1 Introduction

Space-based observations reveal the presence of populations of plasma particles in space and astrophysical environment which are not in thermal equilibrium. This deviation of the plasma particles from thermal equilibrium is in general well explained with the help of wave–particle interactions. Such type of particles is common in the auroral region of Earth’s magnetosphere, planetary magnetosphere, solar corona, stellar corona, and solar wind. [1,2,3]. The velocity distribution functions (VDF) of such particles are observed to be quasi-Maxwellian up to the mean thermal velocity with non-Maxwellian superthermal tails [4]. It is already observed that the Kappa distribution function is very convenient to model such velocity distribution since it fits both the thermal as well as superthermal part of the observed energy spectra. However, in some other works, Cairns distribution and Tsallis distributions [5, 6] are also used to explain velocity distribution of such particles. Till date, different works have been carried out [7,8,9,10] to study the electrostatic as well as electromagnetic nonlinear excitations (solitons, double layers, and shocks) in the presence of these superthermal particles.

The generalized Lorentzian or the Kappa distribution function was shown for the first time by Vasyliunas [11] and is expressed as;

$$ f^{\kappa } \left( v \right) = \frac{n}{{2\pi \left( {\kappa v_{\kappa }^{2} } \right)^{3/2} }}\frac{{\varGamma \left( {\kappa + 1} \right)}}{{\varGamma \left( {\kappa - 1/2} \right)\varGamma \left( {3/2} \right)}}\left( {1 + \frac{{v^{2} }}{{\kappa v_{\kappa }^{2} }}} \right)^{{ - \left( {\kappa + 1} \right)}} $$
(1)

Here, \( \varGamma \left( x \right) \) is a Gamma function, \( v_{\kappa } \) is the effective thermal velocity and is related to the usual thermal velocity \( \left( {V_{\text{t}} = \sqrt {T/m} } \right) \) by \( v_{\kappa } = \left( {\sqrt {\left( {2\kappa - 3} \right)/\kappa } } \right)V_{\text{t}} \). \( m \) is the mass of the particle with characteristic kinetic temperature (i.e., the temperature of the equivalent Maxwellian with the same average kinetic energy) \( T \). The parameter \( \kappa \) is the spectral index, which is a measure of the degree of superthermal or the slope of the energy spectrum of the superthermal particles forming the tail of the velocity distribution function. It must take a value higher than the critical value, \( \kappa_{\text{critical}} = 3/2 \). At the critical value of the spectral index, the distribution function collapses, and the equivalent temperature is not defined. On the other hand, in the limit, \( \kappa \to \infty \) the Kappa distribution transforms to the Maxwellian. Thus, the spectral index can have values \( 3/2 < \kappa \le \infty \). Low values of \( \kappa \) represent more superthermal particles [12]. The Kappa distribution (Eq. 1) leads to a number density \( n(r) \) decreasing as a power law with the radial distance \( r \) as shown by the expression given below [12]:

$$ n\left( r \right) = n_{0} \left( r \right)\left[ {1 + \frac{R\left( r \right)}{{\kappa v_{\kappa }^{2} }}} \right]^{ - \kappa + 1/2} $$
(2)

\( R\left( r \right) \) being the potential energy containing the effects of the gravitation, the electrostatic, and the centrifugal force.

Space and astrophysical plasma may contain dust particles of different sizes, masses, and charges. Dust particles are charged due to local electron and ion currents. Its charge varies because of the change of the parameters such as temperatures, densities, and potential. So, the dust charge variation affects the collective motion of the plasma. Although to deal with real situations one must consider the mass and charge distribution of dust particles nevertheless, for simplicity, it is considered that mass and charge of the dust particles remain constant throughout. One may also consider a particular case of tiny dust particles having fixed dust charge Zd of the order of 103–104 with dust to ion mass ratio 103–104 [13,14,15]. Under these circumstances, it is appropriate to include the dust dynamics along with the ion inertia.

It is well known that dust ion acoustic (DIA) waves and dust acoustic (DA) waves are two distinct normal modes of unmagnetized dusty plasma. The existence of DA modes was first predicted theoretically by Rao et al. [16], and that of DIA modes was predicted by Shukla and Silin [17]. In case of DA wave, the inertialess electrons and ions provide the restoring force while the dust mass provides the inertia. The phase velocity of this wave is much smaller than the ion acoustic velocity. On the other hand, in case of DIA wave the restoring force comes from the electron pressure while the dust mass and ion mass provide the inertia. The phase velocity of this wave is higher than both ion and dust thermal velocities. Also by treating the electrons as inertialess and ions as inertial in the background of immobile dust, DIA modes can be achieved [18, 19]. The existence of these modes was also proved via laboratory experiments [20, 21]. The presence of different nonlinear coherent structures in DIA mode was already predicted theoretically [22,23,24] and verified experimentally [25].

The essential condition that leads to the formation of a double layer is the balance between nonlinearity and dissipation [19, 26]. In fluid, the dissipation comes from kinematic viscosity. However, in case of dusty plasma Landau damping, dust charge fluctuation, viscosity, and a collision between plasma particles may be the source of dissipation. The DIA shock waves in a dusty plasma associated with a single type of thermal electron were observed by Nakamura et al. [25]. Very recently, a theoretical investigation has been made by Ferdousi et al. [27] on the characteristics of dust acoustic shock waves (DASW) in an unmagnetized multi-ion dusty plasma. They assumed a plasma system composed of arbitrarily charged inertial dust, Boltzmann distributed negatively charged heavy ions, positively charged light ions, and superthermal electrons. They showed the existence of both positive and negative shock waves.

In this work, we focus on the study of small-amplitude dust ion acoustic double layer in multi-component plasmas. The Sagdeev potential method for small-amplitude regime has been used to study the existence of such double layer under the influence of superthermal electrons and dust particles. The variation of the amplitude of the double layer with different parameters has also been discussed. Moreover, in this work we consider the dust particles to be mobile along with the positive ion and come to an important conclusion that the dust density can alter the polarity of the double layer. Though for simplicity, we have neglected many important factors, but this study can help us in understanding nonlinear coherent structures in space and the astrophysical environment in simplest possible form.

This paper is organized as follows. The basic equations are presented in Sect. 2, and the corresponding normalizations have been done in Sect. 3. The double-layer solution is derived in Sect. 4. Results are analyzed in Sect. 5. Double-layer solution with variable dust charge is derived in Sect. 6, and a brief conclusion is provided in Sect. 7.

2 Model equations

The nonlinear dynamics of the proposed DIA wave whose phase velocity is much larger than ion and dust thermal velocities and smaller than that of electrons is governed by the following equations (Eqs. 3–9). Dust particles are considered to be of micron size so that ion to dust mass ratio \( \left( {m_{\text{i}} /m_{\text{d}} } \right) \) is not very small. Under such condition, both ion and dust particles can be assumed as cold fluids. Here, the fluid equations are considered as the basic equations for ion and dust particles. The problem adopted here is one-dimensional and planar.

The ion response is described by;

$$ \frac{{\partial n_{\text{i}} }}{\partial t} + \frac{\partial }{\partial x}\left( {n_{\text{i}} v_{\text{i}} } \right) = 0 $$
(3)
$$ \frac{{\partial v_{\text{i}} }}{\partial t} + v_{\text{i}} \frac{{\partial v_{\text{i}} }}{\partial x} = - \frac{e}{{m_{\text{i}} }}\frac{\partial \phi }{\partial x} $$
(4)

Equations (3) and (4) are the continuity and momentum equations for the ion. Here, \( n_{\text{i}} \), and \( v_{\text{i}} \) stand for number density and velocity of the positive ion with mass \( m_{\text{i}} \), \( e \) is the electric charge, and \( \phi \) is the electric potential.

Similarly, the dust dynamics is governed by;

$$ \frac{{\partial n_{\text{d}} }}{\partial t} + \frac{\partial }{\partial x}\left( {n_{\text{d}} v_{\text{d}} } \right) = 0 $$
(5)
$$ \frac{{\partial v_{\text{d}} }}{\partial t} + v_{\text{d}} \frac{{\partial v_{\text{d}} }}{\partial x} = \frac{{Z_{\text{d}} e}}{{m_{\text{d}} }}\frac{\partial \phi }{\partial x} $$
(6)

In the above equations, \( n_{\text{d}} \) and \( v_{\text{d}} \) are the number density and velocity of dust particles of mass \( m_{\text{d}} \). \( Z_{\text{d}} \) is the fixed dust charge or the number of electrons resides over a dust particle. The theory behind the formation of double layer demands dissipation of energy. When the dissipated energy is balanced by the nonlinearity, the double layer is generated. Viscosity, collision among plasma particles, and Landau damping are the primary source of dissipation in plasma. However, the dissipation is not always necessary. The electrons we have considered here are superthermal, and due to that, the ion thermal velocity is much smaller than that of the electron. Thus, ions cannot cross the potential barrier created by the electrons and reflect back and are trapped. So, in Eqs. (4) and (6), no term for dissipation is included. The densities of the superthermal electrons can be derived from generalized Lorentzian or Kappa distribution. On performing volume integration of the Kappa distribution, one can get the number density of the superthermal particles. The densities of the Kappa-distributed cold and hot component of electrons are described by;

$$ n_{\text{ec}} = n_{0c} \left( {1 - \frac{e\phi }{{\left( {\kappa_{\text{c}} - \frac{3}{2}} \right)T_{\text{ec}} }}} \right)^{{\left( { - \kappa_{\text{c}} + \frac{1}{2}} \right)}} $$
(7)
$$ n_{\text{eh}} = n_{0h} \left( {1 - \frac{e\phi }{{\left( {\kappa_{\text{h}} - \frac{3}{2}} \right)T_{\text{eh}} }}} \right)^{{\left( { - \kappa_{\text{h}} + \frac{1}{2}} \right)}} $$
(8)

Here, \( n_{\text{ec}} \) and \( n_{\text{eh}} \) are the densities of the cold and hot species of superthermal electrons with their equilibrium densities \( n_{{0{\text{c}}}} \) and \( n_{{0{\text{h}}}} \). Their respective spectral indices are symbolized by \( \kappa_{\text{c}} \) and \( \kappa_{\text{h}} \), and their temperatures are \( T_{\text{ec}} \) and \( T_{\text{eh}} \) in energy scale. Generally, due to negligible inertia and higher thermal energy than that of ions, electrons settle down very quickly in thermal equilibrium by losing some energy via collision among themselves. However, in a collisionless scenario where number density is small and the temperature is high, the energetic particles cannot settle down to thermal equilibrium. Instead, a large number of particles are observed to have velocities higher than their average thermal velocity. Under such circumstances, electrons seem to obey Kappa distribution. The typical order of electron density considered here is ~ 105 m−3. The temperatures for the cold and the hot component of electrons are \( T_{\text{ec}} < 100 \) eV and \( T_{\text{eh}} \) ~ 100–10 keV.

The whole set of equations are closed through the Poisson’s equation;

$$ \frac{{\partial^{2} \phi }}{{\partial x^{2} }} = \frac{e}{{\varepsilon_{0} }}\left( {n_{\text{ec}} + n_{\text{eh}} + Z_{\text{d}} n_{\text{d}} - n_{\text{i}} } \right) $$
(9)

Here, \( \varepsilon_{0} \) is the permittivity of free space.

3 Normalization and scaling

The above set of equations have to be normalized using suitable normalization parameter in order to scale them down to plasma scale. The appropriate normalization parameters used here are:

\( N_{\text{s}} = \frac{{n_{\text{s}} }}{{n_{{{\text{i}}0}} }} \), \( V_{\text{s}} = \frac{{v_{\text{s}} }}{{c_{\text{d}} }} \), \( \mu_{s} = \frac{{n_{s0} }}{{n_{i0} }} \), \( \varphi = \frac{e\phi }{{T_{\text{ef}} }} \), \( x_{\text{N}} = \frac{x}{{\lambda_{\text{D}} }} \), \( t_{\text{N}} = \frac{t}{{\omega_{\text{pd}}^{ - 1} }} \)\( c_{\text{d}} = \sqrt {\frac{{T_{\text{ef}} }}{{m_{\text{d}} }}} \), \( \lambda_{\text{D}} = \sqrt {\frac{{\varepsilon_{0} T_{\text{ef}} }}{{n_{{{\text{i}}0}} e^{2} }}} \), \( \omega_{\text{pd}} = \sqrt {\frac{{n_{{{\text{i}}0}} e^{2} }}{{\varepsilon_{0} m_{\text{d}} }}} \), \( \sigma_{\text{s}} = \frac{{T_{\text{ef}} }}{{T_{\text{s}} }} \), \( T_{\text{ef}} = \frac{{n_{{{\text{e}}0}} T_{\text{ec}} T_{\text{eh}} }}{{n_{\text{ec}} T_{\text{eh}} + n_{\text{eh}} T_{\text{ec}} }} \).

Here, \( s \) is used to indicate individual charged species. \( n_{{{\text{i}}0}} \) is the equilibrium ion density and is considered ~ 105 m−3 in this work. All velocities are normalized with the dust acoustic velocity \( c_{\text{d}} \), and space and time coordinates are normalized by Debye length \( \left( {\lambda_{\text{D}} } \right) \) and inverse plasma frequency \( \left( {\omega_{\text{pd}}^{ - 1} } \right) \), respectively. \( T_{\text{ef}} \) is the effective electron temperature. Following normalization with the above parameters, the above set of equations lead to a new set of normalized equations and are:

$$ \frac{{\partial N_{\text{i}} }}{{\partial t_{\text{N}} }} + \frac{\partial }{{\partial x_{\text{N}} }}\left( {N_{\text{i}} V_{\text{i}} } \right) = 0 $$
(10)
$$ \frac{{\partial V_{\text{i}} }}{{\partial t_{\text{N}} }} + V_{\text{i}} \frac{{\partial V_{\text{i}} }}{{\partial x_{\text{N}} }} = - \frac{1}{\delta }\frac{\partial \varphi }{{\partial x_{\text{N}} }} $$
(11)
$$ \frac{{\partial N_{\text{d}} }}{{\partial t_{\text{N}} }} + \frac{\partial }{{\partial x_{\text{N}} }}\left( {N_{\text{d}} V_{\text{d}} } \right) = 0 $$
(12)
$$ \frac{{\partial V_{\text{d}} }}{{\partial t_{\text{N}} }} + V_{\text{d}} \frac{{\partial V_{\text{d}} }}{{\partial x_{\text{N}} }} = Z_{\text{d}} \frac{\partial \varphi }{{\partial x_{\text{N}} }} $$
(13)
$$ N_{\text{ec}} = \mu_{\text{c}} \left( {1 - \frac{{\sigma_{\text{c}} \varphi }}{{\kappa_{\text{c}} - \frac{3}{2}}}} \right)^{{\left( { - \kappa_{\text{c}} + \frac{1}{2}} \right)}} $$
(14)
$$ N_{\text{eh}} = \mu_{\text{h}} \left( {1 - \frac{{\sigma_{\text{h}} \varphi }}{{\kappa_{\text{h}} - \frac{3}{2}}}} \right)^{{\left( { - \kappa_{\text{h}} + \frac{1}{2}} \right)}} $$
(15)
$$ \frac{{\partial^{2} \varphi }}{{\partial x^{2} }} = \left( {N_{\text{ec}} + N_{\text{eh}} + Z_{\text{d}} N_{\text{d}} - N_{\text{i}} } \right) $$
(16)

In Eq. (11), \( \delta \) is the ion to dust mass ratio (\( m_{\text{i}} /m_{\text{d}} \)). The quasineutrality condition demands \( \mu_{\text{c}} + \mu_{\text{h}} + \mu = 1 \), where \( \mu = Z_{\text{d}} n_{{{\text{d}}0}} /n_{{{\text{i}}0}} \) is the normalized equilibrium dust density.

4 Double-layer solution

To get a steady-state solution, we transform the independent variable \( x_{\text{N}} \) and \( t_{\text{N}} \) into another set of variables \( \xi = x_{\text{N}} - Mt_{\text{N}} \), \( \tau = t_{\text{N}} \) where \( M = V/c_{\text{d}} \) is the Mach number with \( V \) being the speed of the nonlinear structure with respect to an inertial frame. In this new frame where the nonlinear structure is stationary \( \left( {\partial /\partial \tau = 0} \right) \) and all variables tend to their equilibrium values at \( \xi \to \pm \infty \), by integrating Eqs. (10–13) the normalized ion and dust densities have been calculated and are

$$ N_{\text{i}} = \left( {1 - \frac{2\varphi }{{\delta M^{2} }}} \right)^{{ - \frac{1}{2}}} $$
(17)
$$ N_{\text{d}} = \left( {1 + \frac{{2Z_{\text{d}} \varphi }}{{M^{2} }}} \right)^{{ - \frac{1}{2}}} $$
(18)

After substituting the densities in Poisson’s equation and expanding in Taylor series expansion (for \( \varphi < < 1 \)), the normalized form of Poisson’s equation can be expressed as;

$$ \frac{{\partial^{2} \varphi }}{{\partial \xi^{2} }} = B_{1} \varphi + B_{2} \varphi^{2} + B_{3} \varphi^{3} = - \frac{\partial V\left( \varphi \right)}{\partial \varphi } $$
(19)

Here, \( V\left( \varphi \right) \) is the Sagdeev potential.

Integrating Eq. (19), the energy law \( \left( {\frac{1}{2}\left( {\frac{{{\text{d}}\varphi }}{{{\text{d}}\xi }}} \right)^{2} + V\left( \varphi \right) = 0} \right) \) can be obtained. Further simplification implies;

$$ \frac{1}{2}\left( {\frac{\partial \varphi }{\partial \xi }} \right)^{2} = \frac{1}{2}B_{1} \varphi^{2} + \frac{1}{3}B_{2} \varphi^{3} + \frac{1}{4}B_{3} \varphi^{4} = - V\left( \varphi \right) $$
(20)

where

$$ B_{1} = \frac{{\mu_{\text{c}} \sigma_{\text{c}} \left( {\kappa_{\text{c}} - \frac{1}{2}} \right)}}{{\left( {\kappa_{\text{c}} - \frac{3}{2}} \right)}} + \frac{{\mu_{\text{h}} \sigma_{\text{h}} \left( {\kappa_{\text{h}} - \frac{1}{2}} \right)}}{{\left( {\kappa_{\text{h}} - \frac{3}{2}} \right)}} - \frac{{\mu Z_{\text{d}} }}{{M^{2} }} - \frac{1}{{\delta M^{2} }} $$
(21)
$$ B_{2} = \frac{{\mu_{\text{c}} \sigma_{\text{c}}^{2} \left( {\kappa_{\text{c}} - \frac{1}{2}} \right)\left( {\kappa_{\text{c}} + \frac{1}{2}} \right)}}{{2\left( {\kappa_{\text{c}} - \frac{3}{2}} \right)^{2} }} + \frac{{\mu_{\text{h}} \sigma_{\text{h}}^{2} \left( {\kappa_{\text{h}} - \frac{1}{2}} \right)\left( {\kappa_{\text{h}} + \frac{1}{2}} \right)}}{{2\left( {\kappa_{\text{h}} - \frac{3}{2}} \right)^{2} }} + \frac{{3\mu Z_{\text{d}}^{2} }}{{2M^{4} }} - \frac{3}{{2\delta^{2} M^{4} }} $$
(22)
$$ B_{3} = \frac{{\mu_{\text{c}} \sigma_{\text{c}}^{3} \left( {\kappa_{\text{c}} - \frac{1}{2}} \right)\left( {\kappa_{\text{c}} + \frac{1}{2}} \right)\left( {\kappa_{\text{c}} + \frac{3}{2}} \right)}}{{6\left( {\kappa_{\text{c}} - \frac{3}{2}} \right)^{3} }} + \frac{{\mu_{\text{h}} \sigma_{\text{h}}^{3} \left( {\kappa_{\text{h}} - \frac{1}{2}} \right)\left( {\kappa_{\text{h}} + \frac{1}{2}} \right)\left( {\kappa_{\text{h}} + \frac{3}{2}} \right)}}{{6\left( {\kappa_{\text{h}} - \frac{3}{2}} \right)^{3} }} - \frac{{5\mu Z_{\text{d}}^{3} }}{{2M^{6} }} - \frac{5}{{2\delta^{3} M^{6} }} $$
(23)

To get the double-layer solution, the Sagdeev potential function \( V\left( \varphi \right) \) must satisfy following conditions;

  1. (i)

    \( V\left( \varphi \right) = \frac{\partial \varphi }{\partial \xi } = 0 \) at \( \varphi = 0 \), \( \pm \psi \).

  2. (ii)

    \( V^{\prime}\left( \varphi \right) = 0 \) at \( \varphi = 0 \), \( \pm \psi \).

  3. (iii)

    \( V^{\prime\prime}\left( \varphi \right) < 0 \) at \( \varphi = \psi \); here, primes denote derivatives of \( V\left( \varphi \right) \) with respect to \( \varphi \).

Following the above conditions, finally the double-layer solution is derived;

$$ \varphi = - \frac{\psi }{2}\left[ {1 + \tanh \left( {\sqrt {\frac{{B_{3} }}{8}} \psi \xi } \right)} \right]\;{\text{with}}\;{\text{amplitude}}\;\psi \;{\text{and}}\;{\text{width}}\;\left( {\sqrt {8/B_{3} } } \right)/\psi $$
(24)

From Eq. (24), it is observed that the electrostatic double layer only exists when \( B_{3} \) is positive. The negative sign in front of the amplitude in the expression of the potential indicates the existence of negative (rarefactive) double layer.

Again, expanding the right-hand side of the Poisson’s equation in Taylor series expansion (\( \varphi < < 1 \)) and considering only up to the second-order term of \( \varphi \), the soliton solution is derived;

$$ \varphi = - \psi \text{sech}^{2} \left( {\sqrt {\frac{{B_{2} \psi }}{6}} \xi } \right),\;{\text{with}}\;{\text{amplitude}}\;\psi \;{\text{and}}\;{\text{width}}\;\sqrt {6/\left( {B_{2} \psi } \right)} . $$
(25)

Here, the parameters \( \mu_{\text{c}} \), \( \mu_{\text{h}} \), \( \sigma_{\text{c}} \), \( \sigma_{\text{h}} \), \( \kappa_{\text{c}} \), and \( \kappa_{\text{h}} \) specify the precise composition of the plasma and can be assumed to be given. However, the value of \( M \) cannot be pre-assumed and is to be determined so that nonlinear structures can exist. From the third of the conditions for the formation of a double layer (as already mentioned), i.e., \( V^{\prime\prime}\left( \varphi \right) \le 0 \) one can determine \( M \). Physically, \( V^{\prime\prime}\left( \varphi \right) \le 0 \) or the proper convexity condition ensures that the nonlinear structures are supersonic and this yields the minimal value of \( M \) for their existence. Thus,

$$ M \ge M_{\text{s}} = \left[ {\frac{{\mu Z_{\text{d}} + \frac{1}{\delta }}}{{\frac{{\mu_{\text{c}} \sigma_{\text{c}} \left( {2\kappa_{\text{c}} - 1} \right)}}{{\left( {2\kappa_{\text{c}} - 3} \right)}} + \frac{{\mu_{\text{h}} \sigma_{\text{h}} \left( {2\kappa_{\text{h}} - 1} \right)}}{{\left( {2\kappa_{\text{h}} - 3} \right)}}}}} \right]^{{\frac{1}{2}}} $$
(26)

Here, \( M_{\text{s}} \) which satisfies \( V^{\prime\prime}\left( {0,M_{\text{s}} } \right) = 0 \) is the normalized acoustic velocity in the plasma. So, the ratio \( \left( {M/M_{\text{s}} } \right) \) is the true Mach number in the system.

5 Results and discussions

The existence, nature, and size of a double layer in the proposed plasma model are examined here. Along with that, the effect of the superthermal electrons on the property of the double layer is also investigated. The potential profile of the double layer and its corresponding Sagdeev potential are depicted in Figs. 1 and 2, respectively, with a typical set of values of parameters for dusty plasma [26] \( \kappa_{\text{c}} = 1.6 \), \( \kappa_{\text{h}} = 7 \), \( \mu_{\text{c}} = 0.6 \), \( \mu_{\text{h}} = 0.33 \), \( \sigma_{\text{c}} = 2.5 \), \( \sigma_{\text{h}} = 0.1 \), \( \delta = 10^{ - 3} \), and \( Z_{\text{d}} = 10^{3} \). (The minimum value of Mach number for the existence of soliton and double layer as calculated by Eq. (26) is \( M_{\text{s}} = 8.0434 \) and for the double-layer potential, \( M/M_{\text{s}} = 1.047 \)). The double layer is found to be of negative polarity (rarefactive) with maximum negative potential \( \psi = - 0.028 \). From the Sagdeev potential profile, it is observed that the Sagdeev potential has double roots at \( \varphi = 0 \) and \( \varphi = \psi = - 0.028 \), respectively, which shows that the potential profile depicted in Fig. 1 is that of a double layer. Moreover, the potential profile along with its corresponding Sagdeev potential for soliton is shown in Figs. 3 and 4. Like the double layer, the soliton is also found to be of negative polarity.

Fig. 1
figure 1

Potential profile of the double layer in the system with parameters \( \kappa_{\text{c}} = 1.6 \), \( \kappa_{\text{h}} = 7 \), \( \sigma_{\text{c}} = 2.5 \),\( \sigma_{\text{h}} = 0.1 \), \( \mu_{\text{c}} = 0.6 \), \( \mu_{\text{h}} = 0.33 \), \( \delta = 10^{ - 3} \), \( Z_{\text{d}} = 10^{3} \) and \( M = 8.42444 \)

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Fig. 2
figure 2

Sagdeev potential associated with the double layer of Fig. 1

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Fig. 3
figure 3

Potential profile of the soliton

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Fig. 4
figure 4

Sagdeev potential of the soliton

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The effect of temperature ratio of the superthermal particles on the double-layer potential is also examined [given in Fig. 5(a, b)]. It is perceived that on increase of the value of \( \sigma_{\text{c}} \), the amplitude of the double layer decreases while on increase of the value of \( \sigma_{\text{h}} \) the amplitude of the double layer increases monotonically. Thus, if the temperature of the hot (cold) electron species is decreased (increased), the magnitude of the amplitude of the double layer will be increased or vice versa. In other words, if the temperature ratio of hot to cold electron species (i.e., \( T_{\text{h}} /T_{\text{c}} \)) is decreased, the amplitude will increase. So, when \( T_{\text{ec}} = T_{\text{eh}} \), or instead of two species of different temperatures the plasma contains a single species of the superthermal electron, the amplitude of the double layer will be maximum. In the investigation, the temperature of the cold electron is varied from 2.4 \( T_{\text{ef}} \) to 10, and that of hot electron is varied from 0 to effective electron temperature (\( T_{\text{ef}} \)). It is observed that if \( T_{\text{ec}} < 2.4T_{\text{ef}} \), then no double layer will generate.

Fig. 5
figure 5

(a) Variation of the amplitude of the double layer with associated superthermal electron temperature. Here, \( \sigma_{\text{c}} \) is varied by keeping \( \sigma_{\text{h}} \) fixed at 0.1. Other parameters are same. (b) Variation of the amplitude of the double layer with associated superthermal electron temperature. Here, \( \sigma_{\text{h}} \) is varied by keeping \( \sigma_{\text{c}} \) fixed at 2.5. Other parameters are same

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The dependency of the polarity, amplitude, and width of the double layer on the densities of superthermal electrons is depicted in Fig. 6(a, b). From the figure, it is observed that the amplitude, as well as the width of the double layer, will be maximum when \( \mu_{\text{c}} = \mu_{\text{h}} \). On further increase of the concentration of the cold species of the superthermal electron, the amplitude gradually decreases with a corresponding decrease in its width. Though initially (\( \mu_{\text{c}} \to 0 \) to \( \mu_{\text{c}} < \mu_{\text{h}} \)) the amplitude seemed to increase, the width is zero [Fig. 6(a, b)]. So, from those two figures, it is perceived that no double layer will form for \( \mu_{\text{c}} < \mu_{\text{h}} \). Hence, for the formation of the double layer, number density of cold species of the superthermal electron should be equal to or greater than that of hot species.

Fig. 6
figure 6

(a) Variation of the amplitude of the double layer with cold electron density \( \left( {\mu_{\text{c}} } \right) \). Here, the densities of the electrons are varied by keeping dust density fixed. (b) Variation of the width of the double layer with cold electron density \( \left( {\mu_{\text{c}} } \right) \) by keeping dust density fixed

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The effect of the dust density on the formation of the double layer is also observed. It is found that if we increase the dust grain concentration, the amplitude of the double layer will increase [Fig. 7(a)] with a corresponding decrease in its width [Fig. 7(b)]. For a critical value of the dust density \( \mu = \mu_{\text{critical}} = 0.28 \), the amplitude \( \psi > 1 \). So the double layer is no longer valid at that point (because we have considered \( \varphi \) being much less than a unity). On the other hand, it is observed that for \( \mu < \mu_{\text{critical}} \) the dusty plasma can support negative (rarefactive) double layer while for \( \mu \ge \mu_{\text{critical}} \) the polarity of the double layer changes. Again from Fig. 7(b), it is clear that the width of the double layer dissolves (\( \Delta \to 0 \)) for \( \mu \ge \mu_{\text{critical}} \). So, from Fig. 7(a, b), it can be concluded that the double layer depends upon dust grain concentration. It should be less than 28% of the total plasma density.

Fig. 7
figure 7

(a) Variation of the amplitude of the double layer with dust concentration. (b) Variation of the width of the double layer with the dust concentration

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5.1 Double layer with variable dust charge

The double-layer solution with variable dust charge is determined here. By employing reductive perturbation technique and the stretched coordinate \( \xi = \varepsilon \left( {x_{\text{N}} - Mt_{\text{N}} } \right) \) and \( \tau = \varepsilon^{3} t_{\text{N}} \), we finally get the modified KdV equation [28].

$$ \frac{\partial \varphi }{\partial \tau } + a\varphi^{2} \frac{\partial \varphi }{\partial \xi } + b\varphi \frac{\partial \varphi }{\partial \xi } + c\frac{{\partial^{3} \varphi }}{{\partial \xi^{3} }} = 0, $$
(27)

where a, b, and c are constant quantities as given in “Appendix.” The double-layer solution of Eq. (27) is

$$ \varphi = \frac{{\varphi_{\text{m}} }}{2}\left[ {1 - \tanh \left( {\frac{b}{{2\sqrt { - 6ac} }}X} \right)} \right] $$
(28)

with \( \varphi_{\text{m}} = - \frac{b}{a} \) and width \( \Delta = \frac{{2\sqrt { - 6ac} }}{b} \).

The double-layer solution exists only when \( a < 0 \). The potential profile for the double layer is depicted in Fig. 8 and is observed to be of the same polarity with slightly higher amplitude for the same set of value of the parameters.

Fig. 8
figure 8

Potential profile of the double layer with variable dust charge with parameters \( \kappa_{\text{c}} = 1.6 \), \( \kappa_{\text{h}} = 7 \), \( \sigma_{\text{c}} = 2.5 \),\( \sigma_{\text{h}} = 0.1 \), \( \mu_{\text{c}} = 0.6 \), \( \mu_{\text{h}} = 0.33 \), \( \delta = 10^{ - 3} \), and \( M = 11.6092 \)

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6 Conclusions

The existence of double layer in an unmagnetized, collisionless plasma containing positive ion, negatively charged dust and two species of the superthermal electron is studied here. From the analytical study, it is observed that only double layer with negative potential can exist in such type of plasma. The effect of various parameters of the superthermal electrons is also analyzed here and is observed that they alter the amplitude and width of the double layer. With the variation of \( \sigma_{\text{c}} \) and \( \sigma_{\text{h}} \), the amplitude is varied and it is observed that instead of two components of the superthermal electron if the plasma contains a single component, it can support double layer of higher amplitude.

In the generation of the double layer in the proposed model, the densities of the hot and cold component of electrons also play an essential role. It is observed that for the formation of the double layer, the number density of cold species of the superthermal electron should be equal to or higher than that of the hot species of the electron.

The effect of the dust density is also observed, and it is seen that an increase of dust density results in an increase of amplitude with a corresponding decrease in its width. Alam, Masud, and Mamun [26] also concluded that the essential features (polarity, amplitude, and width) of an electrostatic shock depend on the relative temperature ratio of electrons as well as relative dust number density. They have considered plasma with cold fluid positive ion, negatively charged static dust, and two species of superthermal electrons.

Considering the charge variation of the dust particle, the possible change in double-layer potential is also checked and it is found that for the same set of values of the parameters the polarity of the double layer remains the same with slightly higher amplitude.

The results obtained from this investigation may be useful in understanding the nonlinear structures in space dusty plasma, viz. Saturn’s magnetosphere and pulsar magnetosphere where the ions, superthermal electrons of two different temperatures (hot and cold), and negatively charged dust are the dominant species. Moreover, Schippers et al. [29] analyzed the CAPS/ELS and MIMI/LEMMS data from the Cassini spacecraft orbiting Saturn over a range of 5.420 Rs (Rs ≈ 60,300 km being the radius of Saturn) and found the best fit for the electron velocity distribution using a combination of a hot and cold electron components. The model considered here has a good match with the data provided by them for outer magnetosphere of Saturn (~ 14 Rs). So, the results obtained here may be useful for the study of nonlinear structure formation in Saturn magnetosphere.