The propagation features of dust acoustic waves in a threecomponent plasma system composed of regularized Kappa distributed electrons, Maxwellian ions, and dust grains carrying positive charges are investigated. The reductive perturbation technique is employed to derive the KdV equation. A generalized expression for the polarization force is derived and the effect of the polarization force is taken into consideration as well. The bifurcation analysis is used, and the solitary wave solution was investigated. The critical value of the superthermal spectral index κ is introduced at which the solitonic structure turns up from rarefactive to compressive. It is found that in the range $ 0 < \kappa < 2.23$, a rarefactive structure is obtained while the compressive structure appears for $ \kappa > 2.23$. In addition, it is found that by increasing the value of cutoff parameter α, the polarization strength increases too. All the obtained results are helpful to investigate the characteristics of the nonlinear wave propagating in the mesosphere region.
I. INTRODUCTION
Nonlinear waves and their unforeseen evolution with both time and space attract the physicists' interest in recent years because of their ability to interpret many events in space. Dusty plasma is one of the complex nonlinear systems that represent a rich environment of nonlinearity. Many of the observed space environments such as interplanetary space, solar nebulae, and Earth's magnetosphere contain dusty plasma.^{1,2} The existence of large charged dust grains in plasma medium participate in creating new eigen modes, such as dust ion acoustic mode^{3,4} and dust acoustic (DA) mode.^{5,6} The DA mode has been theoretically predicted by Rao et al.^{5} and then approved experimentally by Barkan et al.^{6} Most of the studies are carried out on space regions that contains negative dust^{2,7–10} where the negative dust is existed in some laboratory environments and space regions, such as the upper part of ionosphere or the lower part of magnetosphere.^{11,12} Dust particles, that spreaded in plasma media, acquire either positive or negative charges according to different mechanisms. Dust particles can gain negative charges from the currents of electrons of plasma medium. On the other hand, dust grains can gain positive charge by different physical mechanisms, such as photoelectron, secondary electron emissions, and radiation in the UV/xray spectral regions from nearby sources.^{2,13–16}
A number of studies have been done about positive dust plasma media.^{17–19} Tolba et al.^{17} investigated the DA cnoidal waves in a plasma medium consisting of positive dust grains, Maxwellian electrons, and ions. They found that both positive and negative cnoidal waves can be formed and the ions and electrons streaming velocities do not affect the profile of such waves. Slathia et al.^{18} investigated the evolution of solitons, rogons, and breathers' waves that propagate in a five components plasma consisting of positive dust grains, superthermal electrons, ions, solar wind protons, and electrons. A Korteweg–de Vries (KdV) equation is obtained, and after employing the suitable transformation, they obtained a nonlinear Schrödinger equation (NLSE). They concluded that the superthermality of the charged particles has a drastic effect on the solitons, rogons, and breathers structures.
There are a number of remarkable nonlinear structures that are more informative regarding the nonlinearity of the propagating waves, such as solitons, shock, and periodic waves. Over the past few decades, there are a number of studies that have been done and discussed these structures.^{20–22} Soliton wave solution can be obtained from the KdV equation in the case of small but finitewave amplitude. Solitons are the solitary nonlinear waves that preserve their shape while moving with constant speed for a long distance. The soliton wave is basically a localized stable structure generated as a result of the balance between dispersion properties and the nonlinear wave steepening in plasma medium.
Charged dust grains can suffer considerable electrostatic forces, such as electrostatic force, that generated because of the separation of charges in plasma media. Additionally, the polarization force is produced due to a kind of deformation in the Debye sheath around the dust particles in the background of plasma. Polarization force has significant effects^{23–25} in many space regions and laboratory environments as it helps in explaining various properties of the propagating waves in such region. The polarization force concept was first investigated and mathematically derived by Hamaguchi and Farouki.^{23,24} A lot of theoretical attempts^{10,26–30} have been carried out to examine the effects of polarization force on the structure of dust acoustic waves (DAWs) propagating in various dusty plasma media. Singh et al.^{26} investigated the effect of the polarization force on the propagation of DA periodic waves in dusty plasma medium with negatively charged dust particulates, Maxwellian electrons, and superthermal ions. They found that the polarization force modifies the properties of DA periodic waves as the polarization strength increases the periodic wave amplitude increases as well. Abdikian and Sultana^{27} studied the effect of polarization force DAWs propagating in a plasma medium composed of trapped degenerate electrons, and nonthermal ions. They concluded that the soliton amplitude get smaller by increasing the polarization force. The previous studies have been done using approximations according to the charge type of the proposed dust grains. In highly negatively charged dust grains, the studies commonly use the approximation $ n i T e \u226b n e T i$, while in the case of highly positively charged dust grains, they use $ n e T i \u226b n i T e$. The previous approximations are considered as good approximations. However, always the general expressions are preferable so, in our proposed model, we derive a general expression for polarization force without any approximation. To investigate the full nonlinear structures in any plasma medium, the bifurcation method is employed since it has many advantages: (i) it is a preferable method to distinguish between different nonlinear waves generated in any plasma medium, (ii) it examines the effect of different physical parameters on the characteristics of the propagating nonlinear waves, and (iii) it defines the existence range in which different nonlinear structure can be formed. ElMonier and Atteya^{31} employed the bifurcation theory in investigating a fourcomponent dusty plasma medium composed of cold positively and negatively charged dust particulates, with Maxwellian ions and electrons. They revealed the existence of nonlinear solitonic structure. Ellabany et al.^{32} studied a plasma medium composed of (r, q) distribution electrons and positive and negative ions. They employed the bifurcation method to their proposed system, and they confirmed the existence of Alfvénic solitary and periodic waves.
The nonthermal particles with superthermal tails that departs from the Maxwellian equilibrium are detected in space plasma.^{33,34} These highenergy tails are more appropriate to be modeled by standard kappa distribution (SKD), which was investigated empirically by Vasyliunas.^{35} Recently, the researchers have used to employ the SKD in many planetary and space environments as well as solar wind investigations.^{36–38} Despite the effectiveness of the SKD to model particles in space plasmas, it still has unphysical features that makes its application, somehow, questionable and doubtful. On the top of the issue, the divergent velocity moments for the low values of the spectral index κ hinders our ability to reach to macroscopic description for the plasma system. In SKD, the spectral index κ must fulfill $ \kappa > ( l + 1 ) / 2$ where l is the order of moments. So at the secondorder moment (l = 2), κ should satisfy $ \kappa > 3 / 2$; however, the observations display the existence of harder suprathermal tails which correspond to lower power exponents. Another unphysical feature is the entropy of a certain plasma system and the existence of unphysical particles that have a speed exceeding the speed of light in vacuum. So, the regularized kappa distribution (RKD) has been introduced to resolve such problems^{39,40} as it can be employed to model particles with $ 0 < \kappa < 1.5$. It also removes the nonphysical features of the superluminal particles that appear with SKD.^{41} Furthermore, the RKD introduces a macroscopic description for plasma fluids, keeping all nonthermal characteristics of plasma particles.^{42} A few theoretical investigations have been done using the RKD.^{43–45} Liu^{43} proposed a plasma system which is composed of cold ions and regularizedkappa distribution. He studied the properties of ion acoustic solitary waves through applying Sagdeev method. He revealed that ion acoustic solitary speed has to be larger than the realistic propagation speed of ion acoustic wave. A number of studies have been done to study the ion temperature gradient mode^{46,47} while Zhou et al.^{44} were the pioneers in studying the linear and nonlinear characteristics of ion temperature gradient mode using regularized kappa distributed electrons. They found that the ion temperature gradient mode growth rate decreases by decreasing both κ and the cutoff parameter α.
Most of the studies have been carried out using the classical kappa distribution and an approximated expression for polarization force. So it was motivating for us to employ the RKD to investigate the properties of DAWs generated in the mesosphere region. In addition, a generalized expression for the polarization force is derived. This makes our study more informative and precise due to the physical importance of RKD and the generalized expression for polarization force as discussed through the work.
The present work is organized as follows: Sec. II is devoted to theoretical model, basic equation, and derivation of the generalized polarization force. Section III contains the derivation of the KdV equation. Section IV includes bifurcation analysis and soliton solution while the numerical analysis and discussion are presented in Sec. V. Finally, the last Sec. VI is devoted for conclusions.
II. THEORETICAL MODEL AND BASIC EQUATIONS
A. Expression for regularized polarization force
B. Fluid model
III. DERIVATION OF KDV EQUATION
IV. BIFURCATION ANALYSIS AND SOLITON SOLUTION

$ V ( \Phi ) = d \Phi / d \Phi = 0$ at Φ = 0,

$ d 2 V ( \Phi ) / d \Phi 2 < 0$ at Φ = 0.
V. NUMERICAL ANALYSIS AND DISCUSSION
In this section, we investigate the characteristics of DAWs in a plasma medium with regularized kappa distributed electrons, Maxwellian ions, and positively charged dust particles. The investigations describe some nonlinear phenomena that happens in the mesosphere region where a 5 kmthick layer of the positively charged particles exists.^{58} The ranges of the physical parameters were chosen to fit the mesosphere region as $ n d 0 \u2243 7 \xd7 10 2 \u2009 cm \u2212 3 , \u2009 n e 0 \u2243 2 \xd7 10 5 \u2009 cm \u2212 3 , z d \u2243 100 \u2013 500 , \u2009 T i = 200 \u2013 400 \u2009 Kelvin$ and $ T e = 200 \u2013 400 \u2009 Kelvin$.^{12,58,59} Figure 1(a) shows the variation of speed velocity with the cutoff parameter α. It is depicted that by decreasing the value of α and the phase velocity increases. Then, it is possible to conclude that the velocity of DAWs in the case of RKD is more slower than the case of SKD. Additionally, panel (b) shows that, by increasing the value of regularized electron temperature (T_{e}), the phase speed increases too. This can be interpreted physically as follows: when the temperature increases, the kinetic energy increases and this makes more vibrations, so the probability of increasing the phase velocity increases as well. The nonlinear structure of negative potential is described through Fig. 2 as it shows the variation of $ V ( \Phi )$ vs $\Phi $ for different values of polarization force parameter (R). It is depicted that by decreasing the value of R, and the width and the depth of the potential increase. It is noteworthy that the potential profile is so informative as the potential well width measures the solitary wave amplitude. On the other hand, the depth of potential well reads the steepness of solitary waves. Furthermore, the corresponding phase plane is depicted in Fig. 2(b) where the trajectories express the existence of solitary wave. Such trajectories own a starting (Saddle) point then they circles and return to the same starting point forming a closed separatrix. Mechanically, the pseudoparticle speed is enhanced from zero (Φ = 0) until it reaches the maximum value at the $\varphi $axis and then gradually decreases until reaching the origin again.
Furthermore, the variation of $ V ( \Phi )$ vs $\Phi $ at different values of cutoff parameter α is described through Fig. 3. It shows that by decreasing the value of α, the amplitude and the width of the potential increase. The solid black curve represents the potential well for SKD as α = 0. Starting from the fact that at α = 0 the RKD approaches the SKD, Fig. 3(b) is introduced with α = 0 and shows that by increasing the value of κ, the width and the depth of the potential decrease and this result agrees with the results deduced by.^{60}
The soliton structure is also investigated. In panel 4(a), it can be observed that by decreasing R the amplitude of the soliton increases while the width decreases slightly. Additionally, when the polarization force is neglected, there is a notable increment in the soliton amplitude. This is due to the fact that the polarization force often leads to charge separation and generating electric fields within the plasma medium. Such electric fields can affect the soliton amplitude. When the polarization force is neglected, the amplitude of solitons may decrease as the mechanism that enhances and sustains the soliton structure is not fully considered. So neglecting the polarization effect leads to inaccurate results. On the other hand, panel 4(b) shows that by increasing the value of σ, the soliton amplitude increases sharply while there is a slight increment in the amplitude by increasing the value of α. Furthermore, Fig. 4(c) shows that there is a rarefactive soliton structure in the range $ 0 < \kappa < 2.23$ while the compressive structure appears in the range of $ \kappa > 2.23$. The effect of varying κ on the soliton profile is monitored as well via Fig. 5(a). This figure shows that the amplitude decreases by increasing the value of κ while the width increases slightly.
Furthermore, Fig. 5(b) shows that the soliton width decreases sharply by increasing the value of σ and increases by increasing the value of α. Additionally, Fig. 5(c) assures the existence of both compressive and rarefactive soliton as the polarity changes at a certain value of κ. Generally, the soliton waves can be structured during balancing the nonlinear properties (represented by the coefficient P) with the dispersion properties (represented by the coefficient Q) in the plasma medium.
Figure 6(a) shows the relation between the polarization force parameter (R) and the number of unperturbed electrons $ n e 0$. The figure shows that by increasing $ n e 0$, the polarization force increases too. Physically, it can be interpreted as follows: increasing number of electrons causes an increment in the charges and hence increasing the deformation in Debye sheath around particles. This, in turn, helps in increasing the polarization force. Furthermore, Fig. 6(b) shows that by increasing both κ and α, the polarization force increases too. This means that by increasing the energy spectrum hardness of the suprathermal particles that are occupying the tail of the distribution function, the polarization force increases too.
Figure 7 shows the dependence of soliton energy on R and σ. It indicates that the energy carried by soliton waves decreases by increasing both R and σ. This means that the soliton energy is depleted due to approaching the polarization force magnitude from the electrical force, i.e., decreasing the net force acting on the dust particulates. This result agrees with that obtained by Mayout et al.^{61} Moreover, the electric field profile is displayed in Fig. 8. It is obvious that the electric field profile shows a bipolar signature. Additionally, the figure displays that decreasing the value of α leads to an increase in the electric force.
VI. CONCLUSIONS
The characteristics of DAWs propagating in a plasma medium composed of regularized Kappa distributed electrons, Maxwellian ions, and positively charged dust grains are investigated. The polarization force effects are considered too. The reductive perturbation technique is employed to obtain the KdV equation, and the corresponding Sagdeev potential equation is derived as well. Here are the main results.

It is found that the width and the amplitude of the potential decrease by increasing the value of R.

In the limit α = 0, the Sagdeev potential profile is studied. By taking this limit, the electron distribution approaches the kappa distribution and it turns out that increasing the superthermality of electrons causes a decrement in the width and the amplitude of the potential profile.

When the polarization strength is enhanced, the soliton amplitude gets decreasing.

It is found that the soliton width gets wider by increasing both of κ and α. On the other hand, there is a sharp decrement in the width by increasing σ

The polarization strength increases by increasing the values of $ n e 0$, κ and α.
Finally, our findings contribute to interpret the properties of the propagated DAWs in the space especially the mesosphere region where the micrometersized positive dust grains are existed and consequently the polarization force are existed too.^{58,62,63} In the future, we plan to consider the magnetic field effects.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
A. A. ElTantawy: Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). W. F. ElTaibany: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Metholodogy (equal); Supervision (equal); Validation (equal); Visualization (equal). S. K. ElLabany: Conceptualization (equal); Methodology (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). A. M. Abdelghany: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Supervision (equal); Validation (equal); Visualization (equal).
DATA AVAILABILITY
The data that support the findings of this study are available upon request from the authors.
Appendix: THE MATHEMATICAL DETAILS CONCERNING THE SOLITON AND REGULARIZED KAPPA DISTRIBUTION
More details about soliton solution