In a plasma composed of electron populations with different temperatures, an electron acoustic wave (EAW) is generated when inertial cold electrons oscillate against comparatively inertialess hot electrons [1]. They are generally high-frequency electrostatic waves (EWs) with a phase speed between the cold and hot electron thermal speeds relative to the ion plasma frequency. Positive ions do not contribute to the dynamics at these high frequencies; instead, they serve as an evenly distributed charge background, guaranteeing charge neutrality. The phase speed of the EAWs is significantly higher than the thermal speed of a cold electron and significantly lower than the thermal speed of a hot electron. The restoring force is provided by the pressure of the hot electrons, and the inertial effects needed to sustain the EAWs are provided by the cold electrons [1].

EAWs are frequently observed in laboratory plasmas [2–4], as well as in space plasmas like the Earth's bow shock [5–7] and the auroral magnetosphere [8, 9]. Broadband electrostatic noise (BEN) is another example. Spacecraft endeavors have discovered BEN as a common wave phenomena in the plasma sheet boundary layer zone [10, 11]. Bipolar waves, which range in frequency from about 10 Hz to the local electron plasma frequency (about 10 kHz), make up BEN emissions, often referred to as EAWs. This suggests that electron motion rather than ion movement is what causes the emissions [12, 13].

A thermal Maxwellian distribution is commonly used to characterize the two-electron population in plasmas with two electron temperatures [14–17]. Alternatively, a suprathermal electron population in some space and laboratory plasmas exhibits behavior somewhat distinct from a Maxwell distribution. The inverse power law describes the speed at which electrons travel, which is faster than their thermal velocity. We employ a double spectral index distribution, or generalized (r, q) distribution, to describe this suprathermal population [1].

Nonplanar plasma waves have been the subject of extensive research in recent years [18–21]. Sabry along with colleagues [22] studied nonplanar ion acoustic waves (IAWs) in an unmagnetized plasma with two-electron temperature distributions to analyze their modulational instability period, which was absent in planar geometry. Javidan and Pakzad [23] investigated the dynamics of EAWs including superthermal electrons in nonplanar geometries. They discovered that the existence of superthermal particles has a significant impact on the occurrence of solitons in spherical coordinates. El-Labany and his colleagues investigated nonplanar EAWs in unmagnetized plasma with nonthermally distributed hot electrons and demonstrated the effect of different plasma factors on their analytical phase shift [24]. Demiray and El-Zahar investigated nonplanar electron waves having vortex-like hot electron distributions [25]. The key objective of their investigation was to apply the analytical approximate approach to solve the Korteweg–de Vries (KdV) equation and then juxtapose it to the numerical solution. They efficiently rationalized their conclusions, which were consistent with numerical solutions. In the context of the damped nonplanar Korteweg–de Vries (dnKdV) equation, the evolution of dissipative planar, along with nonplanar, acoustic waves in a collisional electronegative complex plasma made up of cold positive ions, Maxwellian negative ions, electrons, and stationary negatively charged dust grains were investigated both analytically as well as numerically by Kashkari and colleagues [26]. An appropriate numerical approach was offered to determine the numerical solution of the dnKdV, since it is recognized not to permit analytical solution. This was accomplished by using the homotopy perturbation method (HPM). In order to analyze the dnKdV equation and properly identify the boundary requirements for the occurrence of acoustic waves, the analytical solution of the KdV equation at a specific/initial time was employed as the starting solution. The HPM was enhanced by noticing that the analytical, along with numerical, outcomes have an excellent agreement as well as an excellent compatibility. The key plasma factors affecting the dissipative nonplanar acoustic wave shape were explored, and it was discovered that the plasma parameters substantially alter the properties of the acoustic waves. The planar KdV equation is unsuitable to explain the motion of nonlinear waves due to the nonplanar geometry. For modeling these waves, the cylindrical KdV (CKdV) equation is a suitable choice in numerous scientific areas such as communications, oceanography, nonlinear optics, and plasma physics. Shan and his coworkers were inspired by the above applications to successfully employ the Bäcklund transformation to the analysis and analytical solution of the CKdV equation [27].

In both space as well as astrophysical plasmas, such as magnetospheres, solar wind, the magnetosheath, the ionosphere, and the interstellar medium, non-Maxwellian velocity distributions have recently been discovered due to a significant deviation from the Maxwellian distribution. The generalized Lorentzian distribution, or kappa, and superthermal tails, together with the flat-top pattern of bow shocks, magnetosheath, and magneto-tails are examples of the velocity distribution deviations [28]. Qureshi along with his colleagues [29] proposed a more effective and appropriate velocity distribution function (VDF) in terms of the double spectral indices r and q. This VDF is recognized in scholarly work as the generalized (r, q) VDF, where parameter r signifies the low-energy particles on the wide shoulder of the velocity curve, whereas parameter q depicts nonthermality on the tail of the velocity curve. In a number of research projects, the (r, q) distribution has been successfully employed to describe density humps as well as depletion nonlinear patterns in electrostatic solitary waves (ESWs) [30]. Using the (r, q) distribution function to analyze the impact of spectral indices on the waves, several investigators have studied the linear and nonlinear evolution characteristics of different waves in plasma physics. In their investigation of the terrestrial lion roars, which were mostly investigated using a bi-Maxwellian distribution, Qureshi and coworkers [31] used the generalized (r, q) distribution and demonstrated that it matched with the observed data for a (r, q) distribution. Periodic patterns and ion acoustic solitons in upper ionospheric plasmas were studied by Sidra and collaborators [32]. To explore the impact of spectral indices on interaction regime, the interaction of dust ion acoustic solitary waves (DIASWs) has been examined for Saturn's rings with (r, q)-distributed electrons having cubic nonlinearity [33]. We will now look at nonplanar EAWs with a generalized (r, q) distribution in a two-population electron plasma, which is a more practical plasma form. Shumaila and her colleagues explore the properties of nonlinear dust-ion acoustic waves (DIAWs) in a magnetized, collisionless dusty plasma employing the generalized (r, q) distribution. They discovered that the dynamics of DIAWs are significantly altered by the double spectral parameters, r and q [28]. The successful application of the generalized (r, q) distribution is supported by numerous investigations as examples [34, 35].

In a homogeneous, tightly coupled dusty plasma, Gogoi and his collaborators performed an experimental investigation recently on the production and evolution of nonplanar dust acoustic solitary waves (DASW) in a capacitively correlated RF argon plasma [36]. To generate nonplanar waves, a small cylindrical pin known as an exciter is utilized. A negative direct current pulse activates it, and it is situated in the center of the dust cloud. Applying the excitation waveform causes the sheath around the pin to spread, generating a dust hole and a diverging cylindrical dust density variation around the pin. A high-speed laser-illuminated camera monitors the density perturbations. The width, Mach number, and amplitude of the perturbations are found as a function of time. The theoretical study predicts that the characteristics of nonplanar DASWs are the same as those of nonplanar ion acoustic solitary waves (IASWs). They also observed that the profiles that came from solving the nonplanar equations numerically with an additional geometrical factor matched the profiles of the wave experiments exactly.

In the present investigation, the nonplanar KdV equation will be solved using the weighted residual technique. This technique consists of two key parts. In the initial part, an approximate solution that is based on the typical behavior of the dependent variable is explored. In many circumstances, the proposed solution is selected to fulfill the boundary criteria. This previously selected solution is subsequently substituted to solve the differential equation. The proposed solution often does not satisfy the differential equation, resulting in an error or a residual that remains. This is due to the proposed solution being only an approximation. Following that, a collection of algebraic equations is developed by eliminating the residual in a typical manner across the entire solution domain. The next part is to solve the collection of equations acquired in the first part while adhering to the provided boundary condition in order to obtain the required approximate solution. Numerous research works have employed new approaches to develop analytical approximations for nonintegrable differential evolution equations [37–43]. These approximations have been used to accurately describe dynamical systems in various scientific domains by calculating their global and residual errors. We believe that, in contrast to these methods, the weighted residual technique also accurately models nonlinear systems in numerous scientific fields, particularly those having nonplanar geometries.

To the best of our information, to date, no has attempted to use the double spectral index distribution function to study nonplanar EAWs. We manage to solve the nonplanar KdV equation, which is a difficult problem without an exact solution. Therefore, one of the key purposes of the present study is to employ the weighted residual approach to numerically analyze the effects of the nonplanar geometrical element and the double spectral indices r and q on the dynamics of ion acoustic waves. In order to achieve this, the nonplanar KdV equation is obtained using the reductive perturbation method (RPM) and the fundamental set of fluid equations in the nonplanar coordinates for collisionless electron–ion plasma with ions and non-Maxwellian electrons. Thereafter, the nonplanar KdV equation's exact analytical solution is determined using the weighted residual approach. Additionally, the weighted residual approach is employed to numerically solve the nonplanar KdV equation, and the impact of pertinent plasma factors on the nonplanar acoustic waves is investigated.

Because of their significance in a variety of physical situations, such as astrophysical circumstances, space plasmas, and laboratory studies, nonplanar EAWs in plasmas are an important field of research. These kinds of waves are distinct from classical planar waves because they take into consideration the consequences of nonplanar geometries, which are frequently found in plasma environments and include spherical and cylindrical as well as toroidal forms The significance of nonplanar EAWs can be emphasized in multiple crucial areas: (1) Understanding wave events in Earth's magnetosphere and solar wind relies upon nonplanar EAWs. Curved geometries are frequently seen in these areas, and the motion of EAWs in such environments can affect the transfer of energy, particle motion, and plasma stability collectively [44]. (2) Nonplanar geometries are commonly observed in the plasma density and magnetic fields of astrophysical jets. Within such circumstances, EAWs have the potential to accelerate particles and produce shocks along with instabilities. (3) Nonplanar geometries result in more complicated wave behavior by adding terms to the governing equations. For an understanding of the nonlinear behavior of plasma perturbations, nonplanar EAWs are very relevant. (4) The focusing influence of curvature can amplify the nonlinearity in nonplanar plasmas, resulting in shock wave generation. Particle acceleration and energy dissipation mechanisms in plasmas are significantly influenced by these shocks. (5) In nonplanar geometries, the dispersion equation for EAWs gets modified, which impacts the waves' phase and group speeds. Identifying plasma properties, like density and temperature, in experimental as well as space plasmas requires an understanding of these variations. (6) Because of their curved geometry, nonplanar EAWs can interact with particles in novel manners. By effectively transferring energy from waves to particles, these interactions can cause particles to accelerate to high energies. This holds especially true in space plasmas, where cosmic ray acceleration is facilitated by these mechanisms. (7) Compared to planar waves, the investigation of nonplanar EAWs involves solving more intricate mathematical equations. In order to account for curvature, these mathematical equations frequently include extra terms, which increases the difficulty of the analytical as well as numerical investigation while simultaneously increasing the richness of physical processes. (8) High-performance computational techniques are needed to effectively model and anticipate the dynamics of plasmas in nonplanar geometries. Nonplanar EAW-based simulations aid in understanding plasma dynamics under realistic conditions, which is essential for theoretical study along with real-world potential uses. (9) Different plasma instabilities, including the electron acoustic instability, have different growth rates that can be altered by nonplanar EAWs. The occurrence of turbulent structures, which are crucial for the movement of particles and energy in plasmas, might result from these instabilities. (10) The observation of nonplanar EAWs in experimental investigations facilitates the verification of theoretical models and offers insights into wave behavior relevant to astrophysical plasmas and space. These kinds of investigations are crucial to verifying the predictions of nonplanar wave theory.

This work investigates EAWs using two different electron populations in nonplanar geometries. The paper's organization is as follows: section 2 presents the basic equations, while section 3 uses the perturbation approach to derive the non-planar KdV equation. Section 4 discusses the solution and outcomes. Section 5 contains the conclusions to the results.

Assume the nonlinear evolution of EAWs in a collisionless unmagnetized plasma containing cold electron fluid, stationary ions, and (r, q)-distributed hot electrons. The continuity and momentum expressions for cold electrons govern the motion of nonlinear, one-dimensional (1D) electron acoustic solitary waves (EASWs) in non-planar geometry: Here, n_c and u_c are the number density and velocity of cold electrons, respectively; e is electronic charge; and m_e is mass of the electron. In contrast, the well-known Poisson's equation is used to find the electrostatic potential φ, which is given by where n_h is the number density of hot electrons and Z_i is the charge number. For planar, cylindrical, and spherical geometry, p = 0, p = 1, and p = 2, accordingly. In the relationship described above, the ion density is taken to remain constant, i.e. n_i = n_i0 = const.; however, a generalized (r, q) velocity distribution function, as shown below, is used to describe hot electrons [29]:where where the thermal speed is denoted as ${v}_{\mathrm{th}}{(=2{T}_{e}/{m}_{e})}^{1/2}$, while Γ represents the standard gamma function. The plasma factors r and q, restored in the limits r = 0 and q → ∞ and r = 0 and q → κ + 1, demonstrate a deviation from Maxwellian and kappa VDF, respectively. For physically relevant distributions, keep in mind that q(r + 1) > 5/2 and q > 1 [29]. Integrating equation (4) in the velocity space regarding cylindrical coordinates givesBy using the definition of the gamma function,we have Let then and v_⊥ = ∞ ⇒ X = ∞ and ${v}_{\perp }=0\Rightarrow X\,=\frac{1}{q-1}{\left(\frac{{v}_{\parallel }^{2}-U}{\beta \left(2{T}_{e}/{m}_{e}\right)}\right)}^{r+1}=a$. So, equation (7) becomes The following expression for the total density of hot electrons is obtained from the above equation using Mathematica software where In the limits r = 0 and q → ∞, we get the Maxwellian velocity distribution (VD) coefficients, i.e.and with r = 0 and q → κ + 1, we get the kappa VD coefficients, i.e.At this stage, every parameter should be rescaled by an appropriate variable to produce a dimensionless set of equations. The normalized continuity, momentum, and Poisson's expressions in 1D are represented by the following expressions where n_c, u_c, and φ are normalized as n = n_c/n_c0, u = u_c/v₀, and Φ = eφ/k_BT_h, and α = n_h0/n_c0 is a nondimensional variable that signifies the ratio of the density of the hot to cold electron constituents. Here, k_B and T_h are the Boltzmann constant and the temperature of hot electrons, respectively, and n_c0, n_h0 are equilibrium densities. The Debye length ${\lambda }_{\,\rm{De}\,}={({k}_{\,\rm{B}\,}{T}_{h}/4\pi {n}_{{\bf{h0}}}{e}^{2})}^{1/2}$ and inverse hot electron plasma frequency ${\omega }_{\,\rm{peh}\,}^{-1}={(4\pi {n}_{{\bf{h0}}}{e}^{2}/{m}_{e})}^{-1/2}$ are used to normalize time (t) and spatial (ρ) parameters.

The nonlinear evolution of nonplanar EAWs moving in the x-axis is investigated using the conventional reductive perturbation strategy. The elongated coordinates, τ = ε^3/2t and $\xi ={\epsilon }^{1/2}\left(\rho -{v}_{p}t\right)$, are introduced using a modest expansion element ε (0 < ε < 1). The perturbation is determined by this tiny expansion element, and the nonplanar EAW's phase speed, v_p, is established by the compatibility requirement. This description leads to an expansion of the dependent parameters n_c, u_c, and Φ:Reduced expressions are derived by applying equation (18) to equations (15)–(17) and collecting the lowest orders of ε. The first-order expressions are as follow:To obtain the algebraic equation of phase speed for linear EAWs with (r, q)-distributed hot electrons, we simultaneously solve the set of first-order perturbed quantity equations given by Through the assembly of ε's next order terms, we obtain expressions for second–order perturbed quantities expressed in terms of first-order perturbed quantities, which are given by Implementing straightforward algebra to the previously described equations yields the cylindrical/spherical KdV equation, which is given by where, for sake of simplicity, ${{\rm{\Phi }}}^{\left(1\right)}={\rm{\Psi }}$, and κ₁ and κ₂ are the nonlinear and dispersion coefficients, respectively, and are expressed by and A single-dimensional planar KdV equation results from setting p/τ = 0 in equation (22). It is the nonplanar cylindrical (p = 1) or spherical (p = 2) geometry that causes the second term (p$\Psi$/2τ) to appear. Equation (22) further demonstrates that the nonplanar geometry impact is smaller for bigger values of τ and crucial when τ → 0. It is evident that double spectral parameters r and q have a significant impact on both the dispersion coefficient κ₁ and the nonlinear coefficient κ₂.

In this part of paper, we will try to present an approximate analytical solution for the nonplanar KdV expression (22) using the amended weighted residual technique. Numerous numerical solutions for these kinds of problems have been provided by researchers [45–47]. In generic form, our obtained evolution expression may be expressed as where N₁ and N₂ represent the nonlinearity and dispersion coefficients, respectively. In the planar situation, when p = 0, the evolution equation (25) becomes When τ is very large, i.e. τ > τ₀, the evolution equation (26) might be considered the limiting case of equation (25), where the ratio (mV/2τ) can be negligible. The solitary wave solution for the evolution equation (26) may thus look like this where c₀ is the fixed wave amplitude and η₀ = d₀(ξ − u₀τ). d₀ and u₀ are given by We shall propose the following type of solution for equation (25), which is based on the solution in equation (27) where η = d(τ)(ξ − u(τ)) and the unknown variables c(τ), d(τ), and u(τ) are expressed as It is observed that there is a formal equivalence between equations (30) and (28). Following the substitution of equation (29) into equation (25) and considering equation (30), the residual term is as follows The residue component R(η, τ) cannot be equal to zero since equation (29) is not an exact solution to equation (25). The residual component R(η, τ) will be multiplied by a weighting function, and the result will be integrated with respect to η, from η = − ∞ to η = ∞, with the ultimate result being set to zero. The weighted residual technique, which is widely used in practical mathematics, served as the model for this procedure. By applying the prior described procedure and employing $\sec {h}^{2}\eta $ as the weighting function, the present study gives the following expression Since the integral $\underset{-\infty }{\overset{\infty }{\int }}\sec {h}^{4}\eta $ d η ≠ 0, the following ordinary differential expression may be found for a(τ)This ordinary differential expression has the following solution In this case, c₀ is fixed and τ₀ corresponds to the time parameter τ ensuring that for τ₀ < τ, the ratio mV/2τ may be neglected. Thus, the approximate analytical solution for the generalized nonplanar KdV equation is represented by The speed of propagation may be determined using For p = 0, the solutions in equations (34)–(36) reduce to those in (28) and (27). By choosing the parameters N₁ and N₂ in the following way, the nonplanar KdV equation (22) may be approximated analytically using the generic formulation: assuming N₁ = κ₁ and N₂ = κ₂, we get Those interested in learning more about the weighted residual technique might refer to the literature [48, 49]. Employing the idea of an approximate analytical solution, the progressive wave solution for the nonplanar KdV expression can be examined. The dissipative nonlinear Schrödinger equation and perturbed KdV have actually been solved analytically using this method in the past [50]. These same results are achieved by applying the inverse scattering method. If p = 0, then we recovered the planar KdV equation and its solution as reported by Khan and his colleagues [1].

The main goals of this portion of the paper are to numerically examine the nonplanar KdV equation and nonplanar features of EAWs double spectral indices (r, q) and different plasma parameters. The wave profile modifies with the plasma variables like, the flattening index r, superthermality index q, the hot to cold electron density ratio (α), and nonplanar geometry element p/τ. Observations of both hot and cold populations of electrons in the Earth's magnetosphere have been recorded by Viking in the auroral area of BEN. There are two types of electron densities in this region: hot (n_h0 = 1.5 − 2.0 cm⁻³) and cold (n_c0 = 0.5 cm⁻³) [44]. In a variety of space plasma settings, where the circumstances are favorable for their creation, EAWs have been observed. The magnetosphere of Earth, the solar wind, and planetary magnetospheres are examples of these settings. We thus use our analysis to apply to Earth's magnetosphere.

Spherical KdV acoustic waves spread outward from a point source with a spherical wavefront. The amplitude decays rapidly as 1/ρ² where ρ is the radial distance, while, on the other hand, cylindrical KdV acoustic waves emerge from a line source and propagate with a cylindrical wavefront. The amplitude falls more slowly than for spherical waves, as 1/ρ. Spherical acoustic waves have greater curvature impacts leading to rapid dispersion, causing the soliton structures to dissipate faster, while cylindrical acoustic waves have poorer curvature impacts that allow for more stable solitons, as the wave spreads over a smaller area. There is a balance between dispersive broadening and nonlinear steepening in both geometries. But there are differences in the curvature factors, which affect the way solitons evolve and remain stable. Solitons become less stable in spherical systems because of the dispersive effects, which are more noticeable as a result of radial expansion. On the other hand, at longer distances, cylindrical waves keep their structure. Energy is distributed over a greater surface area in the case of spherical waves, leading to a more notable amplitude reduction. In contrast, the energy of cylindrical waves is distributed over a cylindrical surface, which causes the amplitude to diminish more slowly.

In our study, for any value of α, r, and q, nonlinear coefficient κ₁ < 0 in the nonplanar KdV expression. Therefore, at the tiny amplitude limit, only rarefactive solitons can exist. Figures 1–9 present the findings and illustrate how the EA solitary wave profiles are altered by spherical and cylindrical coordinates. In order to examine how the plasma variables α (relative density), r, and q (spectral indices), and geometry element affect the propagation of cylindrical and spherical EAWs, we have displayed the potential $\Psi$ versus ξ in figures 1−9 at three distinct τ values for different plasma variables. In the present scenario, we examine τ values of 3, 6, and 9 for both spherical and cylindrical cases.

In plasma physics and statistical mechanics, the width of a distribution typically relates to how spread the distribution is with respect to particle energy or velocity in probability distributions. This width represents the range within which the velocities of most particles are found, and it has a strong connection to the dispersion or variance of the distribution. Plots of the electron acoustic waves for spherical (p = 2) and cylindrical (p = 1) geometries at τ = 3, 6, 9 for τ₀ = 10, α = 4, r = 1, and q = 2 are shown in figure 1. As the value of the time factor τ increases, it is observed that the amplitude of the electron acoustic wave reduces and its width widens. Note that in nonplanar geometries, the slope and strength of the EAWs differ significantly from the planar geometry due to the p/τ factor in equation (22). Enhancing τ to a very large value would cause the nonplanar coordinates to converge to the planar coordinate. Nonplanar waves have a larger surface area than planar waves. As the EAW moves, its energy is scattered over a larger volume, resulting in a decrease in the local amplitude of the wave even while its total energy remains unchanged. In some cases, nonlinear effects may lead the nonplanar EAWs' wave fronts to steepen. This could eventually lead to wave rupturing, which is recognized as the wave ruining coherence and scattering its energy, leading to a decrease in the wave's amplitude (intensity). Solitons maintain their patterns, but progressive waves in nonplanar geometries cannot be compared to them since the amplitudes of EAWs vary with time. It is significant to highlight that, in comparison to the cylindrical electron acoustic waves, the spherical electron acoustic wave have a greater velocity and amplitude.

The temporal evolution of nonplanar EAWs for Maxwellian and kappa VDFs is shown in Figures 2 and 3. The width of Maxwellian nonplanar EAWs is a bit wider than the width of generalized (r, q) nonplanar EAWs, as can be seen by comparing figures 1 and 3. As a result, the Maxwellian VDF has the highest width, the generalized (r, q) VDF has an intermediate width, and the kappa VDF has the least width.

The impact of the distribution's flatness, or spectral index r, on nonplanar EAWs is illustrated in figure 4. While the other plasma parameters, τ = 3, α = 4, and q = 2, remain unchanged, figure 4, shows that enhancing the electron density at lower energy in high phase space density (r) enhances the width of the nonplanar EAWs. Actually, a rise in inertia due to an increase in low-energy electron concentration allows waves to widen and spread. Physically, there are more low-energy electrons available for liaison with EAWs when the electron density at lower energy in high phase space density r is enhanced. This leads to amplified nonlinear impacts, expanding the wave width. The nonplanar EAWs have a wider spatial structure due to an increase in wave–particle interactions caused by the greater population of low-energy electrons. Because it controls the distribution of electron velocities and, thus, the dynamics of wave dispersion and broadening, the phase space density, or r, is crucial.

Figure 5 shows how the nonplanar EAWs are affected by the tail of the distribution, or spectral index q. Figure 5, maintaining the other plasma factors fixed, demonstrates that the width of the EAW is enhanced by enhancing the electron density at high energy in low phase space density (q). From a physical point of view, a hot electron pressure provides the essential force to restore the EAW. The electrostatic restoring force for EAWs becomes weaker as a result of partial charge neutralization, which increases the population of hot electrons. This weakening is anticipated to result in a decrease in the width of nonplanar EAWs. The EAW core's amplitude and that of its surrounding vibrations therefore change as q increases, which is a process similar to growing hot electron pressure. However, in the nonplanar KdV equation, a rise in the dispersion coefficient with q leads to a greater EAW. Therefore, reduced nonthermality (greater q) leads to wider EAWs. In a low phase space density (q) domain, increasing the electron density at high energy results in a greater number of energetic electrons interacting with the wave. Because there are fewer electrons occupying lower energies in this instance, there is less chance of electron trapping. As a result, the wave can continue for a longer period of time since the linear Landau damping is less effective. The EAW width broadens as a result of this inadequate damping because the wave extends out spatially and accumulates more energy with time. It is important to remember that the EAW's amplitude and velocity are directly proportional to each other. Since the spherical EAWs' amplitude is broader than that of cylindrical EAWs, it can be observed that they move quickly than cylindrical EAWs.

In figure 6, the effect of spectral index κ on the nonplanar EAWs is shown. As the value of spectral index κ grows, it is seen that the width of the nonplanar EAWs also increases. The same justification as that for figure 5 applies here. The only difference is that the width of nonplanar EAWs for the generalized (r, q) VDF is wider than the width of nonplanar EAWs for the kappa VDF.

In order to examine the impact of α, which represents the ratio of hot to cold electron density, figure 7 depicts both the cylindrical and spherical EAWs. It is found that the width of EAWs decreases with an increase in the ratio of hot to cold electron density. A larger amount of hot electrons physically causes more Landau damping, which decreases the wave's amplitude and spatial extent. Here, the wave interacts with the energetic hot electrons more intensely, causing the total width of the EAWs to decrease and the damping to happen more rapidly. Waves are usually supported by cold electrons; nevertheless, wave width decreases when hot electrons become dominant.

The impact of the hot to cold electron density ratio (α) on the spherical and cylindrical EAWs is depicted in figures 8 and 9 for kappa and Maxwellian VDFs. This follows the same reasoning as in figure 7. The Maxwellian VDF has the greatest width, followed by the generalized (r, q) VDF in the middle, and the kappa VDF has the lowest, as seen in figures 7, 8, and 9.

We have analyzed 1D (one-dimensional) propagation of EAWs in an unmagnetized collisionless plasma in nonplanar geometry including (r, q)-distributed hot electrons. A KdV equation is obtained in nonplanar geometry that characterizes the temporal evolution of EASWs through the use of the reductive perturbation approach. By taking into account the dependence of plasma parameters on radial coordinates, the KdV equation is obtained. The EA wave shape is highly dependent on double spectral indices r and q, density ratio α, and geometry effects p/τ, as demonstrated by numerical studies. We find that as the time variable τ varies, so does the amplitude of the EAWs. The results of the numerical study demonstrate that the amplitudes of these EAWs decrease with increasing time variable τ. The waves are classified as non-solitons since the wave shapes vary with τ. Furthermore, the spherical EAWs have greater amplitudes than cylindrical EAWs. The width of the EAWs increases with higher values of r and q, but the width of the EAWs decreases with higher values of α, and so the opposite is true. Moreover, we discovered that the single wave structure is greatly influenced by the geometry.

In conclusion, while it is extremely challenging but yet not impractical to observe such nonlinear waves in situ in astrophysical plasmas, the nonlinear and nonplanar outcomes obtained in this work are general. Nonplanar acoustic waves have been proven to exist in laboratory plasma environments from recent experiments [36], in which were found that the characteristics of the acoustic waves match those anticipated by theoretical research on nonplanar acoustic solitary waves (ASWs). They also found that, with the inclusion of a geometrical factor, the experimental wave shapes closely resembled those obtained from the numerical solution of the KdV equation. Thus, we hope that the current study's findings offer novel perspectives on the nonlinear characteristics of nonplanar EA waves in astrophysics, space, and laboratory situations, among other plasma environments with two electron densities.