1 Introduction

The interplay between plasmas and charged dust grains has opened up a new and fascinating research area, that of a dusty plasma. Dusty plasmas are encountered in space and in the laboratory, as mixtures of ordinary plasma particles and charged dust grains. Astrophysical dust is present in the solar system and interstellar environments [1–3] where dust grains come with a whole range of characteristics. A fundamental difference from ordinary plasmas is that dust grains are charged due to interactions with the plasma and radiation environments in which they are submersed. However, dust grains surrounded by ionized media and radiative environments can become highly charged, and thus bring in important contributions arising from electromagnetic interactions. Since both interactions are long range, and the gravitational interaction is much weaker than the electromagnetic interaction, the latter can significantly influence the collective dynamics of self-gravitating extended systems. More specifically, the study of the so-called dusty plasmas, which consist of electrons, ions, and charged dust grains, has received wide attention in recent years [4–9]. Even for grain sizes in the micron range, the mass of an individual grain is typically about 10–12 orders of magnitude larger than the ion mass, and hence the mass of dusty plasmas is essentially contained in the dust component. The presence of such massive dust grains opens up a new, ultralow-frequency regime for the existence of different types of collective modes in dusty plasmas, which do not exist in the usual electron–ion plasmas. In particular, there are three fundamental dusty plasma modes. It has been shown both theoretically and experimentally that the dust charge dynamics introduces new eigenmodes [10–14] where the mass of the dust particle provides the inertia and the pressures of electrons and ions give rise to the restoring force, is one of these. Rao et al [10] first predicted theoretically the existence of extremely low-phase-velocity (in comparison with the electron and ion thermal velocities) dust-acoustic waves in an unmagnetized dusty plasma. The laboratory experiments [13, 14] have conclusively verified this theoretical prediction and have shown some nonlinear features of dust-acoustic waves.

The unbounded planar geometry may not be a realistic situation in laboratory devices and space. Recently, theoretical studies [6, 15, 16] show that the properties of solitary waves in bounded nonplanar cylindrical/spherical geometry are very different from those in unbounded planar geometry. However, most of these investigations concern with the KdV equation, which produces DASWs either in planar or in nonplanar geometry in dusty plasmas. Parametric regime may create a situation for which the nonlinear term of KdV equation A~0 , i.e. A is around 0 but not equal to 0. The nonlinear term around 0 gives rise to infinite large-amplitude structures which break down the validity of the reductive perturbation method. To get rid of this circumstance, i.e. to study finite-amplitude DASWs beyond the KdV limit, one may deduce the other type of nonlinear dynamical equation which can be valid for A ~0 . Recently, many researchers started studying the Gardner (mixed modified KdV equation) or modified Gardner (MG) soliton structures and solutions on different plasma systems [17–22]. Recently, Mannan and Mamum [23] have studied the nonlinear propagation of GSs in a nonplanar four-component dusty plasma. They derived the modified Gardner equation and solve it numerically. They also found that the propagation characteristics of nonplanar dust-acoustic GSs significantly differ from those of planar ones.

It is seen from the above discussions that most of these are composed of Boltzmann distributed electrons and ions and it is well known that the Maxwell distribution is taken to be valid for the macroscopic ergodic equilibrium state. However, Maxwell distribution may be inadequate for describing the long-range interactions in unmagnetized collisionless plasma where the nonequilibrium stationary state exists. The distribution function that can better model such particle velocity distribution function is known as the generalized Lorengian or kappa distribution [24] with functional dependence of the form \(f_0(v) \approx [1+v^2/(k \theta ^2)]^{-(\kappa +1)} \). The spectral index κ is a measure of the slope of the energy spectrum of the superthermal electrons forming the tail of the velocity distribution function. Several observations in astrophysical plasmas, namely, solar wind, auroral zone plasma, magnetosphere [25–39] insist one to follow the superthermal or non-Maxwellian distribution. Moreover the kappa (superthermal) distribution has also been used to analyse and interpret spacecraft data on the Earth’s magnetospheric plasma sheet [32], Jupiter [33] and Saturn [34]. When κ→ ∞, the distribution approaches a Maxwellian distribution [ ≈ exp ( − v 2/θ 2)].

Now in this paper we consider the normalized set of equations composed of negatively charged cold dust fluid, superthermal/non-Maxwellian electrons (represented by kappa distribution) and Boltzmann distributed ions. By using the reductive perturbation method we have derived the modified Gardner equation and solved it numerically.

2 Derivation of MG equation

We consider the nonlinear propagation of nonplanar (cylindrical and spherical) dust-acoustic waves in a three-component collisionless, unmagnetized dusty plasma whose constituents are negatively charged cold dust fluid, superthermal/non-Maxwellian electrons (represented by kappa distribution) and Boltzmann distributed ions. In equilibrium, the charge neutrality condition is n i0 = Z d0 n d0 + n e0, where n i0, n d0 and n e0 are the unperturbed number densities of the ion, dust and electron, respectively. The nonlinear dynamics of dust-acoustic (DA) waves in such nonplanar geometries can be described by the following set of normalized equations:

$$\frac{\partial n_{\rm d}}{\partial t}+\frac{1}{r^\nu}\frac{\partial} {\partial r}(r^\nu n_{\rm d}u_{\rm d})=0,$$
(1)
$$\frac{\partial u_{\rm d}}{\partial t}+u_{\rm d}\frac{\partial u_{\rm d}}{\partial r}=\frac{\partial \phi}{\partial r},$$
(2)
$$\frac{1}{r^\nu}\frac{\partial}{\partial r}\left(r^\nu \frac{\partial \phi}{\partial r}\right)=\rho,$$
(3)
$$\rho=n_{\rm d}+n_{\rm e}-n_{\rm i},$$
(4)

where ν = 0 for 1D planar geometry and ν = 1,2 for nonplanar cylindrical (spherical) geometries, respectively. n d is the dust particle number density normalized by its equilibrium value n d0, u d is the dust fluid velocity normalized by dust-acoustic speed \(C_{\rm d}=({Z_{{\rm d}0}k_{\rm B}T_{\rm i}}/{m_{\rm d}})^{{1}/{2}}\), and ϕ is the electrostatic wave potential normalized by k B T i/e, where m d is the dust mass, k B is the Boltzmann constant, Z d0 is the number of electrons at equilibrium on the dust grain surface and ρ is the net normalized surface charge density. The time and space variables are in units of dust plasma period \(\omega^{-1}_{\rm pd}=({m_{\rm d}}/{4\pi n_{{\rm d}0}Z^{2}_{{\rm d}0}e^2})^{{1}/{2}}\) and the Debye length \(\lambda_{{\rm D}m}=(\frac{k_{\rm B}T_{\rm i}}{4\pi n_{\rm d0}Z_{\rm d0}e^2})^{{1}/{2}}\), respectively. To model the effects of superthermal electrons [40, 41], we have

$$\label{eq5} n_{\rm e}=\frac{\mu}{(1-\mu)}\left[1-\frac{\sigma\phi}{\left(\kappa-\frac{3}{2}\right)}\right]^{-\kappa+({1}/{2})}, $$
(5)

where the real parameter κ measures the deviation from Maxwellian equilibrium (recall that κ > 3/2 is needed for defining a physically meaningful thermal speed). Here the ions are assumed to be Boltzmann’s distributed, with the density

$$\label{eq6} n_{\rm i}=\frac{1}{(1-\mu)}{\rm e}^{-\phi}, $$
(6)

where μ = n e0/n i0, σ = T i/T e, and T e, T i are the electron and ion temperature, respectively.

To study the DAGSs in this three-component dusty plasma model using (1)–(4) by the reductive perturbation technique [42], we first introduce the stretched coordinates [19, 43] as

$$\label{eq7} \zeta=\epsilon(r-v_{\rm p}t),~~~~\tau=\epsilon^{3}t, $$
(7)

where ε is a small parameter (0 < ε < 1) measuring the weakness of the dispersion and v p (normalized by C d) is the phase speed of the perturbation mode, and expand all the dependent variables (viz. n d, u d, ϕ and ρ) in power series of ε:

$$ n_{\rm d}=1+\epsilon n_{\rm d}^{(1)}+\epsilon^2 n_{\rm d}^{(2)}+\epsilon^3 n_{\rm d}^{(3)}+\cdots,$$
(8)
$$ u_{\rm d}=0+\epsilon u_{\rm d}^{(1)}+\epsilon^2 u_{\rm d}^{(2)}+\epsilon^3 u_{\rm d}^{(3)}+\cdots,\label{eq9}$$
(9)
$$ \phi=0+\epsilon \phi^{(1)}+\epsilon^2\phi^{(2)}+\epsilon^3\phi^{(3)}+\cdots,\label{eq10}$$
(10)
$$ \rho=0+\epsilon \rho^{(1)}+\epsilon^2\rho^{(2)}+\epsilon^3\rho^{(3)}+\cdots.\label{eq11} $$
(11)

Now, expressing (1)–(4) in terms of ζ and τ (using eqs (5) and (6)), and substituting (8)–(11) into the resulting equations [(1)–(4) expressed in terms of ζ and τ], one can easily develop different sets of equations in various powers of ϵ. To the lowest order in ϵ, we obtain

$$\label{eq12} u_{\rm d}^{(1)}=-\frac{\phi}{v_{\rm p}},\quad n_{\rm d}^{(1)}=-\frac{\phi}{v_{\rm p}^2},\quad \rho^{(1)}=0, $$
(12)
$$\label{eq13} v_{\rm p}=\sqrt{\frac{(1-\mu)(\kappa-\frac{3}{2})}{\mu\sigma(\kappa-\frac{1}{2})+(\kappa-\frac{3}{2})}}, $$
(13)

where ϕ = ϕ (1). The expression (13) represents the linear dispersion relation for the DA waves propagating in the plasma model under consideration. To the next order in ε, one obtains another set of equations, which, after using (12)–(13), can be simplified as

$$\label{eq14} u_{\rm d}^{(2)}=\frac{\phi^2}{2v_{\rm p}^3}-\frac{\phi^{(2)}}{v_{\rm p}},\quad n_{\rm d}^{(2)}=\frac{3\phi^2}{2v_{\rm p}^4}-\frac{\phi^{(2)}}{v_\mathrm{p}^2}, $$
(14)
$$\label{eq15} \rho^{(2)}=\frac{1}{2}A\phi^2=0, \quad A=\frac{3}{v_{\rm p}^4}+\frac{\mu}{1-\mu}\frac{\sigma^2(\kappa-\frac{1}{2})(\kappa+\frac{1}{2})}{(\kappa-\frac{3}{2})^2}-\frac{1}{1-\mu}. $$
(15)

It is obvious from (15) that A = 0 since \(\phi \ne 0\). The solution of A = 0 yields κ = κ c, where κ c is found from the equation

$$\begin{array}{lll}3\left\{\mu\sigma\left(\kappa-\frac{1}{2}\right)+\left(\kappa-\frac{3}{2}\right)\right\}^2\nonumber\\ \quad+\,(1-\mu)\left\{\mu\sigma^2\left(\kappa-\frac{1}{2}\right)\left(\kappa+\frac{1}{2}\right)-\left(\kappa-\frac{3}{2}\right)^2\right\}=0. \end{array}$$
(16)

It is obvious that (15) is satisfied for κ = κ c. So, for κ around its critical value (κ c), i.e. for |κ − κ c| = ε corresponding to A = A 0, one can express A 0 as

$$\label{eq17} A_0 \simeq s\left(\frac{\partial A}{\partial \kappa}\right)_{\kappa=\kappa_{\rm c}}|\kappa-\kappa_{\rm c}|=s A_\kappa \epsilon, $$
(17)

where

$$\begin{array}{rll} A_\kappa&=&\frac{6}{(1-\mu)^2}\left[\frac{(\mu\sigma+1)\{\mu\sigma(\kappa-\frac{1}{2})+(\kappa-\frac{3}{2})\}}{(\kappa-\frac{3}{2})^2}\right.\nonumber\\ &&\left.\quad\quad\quad\quad\quad-\frac{\left\{\mu\sigma(\kappa-\frac{1}{2})+(\kappa-\frac{3}{2})\right\}^2}{\left(\kappa-\frac{3}{2}\right)^3}\right]\nonumber\\ &&+\,\frac{2}{(1-\mu)}\left[\frac{\left\{\kappa(\mu\sigma^2-1)+\frac{3}{2}\right\}}{(\kappa-\frac{3}{2})^2}\right.\nonumber\\ &&\left.\quad\quad\quad\quad\quad-\,\frac{\left\{\mu\sigma^2(\kappa-\frac{1}{2})(\kappa+\frac{1}{2})-(\kappa-\frac{3}{2})^2\right\}}{\left(\kappa-\frac{3}{2}\right)^3}\right] \end{array}$$

and s = 1 for κ > κ c and s = − 1 for κ < κ c. So, for \(\kappa \ne \kappa_{\rm c}\), one can express ρ (2) as

$$\label{eq18} \rho^{(2)} \simeq \frac{1}{2} s \epsilon A_\kappa \phi^2. $$
(18)

This means that for \(\kappa \ne \kappa_{\rm c}\), ρ (2) must be included in the third-order Poisson’s equation. To the next higher order in ε, one obtains the third set of equations:

$$ \frac{\partial n_{\rm d}^{(1)}}{\partial \tau}-v_{\rm p}\frac{\partial n_{\rm d}^{(3)}}{\partial \zeta}+\frac{\partial u_{\rm d}^{(3)}}{\partial \zeta}+\frac{\partial F_n}{\partial \zeta}+\frac{\nu}{v_p\tau}u_{\rm d}^{(1)}=0 $$
(19)
$$ \frac{\partial u_{\rm d}^{(1)}}{\partial \tau}-v_{\rm p}\frac{\partial u_{\rm d}^{(3)}}{\partial \zeta}+\frac{\partial F_u}{\partial \zeta}-\frac{\partial \phi^{(3)}}{\partial \zeta}=0 $$
(20)
$$\begin{array}{lll} \frac{\partial^2\phi}{\partial\zeta^2}+\frac{1}{2} s A_\kappa \phi^2-n_{\rm d}^{(3)}\nonumber\\ \;\;-\frac{\mu}{(1-\mu)}\left[\sigma\frac{(\kappa-\frac{1}{2})}{(\kappa-\frac{3}{2})}\phi^{(3)}+\sigma^2\frac{(\kappa-\frac{1}{2})(\kappa+\frac{1}{2})}{(\kappa-\frac{3}{2})^2}\phi\phi^{(2)}\right.\nonumber\\ \quad\quad\quad\quad\quad\left.+\,\frac{\sigma^3(\kappa-\frac{1}{2})(\kappa+\frac{1}{2})(\kappa+\frac{3}{2})}{6(\kappa-\frac{3}{2})^3}\phi^3\right]\nonumber\\ \;\;+\frac{1}{(1-\mu)}\left[-\phi^{(3)}+\phi\phi^{(2)}-\frac{\phi^3}{6}\right]=0, \end{array}$$
(21)

where \(F_n=n_{\rm d}^{(1)}u_{\rm d}^{(2)}+n_{\rm d}^{(2)}u_{\rm d}^{(1)}\) and \(F_u=u_{\rm d}^{(1)}u_{\rm d}^{(2)}\). Now, using (12)–(15) and (19)–(21), one finally obtains a nonlinear dynamical equation of the form:

$$\label{eq22} \frac{\partial\phi}{\partial\tau}+\frac{\nu}{2\tau}\phi+\alpha_1\phi\frac{\partial\phi}{\partial\zeta}+\alpha_2\phi^2\frac{\partial\phi}{\partial\zeta} +\alpha_3\frac{\partial^3\phi}{\partial\zeta^3}=0, $$
(22)

where

$$ \alpha_1=\frac{s A_\kappa v_{\rm p}^3}{2}, $$
(23)
$$ \alpha_2=\frac{v_{\rm p}^3}{2}\left[\frac{15}{2v_{\rm p}^6}-\frac{\mu\sigma^3(\kappa-\frac{1}{2})(\kappa+\frac{1}{2})(\kappa+\frac{3}{2})}{2(1-\mu)(\kappa-\frac{3}{2})^3}-\frac{1}{2(1-\mu)}\right], $$
(24)
$$\label{eq25} \alpha_3=\frac{v_{\rm p}^3}{2}. $$
(25)

Equation (22) is the MG equation. The modification is due to the extra term, (ν/2τ)ϕ, which arises due to the nonplanar geometry. We have already mentioned that ν = 0 corresponds to a 1D planar geometry which reduces (22) to a standard Gardner (SG) equation. Equation (22) is not a modified KdV indeed. In fact it contains both \(\phi({\partial\phi}/{\partial\zeta})\) (nonlinear term of KdV) and \(\phi^2({\partial\phi}/{\partial\zeta})\) (nonlinear term of modified KdV) in the framework of nonplanar geometry which is known as modified Gardner equation. The second term gives us the effect of nonplanar geometry. Taking separately the nonlinear third term or fourth term along with the rest terms we get the KdV or modified KdV respectively, in the framework of nonplanar geometry. The nonplanar KdV equation can be obtained by neglecting the fourth term, i.e. when \(\phi^2({\partial\phi}/{\partial\zeta})\) tends to 0. Thus nonplanar KdV equation is a particular case of our derived MG equation and this nonplanar KdV equation can be derived using the lower-order stretched variables, viz. \(\zeta=\epsilon ^{1/2}(r-v_{\rm p}t),~~~~\tau=\epsilon^{3/2}t \) rather than the stretching which we have used.

3 Numerical results and discussions

Now we numerically solve the MG equation (eq. (22)). For this we first analyse the stationary GSs solution of Gardner equation (i.e. with ν = 0). To do this, we first introduce a transformation ξ = ζ − U 0 τ which allows us to write eq. (22), under the steady-state condition, as

$$ \frac{1}{2}\left(\frac{{\rm d}\phi}{{\rm d}\xi}\right)^2+V(\phi)=0, $$
(26)

where V(ϕ) is the pseudopotential given by

$$ V(\phi)=-\frac{U_0}{2\alpha_3}\phi^2+\frac{\alpha_1}{6\alpha_3}\phi^3+\frac{\alpha_2}{12\alpha_3}\phi^4. $$
(27)

It is obvious from eq. (27) that

$$ V(\phi)\left|_{\phi=0}\right.=\left.\frac{\mathrm{d}V(\phi)}{{\rm d}\phi}\right|_{\phi=0}=0, $$
(28)
$$ \left.\frac{{\rm d}^2V(\phi)}{{\rm d}\phi^2}\right|_{\phi=0} < 0. $$
(29)

The conditions of eqs (28) and (29) imply that SW solution of eq. (26) exists if

$$ V(\phi)|_{\phi=\phi_m}=0. $$
(30)

The latter can be solved as

$$ U_0=\frac{\alpha_1}{3}\phi_{m1,2}+\frac{\alpha_2}{6}\phi^2_{m1,2} $$
(31)
$$ \phi_{m1,2}=\phi_m\left[1\mp\sqrt{1+\frac{U_0}{V_0}}\right], $$
(32)

where ϕ m  = − α 1/α 2 and \(V_0={\alpha^2_1}/{6\alpha_2}\). Now, using eqs (27) and (32) in eq. (26), we have

$$ \left(\frac{{\rm d}\phi}{{\rm d}\xi}\right)^2+\gamma\phi^2(\phi-\phi_{m1})(\phi-\phi_{m2})=0, $$
(33)

where γ = α 2/6α 3. The SW solution of eqs (26) and (33) is therefore given by

$$ \phi=\left[\frac{1}{\phi_{m2}}-\left(\frac{1}{\phi_{m2}}-\frac{1}{\phi_{m1}}\right){\rm cosh}^2\left(\frac{\xi}{\delta}\right)\right]^{-1}, $$
(34)

where ϕ m1,2 are given in eq. (32) and SWs width δ is

$$ \delta=\frac{2}{\sqrt{-\gamma \phi_{m1} \phi_{m2}}}. $$
(35)

We now look at eq. (22) with the term (ν/2τ)ϕ, which is due to the effects of nonplanar (cylindrical/spherical) geometry. An exact analytical solution of eq. (22) is not possible. Hence, we have numerically solved eq. (22) and have discussed the effects of cylindrical and spherical geometries on time-dependent DAGSs. The numerical results are plotted in figures 1–8. The initial condition, which we have used in our numerical analysis, is in the form of stationary solution of eq. (22) without the term (ν/2τ)ϕ.

Figure 1
figure 1

The effects of cylindrical geometry on positive DAGSs for κ = 1.6, U 0 = 0.1, μ = 0.5 and σ = 0.1.

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Figure 2
figure 2

The effects of cylindrical geometry on positive DAGSs for κ = 2.5, U 0 = 0.1, μ = 0.5 and σ = 0.1.

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Figure 3
figure 3

The effects of spherical geometry on positive DAGSs for κ = 1.6, U 0 = 0.1, μ = 0.5 and σ = 0.1.

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Figure 4
figure 4

The effects of spherical geometry on positive DAGSs for κ = 2.5, U 0 = 0.1, μ = 0.5 and σ = 0.1.

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Figure 5
figure 5

The effects of cylindrical geometry on negative DAGSs for κ = 2.1, U 0 = 0.1, μ = 0.5 and σ = 0.1.

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Figure 6
figure 6

The effects of spherical geometry on negative DAGSs for κ = 2.1, U 0 = 0.1, μ = 0.5 and σ = 0.1.

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Figure 7
figure 7

The effects of cylindrical geometry on positive DAGSs for κ = 1.6, U 0 = 0.1, μ = 0.5 and σ = 0.3.

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Figure 8
figure 8

The effects of spherical geometry on positive DAGSs for κ = 1.6, U 0 = 0.1, μ = 0.5 and σ = 0.3.

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The numerical solution of eq. (22) shows that for a large value of τ, the cylindrical and spherical solitary waves are similar to planar ones. This is because for a large value of τ (e.g., τ = 30), the term (ν/2τ)ϕ, which is due to the effects of the cylindrical or spherical geometry, is no longer dominant. However, as τ decreases, the term (ν/2τ)ϕ becomes dominant, and both cylindrical and spherical solitary wave structures differ from planar geometry. It is found that as τ decreases, the amplitude of these localized pulses increases. It is also found that the amplitude of the cylindrical or spherical DASWs (for both positive and negative GSs) increases as spectral index κ increases, but the width of the cylindrical or spherical DASWs (for both positive and negative GSs) does not vary significantly as spectral index κ increases. We have have also found that the amplitude of positive DAGSs decreases as σ (ratio of ion temperature to electron temperature) increases in both cylindrical and spherical geometries, but the width of the positive GSs does not vary with σ in both the geometries.

From figures 1 and 2, it is clear that the amplitude of the cylindrical DAGSs (positive) increases and the width of the cylindrical DAGSs (positive) slightly decreases as the spectral index kappa (κ) increases. Also, it is clear from figures 3 and 4, that the amplitude of the spherical DAGSs (positive) increases and the width of the spherical DAGSs (positive) does not sufficiently vary as the spectral index kappa (κ) increases. From these figures it is seen that, the solitary wave profiles of the spherical DAGSs (positive) change significantly compared to the solitary wave profiles of the cylindrical DAGSs (positive) when time τ becomes lower (i.e. from τ = 20 to τ = 5).

From figures 5 and 6, it is seen that the amplitude of the cylindrical DAGSs (negative) and spherical DAGSs (negative) increases as absolute value of τ decreases, respectively. Also, from the above figures it is seen that, the solitary wave profiles of the spherical DAGSs (negative) significantly changes compared to the solitary wave profiles of the cylindrical DAGSs (negative) when time τ becomes lower (i.e. from τ = 20 to τ = 5).

From figures 1 and 7, it is clear that the amplitude of the cylindrical DAGSs (positive) decreases as σ (ratio of ion temperature to electron temperature) increases, and the width of the cylindrical DAGSs (positive) does not vary as σ increases. Also, it is clear from figures 3 and 8, that the amplitude of the spherical DAGSs (positive) decreases as the value of σ increases, and the width of the spherical DAGSs (positive) does not vary as σ increases.

4 Conclusions

We have investigated the nonplanar (cylindrical and spherical) DAGSs in three-component dusty plasma by deriving the MG equation. For this, we consider negatively charged cold dust fluid, superthermal/non-Maxwellian electrons (represented by kappa distribution) and Boltzmann distributed ions. The basic features of nonplanar DAGSs are discussed. It is found that the properties of nonplanar DAGSs (both positive and negative) significantly differ as the spectral index kappa (κ) changes. We have seen that as kappa increases, the amplitude of the DAGSs (positive and negative) increases, but width of the DAGSs does not vary significantly in both cylindrical and spherical geometries. We have also seen that as ratio of ion temperature to electron temperature (σ) increases, the amplitude of the DAGSs (positive) decreases and width of the GSs does not vary in both cylindrical and spherical geometries. In conclusion, we propose that a new experiment may be done on our results to observe such waves and the effects of nonplanar geometry on these waves in both laboratory and space plasma systems.