Above the weak nonlinearity: supernonlinear waves in astrophysical and laboratory plasmas
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Abstract
The review summarises recent theoretical achievements and observational manifestations of a new, recently discovered type of nonlinear oscillations in multicomponent plasmas, namely supernonlinear periodic waves and supernonlinear solitary waves (supersolitons). Both are characterised by a nontrivial topology of their phase portraits, highly anharmonic profile shapes, extremely long periods, and large amplitudes. Based upon multifluid magnetohydrodynamic plasma models, examples of ionacoustic and Alfvén supernonlinear waves are considered. A multicomponent nature of the plasma was revealed to be a crucial condition for the existence of these supernonlinear waves, with the complexity of the system growing with the number of plasma species accounted for in the model. A minimum number of plasma components which allow for the existence of supernonlinear waves are also discussed. From the observational point of view, typical signatures of periodic supernonlinear waves are manifested, for example, in the oscillatory processes operating in the magnetised plasma of the solar corona and groundbased plasma machines. Supernonlinear solitary structures (supersolitons) of an electrostatic origin are recognised in the Earth’s magnetosphere, laboratory experiments with chemically active plasmas, and numerical simulations.
Keywords
Supernonlinear wave Supersoliton Multicomponent plasma1 Introduction
Due to their unique properties, such as large amplitudes, long periods, and highly anharmonic profile shapes (see Sect. 2.1 for details), SNWs are certainly among the most intriguing extreme phenomena in real physical systems, and the question of whether they can be detected, in particular, in space and laboratory plasmas attracts a growing interest in the research community. A comprehensive analysis of the previous studies dealing with nonlinear waves of various types in plasmas reveals that there are a number of works where the generalised potential energy (hereafter the Sagdeev or Bernoulli pseudopotentials) was found to have two or even three local minima. For example, Kuehl and Imen (1985) and Cairns et al. (1995) obtained the pseudopotential of the ionacoustic waves in the electron–ion plasma, which may have two minima under certain physical conditions. More specifically, for the existence of a doublewell pseudopotential in these models the electron plasma component must have an essentially nonthermal distribution, which allows the electrons to be separated into several groups of different energies. Similar properties of the pseudopotential energy were found for the electrostatic waves propagating in the electron–positron plasma (Verheest et al. 1996), where both plasma components were taken to consist of cold and hot populations. Another interesting example of two minima appearing in the pseudopotential is of the oblique electrostatic waves in the magnetised fourcomponent plasma with cold and hot electrons and heavy and light ions, as shown in Ghosh and Lakhina (2004) in the application to the auroral regions of the Earth’s magnetosphere. Additionally, ionacoustic and dustacoustic modes in dusty plasmas of various models may also have a pseudopotential energy with two minima (see, e.g. Choi et al. 2007; Dubinov and Sazonkin 2008, 2013; Baluku et al. 2010; AkbariMoghanjoughi 2010; Mamun and Shukla 2010; Dubinov et al. 2011; Verheest 2011 for recent studies). The pseudopotential with two minima also appears in the analyses of electrostatic waves in magnetised electron–positron–ion plasmas (Ansar Mahmood et al. 2005), and in the magnetised electron–ion plasma with two oppositely charged ion components (Kaur and Singh Gill 2010). Three minima in the pseudopotential energy for nonlinear dustacoustic waves in an electron–ion plasma with negatively and positively charged dusty grains, were found in Verheest (2009). This particular example implies an even more complicated topology of the phase portrait of the wave, with two separatrices, one of which fully envelops the other. Hence, a broader set of (super)nonlinear solutions can be naturally expected in this case. We also need to mention that in addition to electrostatic waves in plasmas, the magnetohydrodynamic (MHD) waves can have multiwell pseudopotentials too. For example, Choi and Lee (2007) and Masood et al. (2010) analysed the propagation of Alfvén waves in a dusty plasma addressing magnetospheric and laboratory plasma structures and selfgravitating molecular clouds, and found the corresponding pseudopotentials with two minima. In all the studies listed above, periodic solutions of the SNW type are possible. However, none of these works consider waves with amplitudes larger than that of separatrices. It is the purpose of the present review to rectify this omission.
By studying the details of the plasma models in the above examples, which allow the potential energy to contain several minima, we emphasise the importance of a multicomponent nature of the plasma. Indeed, an obvious correlation of the number of plasma species taken into account in a model with the complexity of the potential energy topology and the nonlinear wave forms, appearing in such a plasma, most likely exists. The latter in turn seems to be a crucial condition for the identification of SNWs in plasma systems, which indicates that only multicomponent plasmas are able to support supernonlinear periodic solutions. Such multispecies plasmas can be adequately described by the multifluid approaches that have been widely used in space physics and aeronomy (see, e.g. Schunk 1977; Barakat and Schunk 1982 and all the references mentioned in the above paragraph). In particular, such multifluid models of the plasma are intensively developed in the application to lower layers of the Earth’s magnetosphere (e.g. Konikov et al. 1989; Demars and Schunk 1994; Ganguli 1996) and solar wind (e.g. Echim et al. 2011; Abbo et al. 2016).
On the other hand, the special phase trajectories, separatrices, are known to correspond to essentially nonlinear solutions describing the evolution of shock waves, double layers, and solitons in plasma. Hence, as a natural extension of the basic idea of periodic SNWs, one can assume the case, when there are several separatrices in the phase portrait, and one of them fully envelops the others (see, e.g. Verheest 2009). Similar to periodic SNWs, this outermost separatrix corresponds to a new supernonlinear type of solitary waves in plasma, which can be referred to as a supersoliton. According to such a definition, the supersolitons represent a group of solitary SNWs that combines the unique properties of solitary structures and extreme characteristics of SNWs. So far, the concept of supersolitons became commonly accepted among researchers and is extensively developed in a number of recent works (see, e.g. Hellberg et al. 2013; Maharaj et al. 2013; Verhees et al. 2013a, b, c, 2014; Dutta et al. 2014; Ghosh and Sekar Iyengar 2014; Lakhina et al. 2014; Rufai et al. 2014, 2015, 2016a; Verheest et al. 2014; Verheest 2014; Olivier et al. 2015; Rufai 2015; Singh and Lakhina 2015; Verheest and Hellberg 2015; Paul and Bandyopadhyay 2016 and Sect. 3.3 for details). In the present review we focus on the results obtained in the pioneer studies of periodic SNWs (Dubinov et al. 2011, 2012) and supersolitons (Dubinov and Kolotkov 2012b, c; Dubinov et al. 2012a) in multicomponent plasmas typical for astrophysical and laboratory conditions. We also demonstrate several observational examples of these supernonlinear structures detected in the plasma of the solar atmosphere, Earth’s magnetosphere, laboratory and numerical experiments.
2 Supernonlinear waves (SNWs)
2.1 Classification and signatures of periodic SNWs
Different oscillatory physical processes have different topologies of their phase portraits, characterised by a certain number of stable equilibrium points (centres) and by a number of special phase trajectories, separatrices, which additionally may vary in shape. In particular, Fig. 2 schematically illustrates the examples of the phase portraits with 2, 3, and 4 centres and several layers of separatrices. The complexity of the phase portraits naturally grows with the complexity of the oscillating system, where the discussed supernonlinear waves (SNWs) always represent the most nontrivial cases.
We now describe the unique signatures of SNWs, allowing for the distinguishing of them among the whole possible oscillating ensemble detected in observations. By definition, these are essentially largeamplitude fluctuations, with typical amplitudes which are sufficiently higher than the background level. Their profiles are always anharmonic, of a symmetric triangular shape, with at least four points appearing in a single oscillation cycle, where the second derivative is equal to zero. Due to such an anharmonicity and their original supernonlinear nature, the Fourier spectral analysis of SNWs shows the appearance of wellpronounced equidistant higher harmonics in a power spectrum provided the noise level is sufficiently low. However, traditional spectral techniques based upon Fourier transforms and wavelets are known to use a priori assigned harmonic basis functions, and, hence, are obviously limited in the analysis of such highly anharmonic signals, as SNWs. Instead, a novel Hilbert–Huang Transform (HHT) spectral method has recently been developed for the analysis of such nonlinear signals (Huang et al. 1998, 2008). It uses the empirical mode decomposition (EMD) technique, which expands the signal of interest into a basis derived directly from the data, iteratively searching for the local timescales naturally appearing in the signal. Hence, due to its adaptive nature, the HHT spectral technique is essentially suitable for the processing of highly nonlinear anharmonic time series typical for SNWs.
Another interesting property of SNWs is related to their temporal evolution, namely in the case where the oscillating system dissipates the energy and SNWs decay, their amplitude decreases gradually until a specific value which corresponds to the amplitude of the nearest separatrix. After that, a very rapid decrease (on the timescale shorter than the oscillation period) of the oscillation amplitude most likely occurs, accompanied by the transformation of the wave type (for example, from \(\text{SNW}_{3,2}\) to \(\text{SNW}_{2,1},\) as illustrated in Fig. 2). Such sudden changes of the oscillation amplitude can be detected in observations. Similarly, in the opposite case of the continuous energy supply or an instability development in the system, the amplitude of SNWs may increase rapidly when it reaches the upper separatrix level, providing the corresponding change of the wave type (for example, from \(\text{SNW}_{2,1}\) to \(\text{SNW}_{3,2}\) or from \(\text{SNW}_{3,2}\) to \(\text{SNW}_{4,3},\) according to Fig. 2). In the following sections we consider a number of distinct examples of various multifluid magnetohydrodynamic plasma models which lead to the possibility of periodic SNWs.
2.2 Ionacoustic periodic \(\text{SNW}_{2,1}\) in a plasma with two oppositely charged ions
As the first illustration we show that the periodic solutions of an ionacoustic \(\text{SNW}_{2,1}\) type can exist in a threespecies plasma with electrons and two oppositely charged ion components. The plasma is assumed to be uniform, collisionless, and unmagnetised.
We recall that by definition, this plasma model is able to support periodic ionacoustic solutions of the SNW type when the pseudopotential energy (14) has at least two local minima. Hence, the values of the dimensionless parameters of the model, providing such a configuration of the function \(U_{\rm S},\) are chosen. Figure 3 shows that the latter is possible for both \(V<V_{\rm s}\) and \(V>V_{\rm s}\) cases, where \(V_{\rm s}=(kT_{\rm e}/n_{0{\rm e}})^{1/2}(Z_1^2n_{0\rm i1}/m_{\rm i1}+Z_2^2n_{0\rm i2}/m_{\rm i2})^{1/2}\) is the ionsound speed in the plasma, obtained from the dispersion relation of the linear theory describing the oscillations in the vicinity of \(\varphi =0\) (see, e.g. Dubinov 2009). In other words, both subsonic (lefthand column of Fig. 3) and supersonic (righthand column of Fig. 3) SNW periodic solutions are possible.
We also need to mention that in addition to the discussed extremely large amplitude \(\text{SNW}_{2,1},\) the potential functions and phase portraits shown in Fig. 3 imply the existence of stationary solutions in the form of ordinary smaller amplitude nonlinear waves, \(\text{NW}_{1,0}\) known as cnoidal waves. Their phase trajectories in turn envelop each separate centre in the phase portraits. However, those \(\text{NW}_{1,0}\) solutions whose trajectories do not correspond to the initial equilibrium of the unperturbed plasma with \(\varphi =0\) (namely the trajectories enveloping the lefthand centre in the phase portrait of the subsonic case and all \(\text{NW}_{1,0}\) solutions obtained in the supersonic regime) should be disregarded.

The first derivative of a pseudopotential function \(U_{\rm S}(\varphi ),\) which is in fact an effective generalised force governing the dynamics of a pseudoparticle, must have at least two real roots, that results in a doublewell form of the pseudopotential;

The edge points (shown with the blank circles in Fig. 3) of the interval where \(U_{\rm S}(\varphi )\) has real values, must be above the local maximum of the function \(U_{\rm S}(\varphi ),\) separating its two local minima.
2.3 Electrostatic periodic \(\text{SNW}_{2,1},\) \(\text{SNW}_{3,1},\) and \(\text{SNW}_{3,2}\) in a fourspecies plasma
Consider a more complex model of the plasma consisting of four components: hot electrons and positrons, and two cold ion species with opposite electric charges. Similar to the previous section, the initial full electrical neutrality condition for the unperturbed state of such a plasma is \(Z_1en_{0\rm i1}Z_2en_{0\rm i2}en_{0{\rm e}}+en_{0\rm p}=0,\) where an additional term \(en_{0\rm p}\) describing the positron contribution appears. Introducing again the dimensionless parameters of the model, as \(\alpha =n_{0\rm i2}/n_{0\rm i1},\) \(\beta =m_{\rm i2}/m_{\rm i1},\) \(\delta =n_{0\rm p}/n_{0\rm i1},\) \(\tau =T_{\rm p}/T_{\rm e},\) and \(v=\sqrt{m_{\rm i1}V^2/kT_{\rm e}},\) where V and k are the phase speed of the electrostatic wave propagating in the plasma and the Boltzmann constant, respectively, the initial neutrality condition can be rewritten as \(n_{0{\rm e}}=n_{0\rm i1}(Z_1\alpha Z_2+\delta ).\)
2.4 Shear Alfvén \(\text{SNW}_{2,1}\) in a fourspecies magnetised plasma
In addition to the electrostatic nature (see Sects. 2.2 and 2.3), SNWs may be of an electromagnetic origin too, and it is worth searching for them, for example, among plasma magnetohydrodynamic waves. For illustration, in this section the relevant model of supernonlinear largeamplitude and highfrequency shear Alfvén waves propagating in a multicomponent magnetised plasma is considered. The developed model can be attributed to the class of models investigating the contribution of highfrequency oscillatory phenomena in the solar atmosphere (with typical periods shorter than one second) to the heating of the solar corona and acceleration of the solar wind. They may be driven, for example, by various microturbulences and spontaneous magnetic reconnection occurring in the photospheric–chromospheric magnetic network (Axford and McKenzie 1992; Tu and Marsch 1997), or by nonlinear cascading from lower frequencies in the corona (Isenberg and Hollweg 1983; Tu 1987).
In the model the plasma is assumed to be of an identical composition as considered in Sect. 2.3, with hot and inertialess electrons and positrons, and two sorts of cold massive ion components. The external magnetic field \(\mathbf{B}_{0}\) is directed along the zaxis. The plasma is taken to be sufficiently magnetised with the plasma parameter \(\beta \ll 1\) that allows one to neglect the ion thermal pressure and the ion velocity components parallel to \(\mathbf{B}_{0}.\) Perturbing the plasma in the xdirection, consider a 2.5D shear Alfvén wave propagating in the xozplane. The schematic sketch illustrating the geometry of the problem is shown in Fig. 8. Following the assumption of a low \(\beta\) plasma, the electromagnetic field in the wave is determined according to the two potentials formalism (see, e.g. Chen et al. 2000; Choi and Lee 2007), with \(E_x=\partial \varphi /\partial x,\) and \(E_z=\partial \psi /\partial z\equiv \partial \varphi /\partial zc^{1}\partial A_z/\partial t,\) where \(A_z\) is a magnetic vector potential directed along the external magnetic field \(\mathbf{B}_{0}\) and c is the speed of light.
Although there are a few indirect observational evidences of the presence of a nonnegligible positron fraction in the solar corona (see, e.g. Shar et al. 2004; Fleishman et al. 2013; Murphy et al. 2014), one should admit that the developed model is still sufficiently far from the actual coronal conditions. Nevertheless, its findings suggest the need for a similar analysis of a pure coronal case. For example, a background thermal plasma penetrated by the energetic electron, proton and alpha particle beams could be considered.
3 Solitary SNW in plasma: supersolitons
As a natural extension of the original idea of periodic supernonlinear waves, proposed in Dubinov et al. (2011, 2012) and described in detail in Sect. 2, whose phase trajectories envelop at least one separatrix loop, one can imagine the case, where there are several separatrix layers in the phase portrait, and one of them fully envelops the others (see, e.g. schematic examples shown in (c) and (e–g) of Fig. 2). In that case the outermost separatrix represents a new form of a largeamplitude solitary wave in a system, which may be referred to as a supersoliton. Similar to periodic SNWs, for the existence of supersolitons in a dynamical system its generalised potential energy must contain several (at least three) local minima separated by two local maxima. For the latter condition to be fulfilled in the electrically active medium, such as a plasma, its multicomponent nature is of a crucial importance. In the following sections we demonstrate a few analytical examples of electrostatic waves in the form of supersolitons, propagating in multicomponent plasmas of various hydrodynamic models. Clear agreement between theoretical solutions and observational records obtained in laboratory experiments, indicating the possibility of the existence of supersolitons in real plasmas, is also achieved.
3.1 Supersolitons in epiiplasma
As a first example, the most trivial model of an unmagnetised, collisionless, and uniform plasma supporting the existence of supersolitons and consisting of four charged species: namely hot and inertialess electrons “e” and positrons “p”, and cold and massive positively “1” and negatively “2” charged ion components is considered. Such a composition of the plasma is identical to that discussed in Sect. 2.3, where the periodic \(\text{SNW}_{2,1},\;\text{SNW}_{3,1},\; \text{and}\;\text{SNW}_{3,2}\) of an electrostatic origin were studied. Similar to the analysis performed in Sect. 2.3, we write the initial full electrical neutrality condition for the unperturbed state of such a fourspecies plasma, as \(Z_1en_{0\rm i1}Z_2en_{0\rm i2}en_{0{\rm e}}+en_{0\rm p}=0,\) where the subscript “0” refers to the parameters of the initial equilibrium of the plasma, \(e<0\) is the electron charge, \((e)>0\) is the positron charge, and \((Z_1e)>0\) and \(Z_2e<0\) are the electric charges of the positive and negative ions, respectively.
The dynamics of the cold and massive ion components in the ionacoustic wave is governed by the set of hydrodynamic equations (3)–(4) and (15), while the hot and inertialess electrons and positrons are distributed in the wave according to the Boltzmann laws (2) and (16), respectively. Repeating the calculations performed in Eqs. (6)–(11), one can obtain a secondorder ordinary differential equation with respect to the function \(\varphi (\xi )\) where \(\varphi\) is an electrostatic potential in the wave, and \(\xi \equiv xVt\) with V being a phase speed of the wave, which has a form of Eq. (12).
Figure 12 allows one to compare stationary profiles of the electrostatic potential \(\varphi (\xi )\) varying in a supersoliton (b) and in an ordinary soliton (a) of an ionacoustic type in epiiplasma, obtained from numerical solutions of the governing Eq. (12). As supersolitons cannot be described by the KdV equation, their profiles highly differ from the ordinary bellshaped soliton solutions determined by the \(\text{sech}^2(xVt)\) function in a weakly nonlinear theory. In particular, variation of the electrostatic potential \(\varphi\) in the ionacoustic supersolitary regime, shown in Fig. 12, has at least four points where its second derivative is zero. Additionally, amplitudes and widths of these supersolitary structures are always sufficiently larger compared to the ordinary KdVsolitons.
We would like to emphasise that electrostatic supersolitons exist in unmagnetised plasmas consisting of at least four electrically active components. Indeed, in simpler models of two and threespecies plasmas, the Sagdeev pseudopotential was empirically found to have up to two or three local extrema, respectively (see, e.g. Sect. 2.2). Hence, the required scenario when the external separatrix envelops the internal one cannot occur and supersolitons are impossible. There are a few additional conditions for the existence of supersolitons in a dynamical system, resulting from the analysis of the generalised potential energy function, \(U(\varphi ),\) namely the second maximum of the function \(U(\varphi )\) must be always lower than its first maximum corresponding to the initial neutrality of the plasma (cf. Verheest 2009), while the point of the reflection of ions from the potential barrier in the wave (i.e. the point “A” in Fig. 11) must be always above it.
The contrast between super and ordinary solitons becomes even more pronounced if one looks at the first derivative of the signal that can be used to distinguish them in laboratory experiments and astrophysical observations. In particular, Fig. 12 shows that the first derivative of the electrostatic potential \(\varphi ,\) which is in fact the electric field magnitude, varying in the ordinary ionacoustic soliton (c) has only a single extremum on a halfperiod of the structure, while the supersolitary regime is characterised by the appearance of additional side extrema on the electric field profile (d). Moreover, the number of the equilibrium points in the phase portrait, enveloped by the corresponding separatrix, is unambiguously determined by the number of extrema in the first derivative of the supersoliton.
3.2 Supersolitons in a warm dusty epiidplasma
In this section we demonstrate the possibility for the existence of ionacoustic supersolitons in a more complicated multispecies plasma model, which accounts for the effect of a nonzero ion temperature and includes an additional negatively charged plasma component, for example static dusty grains. Similar to the previous section, the plasma is assumed to be uniform, collisionless, and unmagnetised. The initial full electrical neutrality condition for the unperturbed state of such a fivespecies plasma takes the form \(Z_1en_{0\rm i1}Z_2en_{\rm 0i2}en_{0{\rm e}}+en_{0{\rm p}}q_{\rm d} n_{\rm d}=0,\) where an additional term \((q_{\rm d} n_{\rm d})\) describes the contribution of the dusty component, with \(q_{\rm d}<0\) and \(n_{\rm d}\) being its electric charge and concentration, respectively. The amount of the spatial electric charge of the dusty fraction relative to other plasma components is characterised by the dimensionless parameter \(\alpha =q_{\rm d} n_{\rm d}/en_{0\rm i1},\) which in turn allows one to rewrite the above neutrality condition as \(n_{0{\rm e}}=n_{0\rm i1}(Z_1\gamma Z_2+\delta \alpha ),\) where additional dimensionless parameters \(\gamma =n_{0\rm i2}/n_{0\rm i1}\) and \(\delta =n_{0\rm p}/n_{0\rm i1}\) are introduced.
Figure 14 shows the Sagdeev pseudopotential \(U(\varphi )\) determined by Eq. (49) and the corresponding phase portrait of nonlinear stationary ionacoustic waves propagating in the warm dusty epiidplasma plotted for certain values of the dimensionless parameters of the model. Similar to previous Sect. 3.1, the geometry of the generalised potential energy \(U(\varphi )\) is clearly seen to correspond to the existence of supersolitary solutions in the discussed fivespecies plasma model, i.e. there are three local minima in the potential energy function \(U(\varphi )\) and there is an external separatrix, which fully envelops the internal one, in the phase portrait. For comparison, Fig. 15 illustrates variations of the normalised electrostatic potential \(\varphi\) and its first derivative (that is an effective electric field in the wave) in the ordinary and supersolitary regimes obtained from numerical solutions of the governing equation (48).
3.3 Further development of the concept of supersolitons in plasma
The idea of supersolitons, originally proposed by Dubinov and Kolotkov (2012a, b, c), has opened a new actively developing research area in the field of nonlinear waves in plasma. Since the discovery of this type of waves in plasma in 2012, tens of followup and original studies were published, indicating the possibility of their existence in various plasma models and analysing their physical properties. The Sagdeev pseudopotential technique was used in all these works as a conventional instrument for analysis. In this paragraph we briefly outline stateoftheart achievements obtained in the theory of supersolitons in plasma during these 5 years.
Verheest et al. (2013) shows that electrostatic supersolitons are not a feature of exotic, complicated plasma models, but can exist even in a threespecies nonthermal plasma and are likely to occur in space plasmas. A methodology is given to derive their existence domains in a systematic manner by determining the specific limiting factors. A model of plasma with two groups of kappadistributed electrons is considered in Verheest et al. (2013), where the ionacoustic supersolitons are shown to exist too. Properties of ionacoustic supersolitons in a plasma with twotemperature electrons, positrons, and ions (treating all the light plasma components as nonthermal) are studied in Dutta et al. (2014). Steffy and Ghosh (2017) explore a transition of an ordinary ionacoustic soliton into a supersoliton in a fourspecies plasma with two groups of electrons and two sorts of ions, delineating the parametric ranges of the existence of supersolitons. Ionacoustic solitary waves with a Wshaped profile are found in a fourcomponent plasma in Paul et al. (2016). Such type of solitons may be also referred to as supersolitons.
A series of publications, Rufai et al. (2014, 2015, 2016a, b) and Rufai (2015), investigate the details of oblique (with respect to the magnetic field) propagation of ionacoustic supersolitons in a magnetised auroral plasma. Ionbeam plasmas with stationary dust grains are also found to support ionacoustic supersolitons (Dutta and Sahu 2017). It turns out that not only ionacoustic waves in multicomponent plasmas can have a form of supersolitons. For example, a possibility for the existence of dustacoustic supersolitons in a fourcomponent dusty plasma is shown in ElWakil et al. (2017).
3.4 Minimum number of plasma species needed for the existence of electrostatic supersolitons
Another illustrative example is given by Rufai et al. (2015), where the analytical model of the magnetised plasma consisting of a cold ion fluid and cool Boltzmann and hot kappadistributed electrons was developed. Again according to Verheest et al. (2013), it represents a twospecies plasma, where the obliquely propagating ionacoustic supersolitons were successfully detected (cf. Verheest et al. 2014). Hence, the correct accounting of the light component populations, included in the multispecies plasma model, is essential when analysing the possibility for the existence of supersolitary solutions. We would like to finalise the current discussion pointing out that unjustified partitioning of plasma electrons into several species of different energies sometimes may lead to erroneous results (Yu and Luo 2008).
4 Periodic SNW and supersolitons in space, laboratory, and numerical experiments
4.1 Signatures of SNW and supersolitons detected in earlier records
Nonlinear oscillatory phenomena in astrophysical and laboratory plasmas are regularly observed with the imaging and spectral instruments throughout the whole electromagnetic spectrum. More specifically, electrostatic waves in plasma are usually measured in experiments as the variations of electrostatic potential or concentration of a certain plasma component. Also, the time variability of a local electric field recorded at the location of a measuring detector is often of interest in plasma experiments. According to the theory developed above, a physical quantity experiencing oscillations in a periodic SNW or supersoliton would have an oscillation profile with characteristic wiggles (see, e.g. Figs. 4, 7, 10, and the bottom left panels in Figs. 12 and 15) and nonmonotonic patterns (see, e.g. the bottom right panels in Figs. 12 and 15), which enable the discussed waves to be identified in experimental records. The latter behaviour is determined by a number of separatrices enveloped by a phase trajectory of the wave (see Sect. 2.1). A comprehensive literature survey shows that there are a great number of earlier works reporting on the experimental detection of periodic and solitary waves in laboratory plasma machines and space missions, which have very straightforward signatures of SNW and supersolitons. However, the presence of those wiggling and nonmonotonic patterns in the observational signals was usually ignored by the authors. In this section we summarise several most illustrative observational examples obtained in space, laboratory and numerical plasma experiments, which can be considered as potential candidates for SNW and supersolitons.
More recent examples clearly illustrating typical signatures of periodic SNW were detected in the magnetised plasma of the solar atmosphere, a natural laboratory for studying fundamental plasma processes including nonlinear waves and oscillations (Nakariakov and Verwichte 2005; Nakariakov et al. 2016). In particular, Nakariakov et al. (2010) detected socalled quasiperiodic pulsations (QPP) in the electromagnetic emission of a powerful X1.2 class solar flare which occurred on 29 May 2003. The oscillations were found in two independent observations made with the Nobeyama Radio Polarimeters and with the ACS instrument onboard the INTEGRAL satellite in the radio and hard Xray band, respectively. Spectral analysis of the observational signals, performed with the use of the empirical mode decomposition technique (EMD), revealed that the oscillations shown in the bottom panels of Fig. 17 have an anharmonic profile of a symmetric triangular shape with several points on a single oscillation cycle, where its second derivative equals zero, which are typical signatures of the discussed SNWs. Furthermore, the following temporal evolution of the oscillation profile shows a wellpronounced rapid decrease of the oscillation amplitude, which may look similar to the transformation of the wave type from SNW to NW through a separatrix, as proposed in Sect. 2.1. Similar anharmonic and nonstationary QPPs are often seen in other flaring events and most likely require involvement of the discussed higher order nonlinear effects for the interpretation (see, e.g. Nakariakov and Melnikov 2009; Van Doorsselaere et al. 2016, for recent reviews).
Expected theoretical profiles of the parallel electric field in an ordinary (KdVtype) ionacoustic soliton, ion double layer, ion phase hole, and ionacoustic supersoliton and their experimental analogies with corresponding references
Localised electrostatic ion structures  Ionacoustic soliton  Ion double layer  Ion phase hole  Ionacoustic supersoliton 

Theoretical profiles of E_{z}  
Recorded profiles of E_{z}  
References  Fragment of Fig. 3 (Bostrom et al. 1988)  Fragment of Fig. 1p (Vasko et al. 2015)  Fragment of Fig. 7 (Goldman et al. 1999)  Fragment of Fig. 7b (Lu et al. 2005) 
Consider another type of experiments, numerical simulations, where possible candidates for supersolitary solutions were detected too. Earlier works (Lu et al. 2005; Kakad et al. 2014) show 1D electrostatic particleincell simulations of nonlinear evolution of electrostatic solitary waves in space plasmas, particularly focusing on the Earth’s auroral regions. The electric fields detected in simulations are shown in Fig. 19 and have a localised bipolar structure with welldistinguished nonmonotonic patterns and wiggles, which is consistent with the form of electrostatic supersolitons predicted by theory (see Table 1).
The question of the detection of supersolitons in laboratory experiments with chemically active plasmas is addressed in detail in Sect. 4.2 and references therein.
4.2 Supersolitons detected in laboratory experiments with \(\text{SF}_{6}\)–\(\text{Ar}\)–plasma with negative ions
Consider a uniform, collisionless, and unmagnetised multispecies plasma consisting of the singlycharged positive \(\text{Ar}^{+},\) \(\text{SF}^{+}_5\) and negative \(\text{F}^{},\) \(\text{SF}^{}_5\) ions, and electrons. The initial neutrality condition for the equilibrium state of such a plasma has a form \(en_{0{\rm Ar}^{+}}en_{0{\rm F}^{}}+en_{0{{\rm SF}^{+}_5}}en_{0{{\rm SF}^{}_5}}en_{0{\rm e}}=0,\) where \(e<0\) is the electric charge of the electron and negative ion components, \((e)>0\) is the electric charge of the positive ions, and the subscript “0” hereafter refers to the unperturbed values of the plasma parameters. Introducing the dimensionless parameters of the model, \(\alpha =n_{0{\rm F}^{}}/n_{0{\rm Ar}^{+}},\) \(\beta =n_{0{{\rm SF}^{}_5}}/n_{0{\rm Ar}^{+}},\) and \(\gamma =n_{0{{\rm SF}^{+}_5}}/n_{0{\rm Ar}^{+}},\) the above neutrality condition can be rewritten as \(n_{0{\rm e}}=n_{0{\rm Ar}^{+}}(1\alpha \beta +\gamma ).\) Absolute values of these parameters can be estimated from the actual experimental conditions used in Ludwig et al. (1984), where the partial pressure of the primary \(\text{SF}_{6}\) molecular gas in the discharge chamber was taken to be low, hence \(n_{0{\rm SF}_{6}}\ll n_{0{\rm e}}.\) In this case, under a constant discharge current the initial concentration of the \(\text{F}^{}\) ion gas is naturally lower than that of the \({{\rm SF}^{}_5}\) ion gas, resulting to \(\alpha \ll \beta .\) Moreover, Ludwig et al. (1984) established additional empirical relations for the relative concentrations of the ion components in the performed experiment: \(0.65<n_{0{{\rm SF}^{}_5}}/(n_{0{{\rm SF}^{}_5}}+n_{0{\rm F}^{}})<0.95\) and \(0.11<n_{0{{\rm SF}^{+}_5}}/(n_{0{{\rm SF}^{+}_5}}+n_{0{\rm Ar}^{+}})<0.77,\) allowing one to obtain that \(0.027<\alpha <0.28\) and \(0.12<\gamma <3.35\) for a certain value of \(\beta =0.52.\) Hence, the following values of the parameters \(\alpha =0.035,\) \(\beta =0.52,\) and \(\gamma =0.2\) were chosen for further analysis.
Figure 22 shows the pseudopotential \(U_{\rm S}(\varphi )\) determined by Eq. (56) of an oscillating pseudoparticle, i.e. of the stationary ionacoustic wave, and the corresponding phase portrait of the whole dynamical system plotted for \(\sqrt{m_{{\rm Ar}^{+}}V^2/kT_{\rm e}}=1.7,\) that provides the phase speed of the wave, V to be certainly supersonic, and for \(m_{{{\rm SF}^{+}_5}}/m_{{\rm Ar}^{+}}=3.18,\) \(m_{{{\rm SF}^{}_5}}/m_{{\rm Ar}^{+}}=3.18,\) and \(m_{{\rm F}^{}}/m_{{\rm Ar}^{+}}=0.476.\) This analysis indicates that indeed a largeamplitude supersolitary solution is possible in the developed \(\text{SF}_{6}\)–Ar–plasma model, whose phase trajectory, the external separatrix in the phase portrait, is of a guitarlike form and envelops the internal separatrices. Variations of the electrostatic potential \(\varphi ,\) electron and \(\text{Ar}^{+}\) ion concentrations, \(n_{\rm e}\) and \(n_{{\rm Ar}^{+}},\) and their first derivatives in the detected supersoliton are illustrated in Fig. 23. They are clearly seen to have a wiggling profile shape (lefthand column of Fig. 23) and a nonmonotonic behaviour with a few extrema (righthand column of Fig. 23), typical for the solitary structures observed in the experiments (see Fig. 21).
Such an excellent agreement between the experimental results and their independent theoretical interpretation presented in the current section strongly supports the idea that the ionacoustic solitary structures of unusual form, detected, in particular, by Nakamura et al. (1985) and Nakamura and Tsukabayashi (1985) in a laboratory fourion \(\text{SF}_{6}\)–Ar–plasma, could be supersolitons with a nontrivial topology of their phase portraits, and a comprehensive search for these structures in multicomponent space and astrophysical plasmas is needed.
5 Conclusions
The review addresses current trends in the analysis of a new type of stationary nonlinear waves in multispecies plasmas, periodic supernonlinear waves (SNW) and solitary supernonlinear waves (supersolitons), characterised by a nontrivial topology of their phase portraits, long periods, and large amplitudes. The scheme allowing for the classification of these supernonlinear solutions and their convenient notation is proposed. Using the multifluid plasma models, SNWs are shown to have both electrostatic and magnetohydrodynamic nature. The presence of at least three electrically active components in the plasma (e.g. thermal and nonthermal electrons, ions, charged dusty grains, positrons, etc.) is found to be among the essential criteria for the existence of SNWs. In the space plasma context the nearest candidates are an electron–proton plasma penetrated by alpha particles (e.g. typical for the physical conditions of the solar corona and solar wind) and dusty electron–ion plasmas represented in the vast majority of the astrophysical and space objects, which can be expected to support SNW. The increase of the number of plasma species results in a more complicated topology of the SNW’s phase portraits, providing a broader set of topologically different SNW types. Alternatively, nonneutral plasma systems with intense beams are also known to support solitary waves (see, e.g. Mo et al. 2013, where the first experimental observation of a KdVtype soliton wave train in electron beams is reported), and thus should be also included into the list of natural plasmas which are expected to sustain supersolitons. First results in this direction have been achieved by Dutta and Sahu (2017), who showed the possibility for the existence of ionacoustic supersolitons in ionbeam plasmas with stationary dust grains.
So far, periodic SNW and supersolitons have been analytically studied by the mean of the mechanical analogy method based on the Sagdeev pseudopotential representation. One should admit that the application of the perturbation reductive procedures introducing a small parameter into the model for the analysis of these highly nonlinear structures is compromised by their naturally large amplitudes. For instance, neither KdV equation (with the secondorder nonlinearity) nor modified KdV equation (thirdorder nonlinearity) is able to describe supersolitons. In contrast, the Sagdeev pseudopotential technique allows for studying arbitrarily largeamplitude fluctuations, which justifies the choice of it as a conventional tool for the analysis of SNW and supersolitons in all considered works. Following its importance in the discussed problem of supernonlinear waves in plasma, we would like to briefly outline some more aspects behind this approximation, namely the role of the momentum and energy integrals of the governing equations and the role of different sonic Mach numbers in a fully nonlinear analysis is emphasised by McKenzie et al. (2004). This approach should be treated as an alternative to the Sagdeev potential technique, and can be useful in describing spiky wave forms associated with choked flows at the sonic points. The aspects of a Hamiltonian description of the travelling waves in plasma are discussed by Webb et al. (2005) and McKenzie et al. (2007). The role of generalised vorticities and Bernoulli integrals in the formulation of travelling waves in multifluid plasmas is studied by Mace et al. (2007). Webb et al. (2007, 2014) show that there are two different Hamiltonian formulations for the travelling waves, which use the xmomentum integral and the energy integral of the system. The momentum integral can be thought of as the Hamiltonian of the system, in which the variables are constrained by the energy integral. An alternative Hamiltonian formulation also results from using the energy integral as the Hamiltonian in which the momentum integral is a constraint. These formulations are related to the multisymplectic view of the equations, as developed by, e.g. Bridges (1992), Bridges et al. (2005), Hydon (2005), and Webb et al. (2015).
Typical signatures of SNWs allowing for the detection of them in astrophysical observations and under laboratory conditions are given. More specifically, their oscillation amplitudes are always large (typically greater than or comparable to a nonoscillating background level) and scale with the oscillation periods, while the oscillation profiles are highly anharmonic of a symmetric triangular shape. Furthermore, several observational signals clearly manifesting such a supernonlinear behaviour and observed in the magnetised plasma of the solar atmosphere, Earth’s magnetosphere, in laboratory and numerical experiments are demonstrated. Despite a significant progress in the field, a further comprehensive search for observational evidences of these supernonlinear structures using modern groundbased and spaceborne instruments and experimental plasma machines, and the developing of corresponding analytical models is of the great interest and importance.
Notes
Acknowledgements
The authors are grateful to Prof. Valery Nakariakov and Prof. George Rowlands for valuable discussions and constructive comments. D.Y. Kolotkov acknowledges the support of the STFC consolidated grant ST/L000733/1. A. E. Dubinov worked in the framework of the Program of Increasing the Competitiveness of NRNU MEPhI.
Compliance with ethical standards
Conflict of interest
The authors would like to state that there is no conflict of interest associated with this publication.
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