it\alpha}}]}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{3}}-\frac{1}{4}(q+1)(3-q)\right\},\\ \displaystyle C:=\frac{1}{4}\left\{\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{4}{\it\rho}_{{\it\alpha}0}P_{{\it\alpha}}}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{5}}-\frac{1}{8}(q+1)(3-q)(5-3q)\right\}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$
Hence, the Tsallis electrons modify the analysis in (2.18) to
$$\begin{eqnarray}\mathop{\sum }_{{\it\alpha}}\frac{z_{{\it\alpha}}^{4}{\it\rho}_{{\it\alpha}0}[3V^{2}+({\it\gamma}_{{\it\alpha}}-2){\it\sigma}_{{\it\alpha}}]^{2}}{(V^{2}-{\it\sigma}_{{\it\alpha}})^{5}}>\frac{1}{8}(q+1)(3-q)^{2},\end{eqnarray}$$
which yields
$$\begin{eqnarray}C>{\textstyle \frac{1}{24}}\,q(1+q)(3-q)>0.\end{eqnarray}$$
In the Maxwellian limit
$q\rightarrow 1$
one recovers the earlier result that
$C>1/6$
, and for the lower limit
$q\rightarrow 1/3$
(
$q=1/3$
not included)
$C$
starts at
$4/81$
, and increases monotonically, almost linearly, but not quite, with
$q$
.
3. Conclusions
In this paper I have generalized an earlier proof (Verheest Reference Verheest1988) that, for the fast acoustic mode in a plasma with a number of fluid species and one kind of inertialess electrons, there can be critical densities (meaning that the coefficient
$B$
of the quadratic nonlinearity in a KdV equation vanishes) but that no supercritical densities exist (in the sense that
$B=C=0$
simultaneously, where
$C$
is the coefficient of the cubic nonlinearity in the mKdV equation that replaces the KdV equation for
$B=0$
). These conclusions also hold for those models admitting a large-amplitude Sagdeev pseudopotential description, in terms of
$S({\it\varphi})$
, since there is an equivalence
$B=0~\Leftrightarrow ~S^{\prime \prime \prime }(0)=0$
and
$C=0~\Leftrightarrow ~S^{\prime \prime \prime \prime }(0)=0$
.
The cool fluid species can have various polytropic pressure–density relations, including adiabatic (
${\it\gamma}_{{\it\alpha}}=3$
) and isothermal (
${\it\gamma}_{{\it\alpha}}=1$
). If there is only one electron species, the hot inertialess one, the fluid ions can include one or more species of negative ions, and the fast mode is of an ion-acoustic nature. If there is an additional cool electron species, with its inertia taken into account, this is equivalent to a fluid negative-ion species, but now the fast mode refers to what is usually called the electron-acoustic mode in a two-temperature plasma. The hot inertialess electrons are modelled by various extensions for non-thermal effects, such as the Cairns, kappa or Tsallis distributions, all including the standard Boltzmann description as a special case. Together, in this way quite a number of plasma models are covered.
Unfortunately, there seems to be no equivalent generic statement for the slow modes, so that these have to be studied on a case-by-case basis, which for a model with more than three species is far from straightforward, given the parameter ranges to be taken into account. There seem to be very recent indications that for the slow modes credible supercritical density regimes might occur (Olivier et al. Reference Olivier, Maharaj and Bharuthram2015), in a specific four-component plasma model, an avenue that certainly needs further exploration.
A final remark is that having two inertialess or two inertial and hot electron species, besides the ions, does not lead to lower bounds on
$C$
that are unambiguously positive. This is again a case where detailed but model-specific investigations are needed to see whether supercritical densities are possible.
Acknowledgement
Interesting discussions with M. A. Hellberg are gratefully acknowledged.
