A pseudopotential analysis is presented for the propagation of nonlinear periodic dust-acoustic waves in a dusty plasma comprising cold negative dust, Boltzmann electrons, and Boltzmann or Cairns nonthermal positive ions, extending thus earlier treatments for ion-acoustic waves in electron–proton plasmas. The dusty plasma model where both electrons and ions are Boltzmann does not admit solitons, but works for nonlinear periodic waves. For consistency in the periodic case, two properties are required: conservation per cycle of species densities and that for very small amplitudes the waves resemble linear waves. The first property has to be imposed through a global perturbation of the undisturbed equilibrium, whereas the second property follows naturally from the formalism. After obtaining the general analytical methodology, a numerical analysis is discussed and illustrated with graphs for the electrostatic potential profile, the Sagdeev pseudopotential, the wave electric field, and the three different species densities, first for the Boltzmann and thereafter for the Cairns ions.

## I. INTRODUCTION

The interest in dusty plasmas comes from several sides, among which are important results from space observations by the Voyager missions of the Jovian and Cronian dust rings. These have drawn attention to the interaction between dust grains and plasmas. Notable examples are the radial spokes in the B ring^{1} and the braiding of the F ring^{2} of Saturn. Here phenomena have been observed that could not be explained by purely gravitational properties but required the presence of an electrically charged dust component. In this context, the physics of dusty plasmas concerns charged micrometer- to nanosized dust grains, which, if sufficiently numerous, can be treated as an additional fluid species that introduces a specific low-frequency component to a multispecies plasma.

The initial model for a dust-acoustic wave (DAW) is a bold transposition by Rao *et al.*^{3} to the physics of dusty plasmas of the concept of an ion-acoustic wave,^{4} first for linear and then solitary waves. To sustain an acoustic wave, at least two elements are needed: an inertial and a hot thermal species, in the earliest ion-acoustic models cold protons plus Boltzmann electrons. Because the charged dust grains are so much heavier than protons and electrons the obvious first model for DAWs consisted of cold negatively charged grains in the presence of proton and electron Boltzmann species, the inertia of which could be neglected compared to the dust grains.

The theoretical description is simple enough, consisting of the cold fluid dust continuity and momentum equations, and for the inertialess hot electrons and protons Boltzmann distributions.^{5} The species densities are coupled by Poisson's equation for the electrostatic potential. This simple dust-acoustic dispersion model has been experimentally verified. Earlier discussions of collective effects in microplasmas by James and Vermeulen^{6} using many-fluid models as in Verheest^{7} were not addressing dusty plasmas as now understood.

All this quickly expanded into a whole new domain of the dusty plasma wave literature and has been summarized in various books, notably by Verheest^{8} and Shukla and Mamun,^{9} containing many relevant references. Coming in particular to the present paper, we will be guided by some properties known from the study of nonlinear solitary waves in dusty plasmas. The original model for the DAWs (cold negative dust and Boltzmann electrons and protons) claimed to admit negative and positive polarity linear and nonlinear solitary DAWs,^{3} although this has not been discussed in sufficient detail.

The parameter range of the negative polarity waves is limited by the infinite dust compression, the positive polarity waves being restricted by encountering a double layer. A worked-out example of nonthermal distributions is based on the so-called Cairns distribution,^{10} but kappa^{11,12} or later Tsallis distributions^{13,14} might presumably also do the trick.

There is an important distinction between nonlinear solitary vs periodic waves and the perturbations of the undisturbed equilibrium needed to generate them. As the soliton profile has a finite area under the curve for the electrostatic potential in a co-moving frame, the energy and related quantities are finite, thus the perturbation necessary to excite the wave can be a local one, confined to a small area of the spatial domain. For nonlinear periodic waves, however, the species densities should be conserved over one cycle, because otherwise an excess or loss of density per cycle becomes, over an infinity of cycles, an infinite excess or loss, which is physically unacceptable. This density conservation requires a global perturbation, even though the relevant changes in the maximum and minimum amplitudes might be small.

In this respect, most of the extant literature on nonlinear periodic waves is not correct and does not conserve the species densities. We will only cite here the deficient papers concerning themselves with DAWs.^{15–20} It is only in four recent papers that the picture has been corrected,^{21–24} all for simple proton–electron plasmas, not yet for DAWs. Another desirable phenomenon is that the small amplitude limit of the nonlinear periodic waves resembles very much what one would obtain from a linearized description in terms of sine or cosine modes. That seems to obtain from the mathematical and numerical analysis without having to be imposed from the outset.

For the DAWs, we will follow the model discussed by Verheest and Pillay^{25} for solitons in plasmas with negative dust, where the electrons are Boltzmann but the protons have a Cairns nonthermal density distribution, which allowed positive polarity solitons. Using this composition for obtaining nonlinear periodic DAWS needs a careful attention to the boundary conditions and to the introduction of restrictions leading to density conservation. Interchanging the model to positive dust, amounts essentially to a mathematical interchange of the polarities in the formalism but engenders no specific difficulties.^{26}

In particular and in contrast with our earlier efforts for electron-ion plasmas, there are now three densities to be separately conserved: for the cold dust grains, Boltzmann electrons, and nonthermal Cairns protons. This will impact on the global initial perturbation, together with the occurrence of several additional compositional parameters, rendering the numerical evaluation rather more complicated. As seen in our earlier papers, it is relatively easy to establish existence domains for periodic nonlinear waves, as long as the wave velocity and obliquity parameters are adequately chosen inside the appropriate domains.

Periodic electric field signals have been reported in many regions of Earth's magnetosphere (see, for example, Pickett *et al.*,^{27} Rufai,^{28} Singh *et al.*^{29} and references therein), and therefore, we will specifically address also the electric field profiles in the numerical discussions. In space observations, electric field profiles are easier to record than electrostatic potentials or species densities.

## II. BASIC SAGDEEV ANALYSIS

We consider the model of a dusty plasma consisting of cold, singly charged negative dust, Boltzmann electrons, and Cairns nonthermal protons.

^{10,25}the normalized proton density $ni$ is given at the macroscopic level by

*f*is the fraction of the negative charge density taken up by the dust relative to the positively charged ions at equilibrium; hence, $(1\u2212f)$ represents the equilibrium electron charged density fraction.

^{25}The model of singly charged negative dust is easily extended to other charge numbers by adapting the normalization.

*V*is the velocity of the nonlinear wave.

*L*(to be defined below) of the periodic structure. This yields

*global*density over one period, written as

^{23}this conservation law is not dependent on the specific species descriptions. The unperturbed equilibrium densities are $ni=1$, $ne=1\u2212f$ and $nd=f$ for all $\xi $ and would give

*L*, $(1\u2212f)L$ and

*fL*, respectively, when integrated over one wavelength

*L*. We thus impose the conservation of species number densities through

Because $d\phi /d\xi =0$ at the minimum and maximum values $\phi =\phi min$ and $\phi =\phi max$, there necessarily is an intermediate value where the tangent $d\phi /d\xi $ reaches a positive maximum $\alpha =d\phi /d\xi |\xi =0>0$. This inflection point has been taken as the origin of the $\xi $-axis, which can be done without loss of generality. Consequently, $d2\phi /d\xi 2|\xi =0=0$. Furthermore, we denote the function values at the inflection point as $ni(0)=\gamma i$, $ne(0)=\gamma e$, $nd(0)=\gamma d$, $ud(0)=0$ and $\phi (0)=0$, to be further elucidated and discussed below. The choice $\phi (0)=0$ then ensures $\phi min<0$ and $\phi max>0$, an essential prerequisite to ensure the possibility of charge density conservation.

For numerical simplicity, we take here $ud(0)=0$ rather than $ud(0)=u0$ as in earlier papers, where the latter choice allowed for the additional conservation of ion flux that then determined $u0$ uniquely. However, $u0\u22600$ amounts to a Doppler shift on *V*, and but for that change the mathematics are essentially the same. For the multispecies model discussed here, there are already enough additional parameters to consider, and we have thus preferred to leave $u0$ out of the analytics. Should the explicit conservation of dust flux be desired, the analytical modifications are straightforward to implement.^{23}

*f*falls away and the conservation of dust density becomes

We have investigated what happens if one were to assume that $\gamma e=\gamma d$ (so that also $\gamma i$ became equal), but the numerical computation showed only one of the species densities to be conserved. There are therefore two distinct values $\gamma $ to consider, $\gamma e$ and $\gamma d$. In this way, (13) and (17) can be worked out for the conservation of the electron and dust densities per cycle, respectively. According to the theoretical analysis, (8) and (18) should then lead to Eq. (16) being obeyed and the ion density also conserved. This will be used later as a check on the consistency of the numerics, when Eq. (16) is computed separately.

^{30}here

*f*, $\beta $, and $\tau $. The structure parameters, $\alpha $ (inclination at inflection point, later measure for the amplitude) and

*V*(velocity), will then determine the properties of the periodic nonlinear structures. For ease of subsequent notation, we introduce

*V*is sufficiently increased, the pair closest to the origin becomes a double root. $S(\phi )$ does not have enough flexibility for positive roots beyond that and the range of positive roots thus ends at the double layer. The double layer equations are derived from Eq. (21) and written for $\phi =\phi dl$ as

*V*as a function of $\alpha $ yields an upper curve, whereby $\phi dl$ serves as a kind of dummy variable. Above this curve, there are no longer double layers to be found and hence also no positive roots, so that periodic nonlinear waves cannot exist.

At this stage, we remark that for $\alpha =0$, not only does the amplitude go to zero (constant equilibrium solution), but the Sagdeev pseudopotential recovers the expression used to discuss solitons, but derived under a different set of boundary conditions.^{25} Indeed, for $\alpha =0$, the origin of the reference frame for the Sagdeev pseudopotential becomes an unstable maximum, precisely what is needed to obtain solitons. In that case, we know that negative polarity solitons occur until the dust compression limit is reached. Positive polarity solitons are limited by the occurrence of positive double layers, requiring a nonzero minimal degree of nonthermality of the proton plasma component. The interpretation of this result is that the very early plasma model for DAWs, cold negative dust and Boltzmann electrons and ions, can only give rise to negative polarity solitons.

However, we are investigating nonlinear periodic DAWs and will show in Sec. III that even in the case of Boltzmann ions ( $\beta =0$) such waves are possible, keeping in mind that for periodic waves in the Sagdeev formalism the pseudopotential needs to have at least one positive and one negative root, otherwise species densities cannot be conserved. Perhaps unexpectedly, this is an interesting and rather unusual model, because although positive nonlinear solitary waves cannot be sustained, nonlinear periodic waves can!

We will also treat the case of $\beta \u22600$, but because of the many parameters to be considered and discussed, we will show how to determine the existence regions in parameter space but confine ourselves to a few proof-of-principle worked-out examples. We have indeed seen in our earlier efforts^{23,24} for ion-acoustic waves in electron–ion plasmas that the parametric discussion quickly generates a series of similar looking figures and numerical tables, without really enlarging our physical understanding.

## III. NUMERICAL ANALYSIS

### A. Model with Boltzmann electrons and ions, plus negative dust

To start the numerical analysis, we plot the two limiting curves in ${\alpha ,V}$ space, as shown in Fig. 1. The blue dashed curve starting from $V=0$ reflects the dust density compression limit, whereas the solid red curve starting from the highest *V* for $\alpha =0$ indicates the occurrence of positive double layers. Outside the triangular region no solutions can exist. The highest *V* corresponds to the correct acoustic velocity $cda$, and because $V<cda$ for admissible *V*, the periodic nonlinear DAWs are subsonic, whereas at $\alpha =0$, we are in the soliton regime and those are known to be supersonic. Small changes in the different $\gamma $ away from 1 will affect the existence triangle, but for simplicity we have used $\gamma e=\gamma d=\gamma i=1$ to produce Fig. 1.

The idea is to pick a pair of ${\alpha ,V}$ values inside the existence triangle, for chosen compositional parameters *f* and $\tau $, and then find the corresponding $\gamma e,\gamma d,\gamma i$ to conserve the species densities. For Boltzmann ions, $\beta =0$. We thus compute in Table I for various *f* and $\tau $ combinations values for $\alpha $ and *V* which can produce periodic nonlinear DAWs.

f
. | $\tau $ . | $\alpha $ . | V
. | $\alpha $ . | V
. |
---|---|---|---|---|---|

0.1 | 0.1 | 0.04 | 0.14 | ||

0.1 | 1 | 0.03 | 0.10 | ||

0.1 | 10 | 0.01 | 0.05 | ||

0.5 | 0.1 | 0.21 | 0.33 | 0.10 | 0.47 |

0.5 | 1 | 0.17 | 0.26 | 0.10 | 0.36 |

0.5 | 10 | 0.08 | 0.12 | ||

0.9 | 0.1 | 0.40 | 0.44 | 0.10 | 0.77 |

0.9 | 1 | 0.39 | 0.43 | 0.10 | 0.72 |

0.9 | 10 | 0.22 | 0.25 | 0.10 | 0.41 |

f
. | $\tau $ . | $\alpha $ . | V
. | $\alpha $ . | V
. |
---|---|---|---|---|---|

0.1 | 0.1 | 0.04 | 0.14 | ||

0.1 | 1 | 0.03 | 0.10 | ||

0.1 | 10 | 0.01 | 0.05 | ||

0.5 | 0.1 | 0.21 | 0.33 | 0.10 | 0.47 |

0.5 | 1 | 0.17 | 0.26 | 0.10 | 0.36 |

0.5 | 10 | 0.08 | 0.12 | ||

0.9 | 0.1 | 0.40 | 0.44 | 0.10 | 0.77 |

0.9 | 1 | 0.39 | 0.43 | 0.10 | 0.72 |

0.9 | 10 | 0.22 | 0.25 | 0.10 | 0.41 |

For $f=0.1,0.5,0.9$, there is more and more dust, which cannot be totally absent because without the dust there is no inertial species to sustain the waves, hence $f\u22600$. On the other hand, $f=1$ means that all electrons are accreted to the dust, and a two-component plasma composed of negative cold dust and Boltzmann (or Cairns) ions is, up to a change in polarities and charge signs, mathematically the same as the case of positive cold ions and Boltzmann or Cairns electrons treated in our earlier Sagdeev papers.^{23,24}

Typical values of $\tau =0.1,1,10$ indicate that $Te=10,1,0.1$ times $Ti$, respectively, so increases in $\tau $ are interpreted as decreases in $Te$, given that the yardstick for the temperatures has been chosen as $Ti$.

We know from previous papers^{23,24} that any such pair ${\alpha ,V}$ will generate graphs which look qualitatively similar, only differing quantitatively. There is not really a great need to produce all these graphs; they will not add much to our physical understanding but require some effort to arrive at the correct $\gamma $ factors to ensure density conservation for the three species.

From Table I, we see that at fixed *f*, the existence region (as in Fig. 1) shrinks when $\tau $ is increased. Therefore, we chose smaller parameters for larger $\tau $ in Table I. An increase in $\tau $ means a decrease in both parameters ${\alpha ,V}$, but an increase in $\tau $ signifies a decrease in $Te/Ti$. At fixed $\tau $, we note that the parameters ${\alpha ,V}$ increase with *f*. This is for Boltzmann ions ( $\beta =0$), but we will see later, in Table II, that for Cairns ions (e.g., $\beta =0.5$) the same trends hold, but at higher values for ${\alpha ,V}$, owing to the nonthermality in the ion component.

f
. | $\tau $ . | $\alpha $ . | V
. | $\alpha $ . | V
. |
---|---|---|---|---|---|

0.1 | 0.1 | 0.05 | 0.20 | ||

0.1 | 1 | 0.03 | 0.13 | ||

0.1 | 10 | 0.01 | 0.05 | ||

0.5 | 0.1 | 0.34 | 0.53 | 0.10 | 0.79 |

0.5 | 1 | 0.21 | 0.32 | 0.10 | 0.48 |

0.5 | 10 | 0.08 | 0.13 | ||

0.9 | 0.1 | 0.73 | 0.84 | 0.10 | 1.33 |

0.9 | 1 | 0.55 | 0.65 | 0.10 | 1.10 |

0.9 | 10 | 0.23 | 0.27 | 0.10 | 0.46 |

f
. | $\tau $ . | $\alpha $ . | V
. | $\alpha $ . | V
. |
---|---|---|---|---|---|

0.1 | 0.1 | 0.05 | 0.20 | ||

0.1 | 1 | 0.03 | 0.13 | ||

0.1 | 10 | 0.01 | 0.05 | ||

0.5 | 0.1 | 0.34 | 0.53 | 0.10 | 0.79 |

0.5 | 1 | 0.21 | 0.32 | 0.10 | 0.48 |

0.5 | 10 | 0.08 | 0.13 | ||

0.9 | 0.1 | 0.73 | 0.84 | 0.10 | 1.33 |

0.9 | 1 | 0.55 | 0.65 | 0.10 | 1.10 |

0.9 | 10 | 0.23 | 0.27 | 0.10 | 0.46 |

After this general discussion, we will pick a specific plasma model with the parameters $\beta =0,f=0.5,\tau =1$ and wave values $\alpha =0.15,V=0.26$ within the existence region. The compositional parameters have been taken for typical values, $f=0.5$ and $\tau =1$, as soliton results have indicated that variations here usually produce little quantitative changes. For those parameters, the density correction factors turn out to be $\gamma d=1.148\u2009756\u20099$, $\gamma e=0.950\u2009844\u20098$, and $\gamma i=1.049\u2009800\u20098$. These are determined by the numerical integration of Poisson's equation (19) under the accompanying assessment of the conservation laws (13), (16), and (17). Remarkably, although (18) has been used in the numerical algorithm, the correct determination of $\gamma d$ and $\gamma e$ from Eqs. (13) and (17), respectively, also annuls (16), serving thus as a useful check on the numerics. The numerical precision on the various density parameters $\gamma $ is of order $10\u22127$.

We show in Fig. 2(a), the electrostatic profile of the periodic nonlinear DAW, for typical compositional parameters $f=0.5,\tau =1$, wave properties $V=0.26,\alpha =0.15$, and inflection point densities $\gamma d=1.148\u2009756\u20099$, $\gamma e=0.950\u2009844\u20098$, $\gamma i=1.049\u2009800\u20098$, needed to conserve the species densities. From the profile in Fig. 2(a), we can deduce the wavelength $L=\xi max\u2212\xi min=2.954\u2009329$, where $\xi max$ is the smallest of the positive roots and $\xi min$ the largest of the negative roots of the Sagdeev potential. In addition, as we are not dealing with linear waves, we define $\Phi =\phi max\u2212\phi min=0.130\u2009615$ as a measure for the amplitude and $\phi min,\phi max$ have been introduced already below (11).

We also show in Fig. 2(b), the corresponding Sagdeev pseudopotential with its three roots, one negative at $\phi min$ and two positive, of which the first is $\phi max$ and the second has no physical meaning, as it cannot be reached from the undisturbed values. The quite spiky electric field is shown in Fig. 2(c). Similar profiles have been observed in many regions of Earth's magnetosphere (see, for example, Pickett *et al.*,^{27} Rufai,^{28} Singh *et al.*^{29} and references therein).

Finally, the three species' densities are combined in Fig. 3, with the ion density indicated in blue, the electron density in red, and the dust density in green. It is remarked that the dust density compression is severely spiked (the top part of this spike is truncated for visual clarity), compensated by a longer but shallower rarefaction.

### B. Model with Boltzmann electrons, Cairns ions and negative dust

Now we deal with Cairns instead of Boltzmann ions, and choose $\beta =0.5$ as appropriate for strong nonthermality, as it is near but smaller than the upper limit $\beta =4/7\u22430.57$ beyond which the underlying microscopic Cairns distribution starts to develop nonmonotonic beam-like characteristics.^{25} The existence region is shown in Fig. 4, for average parameters $\beta =0.5,f=0.5,\tau =1$, and simplified $\gamma i=\gamma e=\gamma d=1$.

Note that the ranges in *V* and $\alpha $ are now larger than for the Boltzmann ions shown in Fig. 1, owing to the effects of nonthermality.

We can thus generate Table II, with an arrangement similar to that in Table I. A comparison between the two tables confirms that the ${\alpha ,V}$ existence region increases with the nonthermal parameter $\beta $.

For the illustration of a typical example, we pick compositional parameters $\beta =0.5,f=0.5,\tau =1$, wave parameters $\alpha =0.2,V=0.3$, and then the density conditions yield $\gamma d=1.153\u2009078\u20097,\gamma e=0.920\u2009749\u20092,\gamma i=1.036\u2009914\u20090$. The wavelength is $L=3.335\u2009177$ and the measure for the amplitude $\Phi =0.197\u2009049$. Due to the nonthermality, these values are larger than the corresponding values obtained for the Boltzmann ion model.

For these values, the results are shown in Fig. 5 for the electrostatic periodic profile, the Sagdeev pseudopotential, and the wave electric field and in Fig. 6 for the species densities. The qualitative agreement with the corresponding results for $\beta =0$ is obvious.

Note that for graphical clarity the serious dust density compression has not been show above $nd=2$, because otherwise the subtleties in the electron and ion densities are obliterated. In reality, $nd$ at maximum compression goes up to 11.45. This high level of compression is indicative of its close proximity to the infinite compression limit that acts as the upper bound for $\alpha $.

### C. Small amplitude quasi-linear waves

There is obviously a clear link between $\alpha $ and wave amplitude Φ, $\alpha $ to be found on ordinate axis. Note the differences in scale between the two parts of Fig. 7, due to the nonthermality induced by the Cairns model, leading to larger velocities and amplitudes. For smaller wave amplitudes, the hodographs tend to the ellipses one gets for linear waves, having a profile $\phi \u221d\u2009sin\u2009\xi $.

Both the Boltzmann and Cairns distributions for the positive ions have been further compared in Table III.

$\beta $ . | $\alpha $ . | V
. | $\gamma d$ . | $\gamma e$ . | $\gamma i$ . |
---|---|---|---|---|---|

0 | 0.15 | 0.26 | 1.148 756 9 | 0.950 844 8 | 1.049 800 8 |

0 | 0.10 | 0.26 | 1.068 274 5 | 0.977 039 2 | 1.022 656 8 |

0 | 0.05 | 0.26 | 1.017 437 0 | 0.994 070 2 | 1.005 753 6 |

0 | 0.01 | 0.26 | 1.000 702 5 | 0.999 760 2 | 1.000 231 4 |

0.5 | 0.20 | 0.30 | 1.153 078 7 | 0.920 749 2 | 1.036 914 0 |

0.5 | 0.10 | 0.30 | 1.042 289 8 | 0.979 074 8 | 1.010 182 3 |

0.5 | 0.05 | 0.30 | 1.010 878 5 | 0.994 357 8 | 1.002 618 2 |

0.5 | 0.01 | 0.30 | 1.000 439 3 | 0.999 772 1 | 1.000 105 7 |

$\beta $ . | $\alpha $ . | V
. | $\gamma d$ . | $\gamma e$ . | $\gamma i$ . |
---|---|---|---|---|---|

0 | 0.15 | 0.26 | 1.148 756 9 | 0.950 844 8 | 1.049 800 8 |

0 | 0.10 | 0.26 | 1.068 274 5 | 0.977 039 2 | 1.022 656 8 |

0 | 0.05 | 0.26 | 1.017 437 0 | 0.994 070 2 | 1.005 753 6 |

0 | 0.01 | 0.26 | 1.000 702 5 | 0.999 760 2 | 1.000 231 4 |

0.5 | 0.20 | 0.30 | 1.153 078 7 | 0.920 749 2 | 1.036 914 0 |

0.5 | 0.10 | 0.30 | 1.042 289 8 | 0.979 074 8 | 1.010 182 3 |

0.5 | 0.05 | 0.30 | 1.010 878 5 | 0.994 357 8 | 1.002 618 2 |

0.5 | 0.01 | 0.30 | 1.000 439 3 | 0.999 772 1 | 1.000 105 7 |

We illustrate the quasi-linear, almost cosinusoidal character also for the electric field in Fig. 8.

## IV. SUMMARY

We have applied an analogous Sagdeev pseudopotential formalism to the description of nonlinear periodic dust-acoustic waves that was before used for the simpler model of nonlinear periodic ion-acoustic modes.^{23,24} The fundamental difference is that with more species than just electrons and positive ions one has to introduce more density parameters in the initial global perturbations in order to conserve the individual species densities per cycle of the nonlinear periodic wave. For the usual dust-acoustic wave model, there are three species, and this requires two density parameters instead of one, given that there is always a global relation between the species densities implied in Poisson's equation. All this renders the analytical and the numerical analyses more complicated, but surmountable.

Nevertheless, the same salient characteristics were imposed or obtained as for ion-acoustic models. These are (a) the conservation of species densities per cycle and (b) mimicking linear wave properties for very small obliquity. Characteristic (a) is an analytical requirement that has to be imposed to arrive at a physically correct description, whereas on the other hand characteristic (b) follows from the numerical analysis at small amplitudes. In erroneous treatments, the small amplitude limits cannot be connected with the undisturbed conditions and, thus, leave a conceptual gap between linear and nonlinear periodic waves which is physically not understandable.

Finally, it is obvious that even more sophisticated plasma models can be treated, by establishing proper extensions of the Sagdeev formalism for nonlinear periodic waves. The repetitivity of the various figures obtained in earlier papers and here illustrates this very well. However, it is not our intention to add a spate of more papers on this topic, as has been the case for the description of solitons in various multispecies plasmas. The essential difference with the soliton theory is that for periodic nonlinear modes, one needs precise global rather than local initial perturbations. That might render solitons easier to generate in nature than really periodic modes. This is rather peculiar, given that after the original observations of solitons on shallow water surfaces, it took quite some time before they were recognized as different from the ubiquitous linear wave pictures.

## ACKNOWLEDGMENTS

This work is based on research supported in part by the National Research Foundation of South Africa (Grant No. 145712).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Frank Verheest:** Conceptualization (equal); Formal analysis (equal); Writing – original draft (lead); Writing – review & editing (equal). **Carel Petrus Olivier:** Conceptualization (equal); Formal analysis (equal); Writing – original draft (supporting); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

## REFERENCES

*Waves in Dusty Space Plasmas*

*Introduction to Dusty Plasma Physics*

**14**, 110702 (2007)]

*Reviews of Plasma Physics*