Nonthermal equilibrium is an intrinsic characteristic of space and astrophysical plasmas, and in many space environments, the velocity distributions of charged particles with suprathermal tails may be well be fitted by the Kappa function, which becomes the Maxwellian distribution for $\kappa \u2192\u221e$. Various studies of ion or dusty acoustic solitons, thus, have considered the Kappa distributed electrons in the model calculations. However, the Kappa velocity distribution (KVD) is theoretically not applicable for $\kappa \u2264 3 / 2$. Alternatively, the recently proposed regularized Kappa distribution with two free parameters, $\kappa $ and $\alpha $, have been shown to be mathematically and physically smooth for all $\kappa $ values, which may recover the standard KVD for $\alpha =0$ and the Maxwellian distribution for $\kappa \u2192\u221e$ and $\alpha =0$. In this study, we examine the characteristics of ion acoustic solitons based on the linear, weakly nonlinear Korteweg–de Vries (KdV) and fully nonlinear theories with the regularized Kappa distributed electrons and warm ion fluids. These approaches may give rise to the dispersion relation with modified characteristic speed of acoustic waves, the analytical KdV solutions, and the Sagdeev's potential as well as the fully nonlinear solutions. It is shown that the model results are mathematically and physically valid for $\kappa \u2264 3 / 2$ and the formulations with the charges being free parameters are applicable for general acoustic solitons.

## I. INTRODUCTION

Nonlinear solitary waves have been widely observed in space environments that exhibit bipolar electric field structures with the width of a few electron Debye lengths, referred to as the electrostatic solitons.^{1–4} The observed electrostatic solitons are mostly classified as electron solitons associated with the electron holes^{1} that, however, are not found in the recent high resolution observations of soliton events.^{4} The theoretical formulations for ion acoustic solitons (IASs) are first developed by Sagdeev^{5} in which the ions are treated as cold fluids and the hot electrons are described by the kinetic equilibrium with Maxwellian velocity distribution (MVD). The model equations can be solved by the Sagdeev's potential method, which have been widely included in the plasma physics textbooks.^{5–8} The formation of IASs is the perfect balance between the dispersive effects and nonlinear steepening of acoustic waves. The necessary condition for the existence of IAS is $M>1$, where $M= U / C s$, or, that the propagation speed $U$ is greater than the phase speed of ion acoustic waves, $ C s= \gamma e k B T e / m i 1 / 2$, where $ \gamma e=1$ for Maxwellian distributed electrons. The assumption of MVDs necessarily leads to the isothermal solitary structures. Unlike the electron solitons, the IAS does not exhibit hole structures associated with the trapped electrons.

In the simplest model, the IAS exhibit density enhancement associated with the positive electric potential. However, there have been some observational evidence for the anomalous solitary structures with positive potential and depleted number density.^{9,10} The first nonthermal model for IAS is proposed by Cairns *et al.*,^{11} in which the regularized MVD with relatively higher tail, referred to as Cairns *et al.*'s distribution, may yield anomalous IAS for certain parameter regimes. The Cairns *et al.*'s distribution is a special class of nonthermal distributions, and the more frequently adopted nonthermal distributions are the Kappa velocity distribution (KVD), which have been observed in the solar wind and magnetosphere.^{12–14} The IAS models developed by Chuang and Hau^{15} are based on the Kappa distributed electrons, which show that the Kappa–IAS do not exhibit anomalous features with depleted density. By allowing the hot components to have the Kappa or Cairns *et al.*'s velocity distribution, Chuang and Hau^{16} have further developed a general formulation for three-component acoustic solitons with the charges being free parameters in the model calculations. For the applications of planetary sciences, the generalized formulations for acoustic solitons with multiple components are developed by Wang and Hau,^{17} in which the combinations of Kappa and Cairns *et al.*-like distributions are considered for hot components with the charges being unspecified. The model calculations may recover the various existing models for ion, dusty, and electron solitons. Note that there exist various multiple component models of acoustic solitons for the applications of space and planetary environments.^{2,18–23} In the context of dynamic evolution, the effect of KVD electrons on the formation of IASs has been examined by the numerical simulations of the fluid models.^{24}

In most of the existing models for IASs with kinetic electrons, the ions are described by the fluid equations with zero temperature which, however, do not correspond to the real space plasma environments. In such models, the Sagdeev's potential method is used to obtain the soliton solutions. By assuming the adiabatic law with $\gamma =3$ corresponding to one degree of freedom for the ions, the modified Sagdeev's potential have been derived for the ion and dust acoustic solitons with Kappa and Cairns *et al.* distributed electrons.^{2,22,23,25} Note that there exist some fluid models of acoustic solitons that are not limited to cold ions, but in those studies, the electrons are not described by the kinetic equilibrium.^{26,27} In the fluid models of IASs, in which the electrons are not in kinetic equilibrium, the Sagdeev's potential method has also been adopted to derive the soliton solutions.^{26,27} More recently, the general formulations are developed by Jao and Hau^{28} for the acoustic solitons with both Kappa and Cairns *et al.*-like distributions for the hot components and with the charges being free parameters in the model calculations. The fluid components possess finite temperature and comply with the adiabatic energy law with arbitrary ratios of specific heats. In the studies of Chuang and Hau,^{15,16} Wang and Hau,^{17} and Jao and Hau,^{28} the characteristics of IASs are examined systematically from the perspectives of linear, weakly nonlinear and fully nonlinear theories. The characteristic speeds of acoustic waves are derived first which may vary with different model equations, and only for the Maxwellian electrons and cold ions, the phase speed has the simple form of $ \omega / k= C s$. Note that the lack of linear analyses may yield incorrect definition of Mach number.^{28} The weakly nonlinear Korteweg–de Vries (KdV) approach may give rise to the analytical forms for IASs, which may conveniently be applied to the space observations and compared to the fully nonlinear solutions. As shown by Jao and Hau,^{28} the peak potentials of the KdV solitons are about $40%$ of the fully nonlinear solutions. The fully nonlinear model equations for warm fluids with arbitrary ratios of specific heats, however, cannot be solved based on the Sagdeev's potential method but need to be solved numerically along with the fixed point analysis method.^{28} The applicability of the Sagdeev's potential method in the study of IASs with warm ions and kinetic electrons will be discussed in detail in Sec. IV.

While the KVD has been widely applied to various plasma physics problems,^{13,14} a few noticeable limitations associated with the model results do exist. In particular, the temperature and thermal pressure may become singular at $\kappa = 3 / 2$ and negative for $\kappa < 3 / 2$.^{29} As a result, the KVD is theoretically not applicable for $\kappa \u22643/2$. Moreover, as pointed out by Scherer *et al.*,^{29} the higher moments such as the heat fluxes are undefinable for $\kappa \u22645/2$. The KVD, thus, cannot serve as the basis of complete plasma fluid theory.^{29} Note that the $\kappa $ values typically observed in the solar wind are in the range of 2–17.^{30} The possibility of even smaller $\kappa $ values remains an open issue for some observational events.^{31} Thus, a modified version of the KVD, referred to as the regularized Kappa distribution (RKD or RK), is proposed by Scherer *et al.*,^{29} which do not possess the mathematical and physical singularities and have been applied to various plasma physics problems in recent years.^{32–34} The RK function contains two free parameters, $\kappa $ and $\alpha $, which for $\alpha =0$ may recover the Kappa function and for $\kappa \u2192\u221e$ and $\alpha =0$ may describe the Maxwellian distribution. The comparisons between the two distribution profiles show clearly that the effect of $\alpha $ terms in the RKD is to regularize the suprathermal tails in the KVD which becomes significant for $\kappa $ close to $ 3 / 2$.^{13} Theoretically, the moments of RKD are continuous for all positive $\kappa $ values, in contrast to the KVD with undefinable macroscopic temperature for $\kappa \u2264 3 / 2$ and heat fluxes for $\kappa \u2264 5 / 2$. In the calculations, both $\kappa $ and $\alpha $ values are the free parameters used to examine the effects of various velocity distributions including the MVD, KVD, and RKD on the model results.

The IAS with RK distributed electrons are first studied by Liu^{35} who has attempted to derive the corresponding Sagdeev's potential and analyzed the existence condition of IAS. The analyses and calculations, however, are limited by certain approximations and due to the lack of linear wave analyses the characteristic speed of ion acoustic waves and the Mach number are not properly defined. The KdV model of the IAS with RK distributions is later developed by Lu and Liu,^{36} which is limited to cold ion fluids. In light of the generality of the RK distribution as compared to the standard Kappa distribution, it is important to carry out a systematic analysis for the IAS with RK distributed electrons based on the linear, weakly nonlinear KdV and fully nonlinear theories in analogy with our previous studies for various nonthermal distributions.^{15–17} In addition, the effects of ion to electron temperatures are incorporated in the model calculations, which may vary in a wide range in space environments.^{28} The present study will focus on two-component plasmas with ion fluids and RK distributed electrons, which may easily be extended to multiple hot components with arbitrary distribution functions.^{17,28} The charges are unspecified such that the formulations are applicable for ion, dusty and electron acoustic solitons, etc. The major results of the present study based on the RK distributed elections include the derivations of the characteristic speed of ion acoustic waves, the KdV equations of weakly IASs for warm ions, the Sagdeev's potential of IASs for cold ions, and the fully nonlinear IASs for warm ions. Moreover, the solutions of IASs are for the first time constructed for the parameter regimes of $\kappa \u2264 3 / 2$ based on the RK distributions, which preserve the nonthermal features of KVDs. This paper is organized as follows. The model equations are introduced in Sec. II and the linear dispersion relation is analyzed in Sec. III for the phase speed of ion acoustic waves which is essential for the correct definition of Mach number. The KdV equations are then derived in Sec. IV for weakly nonlinear solitons which are convenient for the analytical calculations. The fully nonlinear equations are solved in Sec. V which includes the Sagdeev's potential method for the special case of cold ion fluids and the numerical calculations for warm ion fluids.

## II. MODEL EQUATIONS

^{28}The reference values with the subscript 0 for the number density, pressure, and temperature are referred to the uniform backgrounds, which correspond to the upstream and downstream states of the solitary structures in the following sections.

*T*$(\Phi )$ for various $\kappa $ and $\alpha $. As indicated, the curves for $ \alpha 2=0$ and $\kappa = 3 / 2,1$ are missing from the density, pressure, and temperature plots. The MVD case corresponding to $\kappa \u2192\u221e$ and $\alpha =0$ is also marked by the red curves. As indicated, for the same $\kappa $ values, the effects of positive $ \alpha 2$ may yield relatively smaller number density, thermal pressure, and temperature. The differences between the KVD and RK results are more pronounced for small $\kappa $ and large $\alpha $ values. For $\kappa \u2264 3 / 2$, the density profiles, which for $\alpha =0$ can be calculated based on Eq. (10) but are not theoretically applicable, are similar for the KVD and RK cases with small $\alpha $ values, whereas the singularities and unphysical $p(\Phi )$ and

*T*$(\Phi )$ for the KVD may be removed by finite $\alpha $ values in the RK models.

## III. DISPERSION RELATIONS

## IV. KORTEWEG–DE VRIES THEORY

^{28}where the fluid equations for warm ions are considered. In the present study, the first- and second-order perturbed electron number density need to be replaced by the following relations:

^{36}We first write the model equations in dimensionless forms; in particular, the various physical quantities are $ x \u2032\u2261x/ \lambda D , h$, $ t \u2032\u2261t \omega p , c$, where $ \omega p , c 2= n 0 , c q c 2 / m c \epsilon 0$, $ \Phi \u2032\u2261\Phi / k B T 0 , h eff / q c$, $ u c \u2032\u2261 u c/ C S , h$, $ p c \u2032\u2261 p c/ n 0 , c k B T 0 , h eff$, $ n s \u2032\u2261 n s/ n 0 , c$, $ q s \u2032\u2261 q s/e$ ( $s=c,h$), and the parameter $ \sigma c\u2261 T 0 , c/ T 0 , h eff$ measures the relative degree of the ion temperature. The velocity is normalized by $ C S , h$ due to the fact that $ \lambda D , h \omega p , c= C S , h$. Note that the normalization parameters are not fixed for different RK cases due to the fact that each case has different characteristic Debye length and acoustic speed. The choices of these normalization parameters are essential for the coordinate transformations used in the following KdV derivation. The set of dimensionless equations are then expressed in the coordinates of $ \u2009 x \u2032 , \u2009 t \u2032$. The dimensionless variables $ n c \u2032$, $ u c \u2032$, $ p c \u2032$, and $ \Phi \u2032$ can be expanded in terms of the smallness $\u03f5$ as

^{16}and the effects modified by the RK distribution may be examined by varying the parameters, $\alpha $ and $\kappa $. Equation (43) can be solved for $ \Phi \u2032 ( 1 )$ as function of $\zeta $, where $\zeta $ and $ \Phi \u2032 ( 1 )$ are normalized by $ \lambda D , h$ and $ k B T 0 , h eff/ q c$, respectively. For the comparison among different RK cases, the spatial coordinates and various physical quantities shown in Figs. 3 and 4 possess the same units; in particular, $x/ x 0= ( T 0 , h eff / T 0 , h M ) 1 / 2 x \u2032$ and $\Phi / \Phi 0= T 0 , h eff / T 0 , h M \Phi \u2032$, where $ x 0= \epsilon 0 k B T 0 , h M / n o , c q c 2 1 / 2$ and $ \Phi 0= k B T 0 , h M/ q c$. In the calculations, $ q h=\u2212e$, $ q c=e$, $ x 0=1$, and $ \Phi 0=1$ are adopted. We first consider the cold fluid limit, i.e., $ \sigma c=0$, in the calculations, as shown in Fig. 3 for various parameter values of $\kappa $ and $\alpha $ with $M=1.1$, where $M= U 0/ C S , eff$, $ U 0= C S , h u 0 + \u03f5 v 0$, $ u 0= 1 + C S , c 2 / C S , h 2 1 / 2,$ and $\u03f5 v 0$ is a given parameter in the KdV formulation. For various RK cases with fixed $M$ values, $\u03f5 v 0$ can be determined for each case and the upstream velocity $ U 0$ may vary with different cases. Note that the KdV equation possesses a singularity at $\kappa = 1 / 2$ for the standard KVD ( $\alpha =0$). Therefore, there still exist smooth profiles for the electric potential for $\kappa \u22651$ and $\alpha =0$. Nevertheless, the KVD cases with $\kappa \u2264 3 / 2$ are not shown in Figs. 3 and 4 since the pressure and temperature are unphysical for those cases. However, the solutions for RK solitons are smooth and physical for all $\kappa $ values as a result of introducing a finite $\alpha $ value in the RK models. Similar results are shown in Fig. 4 for the warm ion fluid model with $ T 0 , i/ T 0 , e M=0.1$. The examples show that for the same $\kappa $ values, the cases with larger $\alpha $ values have relatively weaker solitary structures with smaller potential jumps and weaker electric field.

^{36}In Figs. 3 and 4, the upstream Mach number is the same for both cold and warm ions, and the differences between the amplitudes and widths of the solitary structures for cold and warm models are relatively minor. Whereas, for the same upstream flow velocity, the ion temperature may yield apparently weaker IASs.

^{28}As indicated, for the same panels with the same $\kappa $ but different $\alpha $ values, the amplitudes of KdV solitons decrease with increasing $\alpha $ values, which may reduce the suprathermal populations of the original KVD. However, for the same other parameter values, the cases with smaller $\kappa $ values may give rise to larger potential peaks.

## V. FULLY NONLINEAR THEORY

^{28}In the frame of reference moving with the time stationary solitons, $\zeta =x\u2212 U ot$, the nonlinear equations become

^{28}

^{16}

^{,}

^{35}which involves the integration of the electron number density and the Kummer functions with infinite terms. Some approximations are, thus, made in the integration of $ n e$ and Kummer functions, which does not yield the exact Sagdeev's potential. Alternatively, by analyzing the electron momentum equation in equilibrium form, $\u2207p=\u2212nq\u2207\Phi $, it can be shown that $ d p h / d \Phi =\u2212 n h q h$. As compared to the relation of $n= d V / d \Phi ( \epsilon 0 / q)$, it becomes clear that $ V h \Phi = \u2212 p h \Phi / \epsilon 0 + C$ and assuming $ V h 0=0$, the Sagdeev's potential for hot components is

^{15,35}which is nevertheless not the focus of the present study on warm ions and kinetic electrons. Note that for $\alpha =0$ and $\kappa \u2264 3 / 2$, the solutions in the number density and electric potential still exist which can be solved based on Eq. (47) along with $ n c \Phi $ and $ n h \Phi $ shown in Eqs. (48) and (49). Figure 6 shows the soliton profiles for various $\kappa $ and $\alpha $ values based on the Sagdeev's potential method with Eqs. (50)–(52) and the Poisson equation method with Eqs. (47)–(49), respectively, indicating the consistency between the two model equations. As indicated, for fixed $\kappa $ values the potential jumps decrease with increasing $\alpha $ values, consistent with the KdV results. The cases for $\alpha =0$ and $\kappa \u2264 3 / 2$ are not shown, though there exist smooth solutions for the electric potential, but the corresponding pressure and temperature are unphysical. For small $\alpha $ values, the differences in the number density and electric potential between the KVD and RK cases are found to be relatively small.

^{28}which can be solved in the same manner as the cold fluid case.

^{27}In general, the full set of nonlinear equations shown in Eqs. (44)–(47) for arbitrary $ \gamma c$ need to be solved numerically along with the fixed point analyses.

^{37,38}In particular, by assuming the perturbations to have the form of $ e \lambda \zeta $, linearization around the upstream or downstream states of the nonlinear structures yields the following perturbed quantities or eigenfunctions and eigenvalues:

^{37}which implies that

The eigenfunctions are then used as the initial perturbations for the integration of the four first-order differential equations starting at the upstream state. Figure 7 shows the values of $ \lambda 2$ for various $M$, $\kappa ,$ and $\alpha $ values with two different ion temperatures. The curves for $\alpha =0$ and $\kappa \u2264 3 / 2$ are marked by the dotted lines in the figure. As indicated, the eigenvalues $ \lambda 2$ increase with increasing ion temperature. Examples of the solitary structures are shown in Figs. 8 and 9 for various $\kappa $ value with $ \alpha 2=0$, $0.01$, $0.05$, $0.1$, and $0.2$ for both cold and warm ions, respectively. In the following figures for various RK cases with the same Mach number, the upstream flow velocity may vary with different cases. For cold ion cases, the model calculations are consistent with the results shown in Fig. 6, which are used to verify the accuracy of the numerical models. Note that the KVD cases of $\alpha =0$ and $\kappa \u2264 3 / 2$ are not shown in Figs. 8 and 9, though there exist smooth profiles for the electric potential and number density but the thermal pressure and temperature are unphysical for those cases. As indicated, for the cases of $\alpha \u22600,$ complete soliton solutions may be constructed for all positive $\kappa $ values in the RK model. However, for small $\kappa $ and $\alpha $ values, the numerical integration becomes difficult to proceed due to the steep structures. The difficulties associated with the numerical integration for very steep structures are intrinsically due to the chaotic natures of the higher-order nonlinear equations for warm fluid models under consideration.^{28,37,38} For cold fluid limit, the set of equations become the second-order forms which may be solved by the Sagdeev's potential method. Nevertheless, the spatial profiles for small $\kappa $ and $\alpha \u223c0$ cases are relatively steep due to the intrinsic singularities for the KVD cases. However, for the cases with $\alpha $ values not too close to $0$, the slopes are less steep and the numerical integrations are smooth with spatially wider profiles. As shown in Figs. 8 and 9, for the same Mach number, the peak values and the widths of the IAS are similar for cold and warm ion cases with the same other parameter values. In the figures, the physical quantities and the spatial coordinates are normalized by the same constants used in Figs. 3 and 4. The calculations shown in the present study have mainly focused on relatively small $\kappa $ and $\alpha $ values which are the parameter regimes not applicable for the standard KVD. The cases of warm ions may impose further numerical difficulties on constructing the equilibrium solutions such that only small values of $ T 0 , i/ T 0 , e M$ are considered in Figs. 8 and 9. In order to see the effects of warmer ions, we have carried out the same calculations for relatively larger $\kappa $ and $ T 0 , i/ T 0 , e M$ shown in Fig. 10, which shows the same tendency as Figs. 8 and 9. In particular, the effects of warm ions are to yield slightly smaller peak values. Consistent with the KdV results, smaller $\kappa $ values yield larger electric potential while smaller $\alpha $ values lead to larger peak potentials. Note that for the same upstream flow velocity, the differences between cold and warm ion cases are relatively significant as compared to the cases with the same $M$. For the applications of IASs to the observed soliton events, it is referred to the paper by Jao and Hau.^{28}

## VI. SUMMARY

In this paper, we have examined the characteristics of electrostatic solitons in nonthermal plasmas with regularized Kappa velocity distributions, characterized by the two parameters, $\kappa $ and $\alpha $, for the hot components (such as the electrons). A systematic study is carried out of the linear acoustic waves, the weakly nonlinear KdV solitons, and the fully nonlinear solutions. The determination of the characteristic speed is essential for analyzing the formation conditions of nonlinear solitons such as the definition of Mach number. The KdV method is adopted to analyze the dynamic evolution of a weakly nonlinear wave and the role of dispersive effects in the formation of acoustic solitons. In the formulation, the relatively cold components (such as the ions) are described by the fluid models with finite temperature. For the special limit of cold ions, the fully nonlinear solutions may be solved based on the Sagdeev's potential method. However, for warm ions, the highly nonlinear set of equations need to be solved based on the numerical integration along with the linear analyses around the fixed points (upstream or downstream states).^{37,38} It is shown that the widths of the solitary structures and the peak electric potential values of the KdV solutions are, respectively, larger and smaller than those obtained from the fully nonlinear models. These effects are more pronounced for the cases with small $\kappa $ and $\alpha $ values. The RK distributions not only may preserve the nonthermal features of the velocity profiles well described observationally by the Kappa functions but also are mathematically and physically correct. In contrast to the standard KVD with singularity and divergence in the temperature and heat flux for small $\kappa $ values, the formulations and the calculations based on the RKD shown in the paper possess no anomalous properties for all $\kappa $ values. Through the regularized parameter, $\alpha $, introduced to the RK functions, the suprathermal tail populations are regularized to prevent the divergence occurring in high moments of the velocity distribution functions such as the thermal pressure and temperature. Note that in both weakly and fully nonlinear models, there exist smooth solutions for the electrical potential and number density for small $\kappa $ values but the thermal pressure and temperatures become unphysical for $\kappa \u2264 3 / 2$ and $\alpha =0$ (KVD cases). Such limitations may be removed by finite $\alpha $ values in the RD distributions as shown in Figs. 3, 4, 6, 8, and 9. For small $\alpha $ values, the density and electrical potential profiles, which are used for comparisons with the observations, are nevertheless similar to the RK results. In contrast to the examples shown in our earlier paper for the KVD cases,^{15} the calculations shown in the present paper are aimed at demonstrating the effects of $\alpha \u22600$ in the RK model for small $\kappa $ values. The developed formulations are applicable for arbitrary ratios of ion to electron temperatures and are, thus, applicable for various space plasma environments.

## ACKNOWLEDGMENTS

This research is supported by the National Science and Technology Council of Taiwan (R.O.C.), National Central University (Grant No. 112-2111-M-008-002), and National Cheng Kung University (Grant Nos. 111-2111-M-006-005-MY2 and 112-2811-M-006-009).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Lin-Ni Hau:** Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Supervision (lead); Validation (equal); Writing—original draft (equal); Writing—review & editing (lead). **Chun-Sung Jao:** Funding acquisition (lead); Project administration (lead); Software (equal); Validation (equal); Visualization (equal); Writing—original draft (equal). **Chun-Kai Chang:** Software (equal); Validation (supporting); Visualization (equal).

## DATA AVAILABILITY

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

## REFERENCES

*Reviews of Plasma Physics*

*Methods in Nonlinear Plasma Theory*

*Advanced Space Plasma Physics*

*Introduction to Plasma Physics and Controlled Fusion*

*Kappa Distributions: From Observational Evidences via Controversial Predictions to a Consistent Theory of Non-Equilibrium Plasmas*

*Kappa Distribution: Theory and Applications in Plasmas*