1 Introduction

According to linear theory, the wave amplitude is assumed to be negligible enough to ignore the consequences of nonlinearity. As the amplitude rises, the linear approximation fails and it necessitates the study of nonlinearities. Nonlinearity is caused by the Lorentz force, ponderomotive force, particle trapping, etc., and it produce various type of waves, including shock waves, double layers, and solitons (Alam et al. 2014; Saha and Chatterjee 2014; Saha and Banerjee 2021; El-Shewy and El-Rahman 2018; Alharbi et al. 2023; Abdelwahed et al. 2023). The dynamic characteristics of the formation and propagation of solitons with strong and weak nonlinearity are inspected by Das et al. (2000). Both the KdV and mKdV equations contain nonlinear and dispersion terms while their balancing effect produces solitons.

The existence of electron–ion (e-i), electron–positron (e-p), and electron–positron-ion (e-p-i) plasmas in space and laboratory (Saitoh et al. 2014; Pedersen et al. 2012; Nakamura and Sugai 1996) are well confirmed. Researchers have analyzed the dynamic behaviors of nonlinear waves in e-i, e-p and e-p-i plasmas (Saitoh et al. 2014; Pedersen et al. 2012; Nakamura and Sugai 1996; Rahman and Haider 2016; Abdelwahed et al. 2021; Pottelette et al. 1999; Shahein and Abdo 2021) with superthermal electrons and positrons. Positrons have the same mass as electrons but the opposite charge and magnetic moment. Positrons are stable in vacuum or in absence of electrons (lifetime is greater than 2 × 1021 years), but rapidly thermalize and annihilate with electrons in materials with a mean lifetime of typically only a few hundred picoseconds (Hakvoort 1993). These superthermal particles are high energetic and they can be described with the help of the Kappa distribution (Alam et al. 2013; Alam 2013; Basu 2009). The superthermal particles (electrons or positrons) can be generated by intense laser beams in hot dense plasmas (Gupta et al. 2023), fusion reaction (Peigney et al. 2014), stochastic heating process (Singh et al. 2021), etc.

Investigation on e-p-i plasmas with the Kappa distributed superthermal electrons and positrons is crucial because they significantly exist in astrophysical space plasmas, such as the pulsar magnetosphere (Shatasvili et al. 1997), the active galactic nuclei (Kirk and Mastichiadi 1992), the quasar’s relativistic jet (Wardle et al. 1998), the inner region of the accretion disk surrounding a black hole (Liang and Dermer 1988), and so on. The plasmas can be produced in laboratory using ultra-intense laser matter interaction (Luo 2018), positron injection into the electron–ion system (Helander and Ward 2003), or relativistic heavy-ion collisions (Baur 1990). There are two basic reasons for conducting this type of research. First, since the properties of the waves in e-p-i plasma systems differ significantly from those of the waves in regular e-i and e-p plasmas, it is important to comprehend the fundamentals of these systems. The second is to comprehend the properties of the plasma’s astrophysical surroundings. Numerous plasma researchers have looked into the propagation of the PA waves in e-p-i plasmas in the recent years (Akter and Hafez 2022; Nejoh 1996; Tribeche et al. 2010, Sahu 2010, Tamang and Saha 2019). Akter and Hafez (2022) investigated the head-on collision between PA solitons while the plasma system contains cold positron fluid, immobile positive ions, (r, q)-distributed hot positrons and hot electrons. Using an electron beam in e-p plasma, the nonlinear wave patterns of large amplitude PA waves is inspected by Nejoh (1996). Tribeche et al. (2009) looked into the dynamic behavior of cold positrons in an e-p-i plasmas that contains positively charged ions (immobile), hot positrons, and hot electrons. Using the same plasma model, PA solitary waves of small amplitude, PA shock waves, head on collision, etc., are examined (Tribeche 2010; Sahu 2010; El-Shamy et al. 2012).

A number of mathematical tecniques including Hirota bilinear method (Ma 2022) are used to derive MSS (Wang and Wang 2001; Ma 2019; Ling et al. 2015; Hirota 2004; Griffiths 2012; Alsayyed et al. 2016; Chen et al. 2019; Ullah et al. 2020; Ismael et al. 2020), while the Hirota bilinear method is one of the powerful methods but have a complexity. In order to create an auxiliary function using this method, the equations must be reduced to a bilinear form, and this procedure may be diffcult for some equations. The simplified Hirota method (that we have used here) is developed by Hereman and Zhuang (1991; 1992) to formalize the simplest process of deriving auxiliary functions without bilinear forms. Wazwaz (2009a; b; 2016) has investigated a large number integrable equations (including the KdV and mKdV equations) for finding MSS while he has used the simplified Hirota method. The solutions can be found when the equations are integrable. The simplified Hirota method is the most effective for determining multi-soliton solutions.

The objectives of this study are to examine the dynamic behavior of multiple-solitons propagated in an e-p-i plasma with highly energetic particles and to investigate the overtaking-collisional and parametric effects on the PA waves. To describe the impact of the plasma parameters on the soliton nature, the KdV and mKdV equations are derived by RPT, and the solutions are obtained by the simplified Hirota method. To the best of our knowledge, the theoretical investigation on overtaking collisions, phase shift and parametric effects is conducted for the first time for the considered e-p-i plasma model.

The research is structured as follows:

The phases of the solution method are provided in Sect. 2. We introduce a plasma theory and construct the wave equations using the RPT in Sect. 3. In Sect. 4, the simplified Hirota method is used to obtain the MSS. Results from a parametric analysis are presented in Sect. 5, and the conclusion is delivered in the last section.

2 The simplified Hirota method

To attain the expected solution, the simplified Hirota method (Hereman 1991; Wazwaz 2016) is applied while the main steps of the technique are described below:

We first set \(u\left(x,t\right)={e}^{{\eta }_{i}}\), \({\eta }_{i}= {k}_{i} x-{\omega }_{i}t (i=1\), 2, …, \(N)\), where \({k}_{i}\) is the wave number and \({\omega }_{i}\) is the wave frequency, into the linear terms of the considered partial differential equation (PDE) which is nonlinear, integrable, and of the form

$$P\left( {u,\frac{\partial u}{{\partial t}},\frac{\partial u}{{\partial x}},\frac{{\partial^{2} u}}{{\partial t^{2} }},\frac{{\partial^{2} u}}{\partial x \partial t},\frac{{\partial^{2} u}}{{\partial x^{2} }}, \ldots \ldots } \right) = 0.$$
(2.1)

The above substitution yields a dispersion relation between \({k}_{i}\) and \({\omega }_{i}\).

Secondly, Cole-Holf transformation (Wazwaz 2016) will be used to acquire the solution of the Eq. (2.1). Specifically, different forms of the transformation will be used for solving different nonlinear PDEs. For example, the following transformation is used for solving the KdV equation

$$u\left( {x,t} \right) = R\frac{{\partial^{2} }}{{\partial x^{2} }}\left( {\ln f} \right).$$
(2.2)

Keeping (2.2) in (2.1) gives \(R\), where \(f\)(\(x\), \(t)\), known as auxiliary function, is expressed as.

  1. (i)

    For single-soliton solution: \(f\)(\(x\), \(t) =1+{e}^{{\eta }_{1}}\), \({\eta }_{1}= {k}_{1} x-{\omega }_{1}t\).

  2. (ii)

    For two-soliton solution: \(f\)(\(x\), t \()=1+{e}^{{\eta }_{1}}+{e}^{{\eta }_{2}}+{a}_{12} {e}^{{\eta }_{1}+{\eta }_{2}}\), \({\eta }_{i}= {k}_{i} x-{\omega }_{i}t\) (\(i=\) 1, 2) and \({a}_{12}\) is the phase factor.

  3. (iii)

    For three-soliton solution:\(f\)(\(x\),\(t)=1+{e}^{{\eta }_{1}}+{e}^{{\eta }_{2}}+{e}^{{\eta }_{3}}+{a}_{12} {e}^{{\eta }_{1}+{\eta }_{2}}+{a}_{13} {e}^{{\eta }_{1}+{\eta }_{3}} +{a}_{23} {e}^{{\eta }_{2}+{\eta }_{3}}+{A}_{123} {e}^{{\eta }_{1}+{\eta }_{2}+{\eta }_{3}}\), \({\eta }_{i}= {k}_{i} x-{\omega }_{i}t\) (\(i=\) 1, 2, 3).

If we get \({A}_{123}={a}_{12} {a}_{23} {a}_{13}\), the three-soliton solution can be found from (2.2) and this gives the surety of getting the soliton solutions of (2.1).

Similarly, the following transformation and auxiliary functions will be used for the mKdV equation.

$$u\left( {x, t} \right) = R\frac{\partial }{\partial x}\arctan \left( {\frac{f}{{\text{g}}}} \right) = R\left( {\frac{{f_{x } {\text{g}} - f{\text{g}}_{x} }}{{f^{2} + {\text{g}}^{2} }} } \right),$$
(2.3)

where the auxiliary functions \(f\)(\(x\), \(t)\) and \({\text{g}}\)(\(x\), \(t)\) are:

  1. (i)

    For single-soliton solution: \(f\)(\(x\), \(t)={e}^{{\eta }_{1}}\), \({\text{g}}\)(\(x\), \(t)=1\).

  2. (ii)

    For two-soliton solution: \(f\)(\(x\), \(t)={e}^{{\eta }_{1}}+{e}^{{\eta }_{2}}\), \({\text{g}}\)(\(x\), \(t)=1-{a}_{12} {e}^{{\eta }_{1}+{\eta }_{2}}\).

  3. (iii)

    For three-soliton solution: \(f\)(\(x\),\(t)={e}^{{\eta }_{1}}+{e}^{{\eta }_{2}}+{e}^{{\eta }_{3}}-{A}_{123} {e}^{{\eta }_{1}+{\eta }_{2}+{\eta }_{3}}\)

and \({\text{g}}\)(\(x\), \(t)=1-{a}_{12} {e}^{{\eta }_{1}+{\eta }_{2}}-{a}_{13} {e}^{{\eta }_{1}+{\eta }_{3}}-{a}_{23} {e}^{{\eta }_{2}+{\eta }_{3}}\).

3 Hydrodynamic fluid model equations

We assume a one-dimensional, collision-less and unmagnetized e-p-i plasma system wherein the plasma composed of inertial cold positrons (Kappa distributed), electrons and positrons (super-thermal) and immobile positive ions. The following are normalized fundamental equations that control the dynamics of the PA waves (Alam et al. 2014; Akter and Hafez 2022):

$$\frac{{\partial n_{pc} }}{\partial t} + \frac{\partial }{\partial x}\left( {n_{pc} u_{pc} } \right) = 0,$$
(3.1)
$$\frac{{\partial u_{pc} }}{\partial t} + u_{pc} \frac{\partial }{\partial x}\left( {u_{pc} } \right) = - \frac{\partial \psi }{{\partial x}},$$
(3.2)
$$\frac{{\partial^{2} \psi }}{{\partial x^{2} }} = \mu_{e} n_{e} - \mu_{ph} n_{ph} - n_{pc} - \mu_{i} ,$$
(3.3)

where \({n}_{e}\), \({n}_{pc}\) and \({n}_{ph}\) denote the hot electron, the cold positron and hot positron number densities respectively; \({u}_{pc}\) is the cold positron fluid speed normalized by \({C}_{pc}={\left(\frac{{K}_{B}{T}_{ef}}{{m}_{pc}}\right)}^{1/2}\), \({T}_{ef}={T}_{e}{T}_{ph}/({\mu }_{e}{T}_{ph}+{\mu }_{ph}{T}_{e})\) is the effective temperature and \({m}_{pc}\) is the mass of cold positron, and \(\psi\) is the wave potential normalized by \(\frac{{K}_{B}{T}_{ef}}{e}\). The time (\(t\)) and space (\(x)\) variables are in the units of \({{\omega }_{pc}}^{-1}=\sqrt{\frac{{m}_{pc}}{4\pi {n}_{pc0}{e}^{2}}}\) and \({\lambda }_{D}=\sqrt{\frac{{K}_{B}{T}_{ef}}{4\pi {n}_{pc0}{e}^{2}}}\) respectively.

The normalized forms of the Kappa distributions of super-thermal positrons and electrons are stated as \({n}_{ph}={\left(1+\frac{{\sigma }_{1} \psi }{{\kappa }_{p}-\frac{3}{2}}\right)}^{-{\kappa }_{p}+\frac{1}{2}}\) and \({n}_{e}={\left(1-\frac{{\sigma }_{2} \psi }{{\kappa }_{e}-\frac{3}{2}}\right)}^{-{\kappa }_{e}+\frac{1}{2}}\) respectively (Alam et al. 2014), where \({\sigma }_{1}=\frac{{T}_{ef}}{{T}_{ph}}\), \({\sigma }_{2}=\frac{{T}_{ef}}{{T}_{e}}\). The parameters \({\kappa }_{p}\) and \({\kappa }_{e}\) are the spectral index parameters which define the strength of the super thermality. The range of these parameters are: \(\frac{3}{2}<{\kappa }_{p},{\kappa }_{e}<\infty\) 3/2. In the limit \({\kappa }_{p},{\kappa }_{e}\to \infty\), the kappa distribution function reduces the well-known Maxwell–Boltzmann distribution. At equilibrium \({n}_{e0}={n}_{pc0}+{n}_{ph0}+{n}_{i0}\) i.e., \({\mu }_{e}=1+{\mu }_{ph}+{\mu }_{i}\), where \({\mu }_{e}=\frac{{n}_{e0}}{{n}_{pc0}}\), \({\mu }_{ph}=\frac{{n}_{ph0}}{{n}_{pc0}}\) and \({\mu }_{i}=\frac{{n}_{i0}}{{n}_{pc0}}\).

To examine the effect of distinct plasma parameters, the KdV equation is acquired by exploiting the stretched coordinates (Alam et al. 2014; Akter and Hafez 2022; Isamel et al. 2020):

$$\xi = \varepsilon^{\frac{1}{2}} \left( {x - V_{p} t} \right)\;{\text{and}}\; \tau = \varepsilon^{3/2} t,$$
(3.4)

where \(\varepsilon\) is a parameter which scales the dispersion weakness; and \({V}_{p}\) is the phase speed.

The perturbed quantities have the following powers of \(\varepsilon\):

$$n_{pc} = 1 + \varepsilon n_{pc}^{\left( 1 \right)} + \varepsilon^{2} n_{pc}^{\left( 2 \right)} + \varepsilon^{3} n_{pc}^{\left( 3 \right)} + \ldots ,$$
(3.5)
$$u_{pc} = \varepsilon u_{pc}^{\left( 1 \right)} + \varepsilon^{2} u_{pc}^{\left( 2 \right)} + \varepsilon^{3} u_{pc}^{\left( 3 \right)} + \ldots ,$$
(3.6)
$$\psi = \varepsilon \psi^{\left( 1 \right)} + \varepsilon^{2} \psi^{\left( 2 \right)} + \varepsilon^{3} \psi^{\left( 3 \right)} + \ldots$$
(3.7)

Use (3.4–3.7) in (3.1–3.3), and then equate the coefficients of the lowermost order of \(\epsilon\). Hence \({u}_{pc}^{\left(1\right)}\) and \({n}_{pc}^{\left(1\right)}\) are found as

$$u_{pc}^{\left( 1 \right)} = \frac{1}{{V_{p} }}\psi^{\left( 1 \right)} \; {\text{and}}\; n_{pc}^{\left( 1 \right)} = \frac{1}{{V_{p}^{2} }}\psi^{\left( 1 \right)} .$$
(3.8)

Besides, the wave phase speed \(V_{p}\) is determined as.

$$V_{p} = \left[ {\frac{{\mu_{ph} \left( {\kappa_{p} - \frac{1}{2}} \right)\sigma_{1} }}{{\kappa_{p} - \frac{3}{2}}} + \frac{{\mu_{e} \left( {\kappa_{e} - \frac{1}{2}} \right)\sigma_{2} }}{{\kappa_{e} - \frac{3}{2}}}} \right]^{ - 1/2} , \mu_{e} = 1 + \mu_{ph} + \mu_{i}$$
(3.9)

Similarly, the subsequent equations are attained by equating the coefficients of \({\epsilon }^{5/2}\) from (3.1) and (3.2), and the coefficient of \(\epsilon\) from (3.3).

$$\frac{{\partial n_{pc}^{\left( 1 \right)} }}{\partial \tau } - { }V_{p} \frac{{\partial { }n_{pc}^{\left( 2 \right)} }}{\partial \xi } + \frac{{\partial u_{pc}^{\left( 2 \right)} }}{\partial \xi } + \frac{\partial }{\partial \xi }\left( {n_{pc}^{\left( 1 \right)} u_{pc}^{\left( 1 \right)} } \right) = 0$$
(3.10)
$$\frac{{\partial u_{pc}^{\left( 1 \right)} }}{\partial \tau } - V_{p} \frac{{\partial u_{pc}^{\left( 2 \right)} }}{\partial \xi } + \frac{{\partial \psi^{\left( 2 \right)} }}{\partial \xi } + u_{pc}^{\left( 1 \right)} \frac{{\partial u_{pc}^{\left( 1 \right)} }}{\partial \xi } = 0.$$
(3.11)
$$\frac{{\partial^{2} \psi^{\left( 1 \right)} }}{{\partial \xi^{2} }} + \left( {\mu_{ph} \frac{{P_{1} P_{2} \sigma_{1}^{2} }}{{2 P_{3}^{2} }} - \mu_{e} \frac{{P_{4} P_{5} \sigma_{2}^{2} }}{{2P_{6}^{2} }}} \right)\left[ {\psi^{\left( 1 \right)} } \right]^{2} + \left( {\frac{{\mu_{ph} P_{2} \sigma_{1} }}{{P_{3} }} + \mu_{e} \frac{{P_{5} \sigma_{2} }}{{P_{6} }}} \right)\psi^{\left( 2 \right)} + n_{pc}^{\left( 2 \right)} = 0.$$
(3.12)

where \({P}_{1}=-\frac{1}{2}-{\kappa }_{p}\),\({P}_{2}=\frac{1}{2}-{\kappa }_{p}\),\({P}_{3}=-\frac{3}{2}+{\kappa }_{p}\),\({P}_{4}=-\frac{1}{2}-{\kappa }_{e}\), \({P}_{5}=\frac{1}{2}-{\kappa }_{e}\), \({P}_{6}=-\frac{3}{2}+{\kappa }_{e}.\)

Now combining Eqs. (3.8)–(3.12), we get

$$\frac{{\partial \psi^{\left( 1 \right)} }}{\partial \tau } + A \psi^{\left( 1 \right)} \frac{{\partial \psi^{\left( 1 \right)} }}{\partial \xi } + B\frac{{\partial^{3} \psi^{\left( 1 \right)} }}{{\partial \xi^{3} }} = 0,$$
(3.13)

where \(A=\frac{{V}_{p}^{3}}{2}\left(\frac{3}{{V}_{p}^{4}}+\frac{{\mu }_{ph}{P}_{1}{P}_{2}{\sigma }_{1}^{2}}{{P}_{3}^{2}}-\frac{{\mu }_{e}{P}_{4}{P}_{5}{\sigma }_{2}^{2}}{{P}_{6}^{2}}\right)\) and \(B=\frac{{V}_{p}^{3}}{2}\) are the nonlinear and dispersive coefficients respectively.

But if there exists a critical condition for which \(A=0,\) then the KdV equation fails to describe the nonlinear solitons, and in this case the mKdV equation is applicable (Alam et al. 2014). Consider the stretched coordinates \(\xi =\epsilon (x-{V}_{p}t)\) and \(\tau ={\epsilon }^{3} t\) to obtain the following mKdV equation \(.\)

$$\frac{{\partial \psi^{\left( 1 \right)} }}{\partial \tau } + C \left[ {\psi^{\left( 1 \right)} } \right]^{2} \frac{{\partial \psi^{\left( 1 \right)} }}{\partial \xi } + B\frac{{\partial^{3} \psi^{\left( 1 \right)} }}{{\partial \xi^{3} }} = 0,$$
(3.14)

where \(C=\alpha B\), \(\alpha =\frac{15}{2{V}_{p}^{6}}+\frac{{\mu }_{ph}{P}_{1}{P}_{2}{P}_{7}{\sigma }_{1}^{3}}{2{P}_{3}^{3}}+\frac{{\mu }_{e}{P}_{4}{P}_{5}{P}_{8}{\sigma }_{2}^{3}}{2{P}_{6}^{3}}\), \(B=\frac{{V}_{p}^{3}}{2}\), \({P}_{7}=-\frac{3}{2}-{\kappa }_{p}\), \({P}_{8}=-\frac{3}{2}-{\kappa }_{e}\).

4 Multiple-soliton solutions

We devote this section for constructing MSS of the KdV and mKdV equations explicitly while the simplified Hirota method is used. Substitution of \({\psi }^{\left(1\right)}\)(\(\xi\), \(\tau ) ={e}^{{k}_{i} \xi -{\omega }_{i} \tau }\) into the linear terms of (3.13) yields the following dispersion relation

$$\omega_{i} = Bk_{i}^{3} .$$
(4.1)

Consequently, the phase variable takes the form \({\eta }_{i}={k}_{i} \xi -B{k}_{i}^{3} \tau\).

Let us assume the MSS (using the Cole-Holf transformation) as

$$\psi^{\left( 1 \right)} \left( {\xi , \tau } \right) = R\frac{{\partial^{2} }}{{\partial \xi^{2} }}\left( {\ln f} \right),$$
(4.2)

where the function \(f\left(\xi ,\tau \right)\) for single-soliton solution is

$$f\left( {\xi ,\tau } \right) = 1 + e^{{\eta_{1} }} , \eta_{1} = k_{1} \xi - Bk_{1}^{3} \tau .$$
(4.3)

Put (4.2) and (4.3) into (3.13) that gives \(R=\frac{12 B}{A}\), and hence the single-soliton solution of the KdV equation is

$$\psi_{1}^{\left( 1 \right)} \left( {\xi , \tau } \right) = \frac{12B}{A}\frac{{k_{1}^{2} e^{{k_{1} \xi - Bk_{1}^{3} \tau }} }}{{\left( {1 + e^{{k_{1} \xi - Bk_{1}^{3} \tau }} } \right)^{2} }} = \frac{3B}{A}k_{1}^{2} {\text{sech}}^{2} \frac{{k_{1} }}{2}\left( {\xi - Bk_{1}^{2} \tau } \right) = \frac{{3U_{1} }}{A}{\text{sech}}^{2} \sqrt {\frac{{U_{1} }}{4B} } \left( {\xi - U_{1} \tau } \right).$$
(4.4)

The solution gives a soliton profile (a similar result is obtained by Alam et al. (2014)), while the wave propagates with speed \({U}_{1}(=B{k}_{1}^{2}\)). Thus, the single-soliton moves with an amplitude \(\frac{3 {U}_{1}}{A}\) and width \(\sqrt{\frac{4B}{{U}_{1}}}\) which implies that the amplitude and width are dependent on index parameters of positrons \(({\kappa }_{p})\) and electrons \(({\kappa }_{e})\), hot positron concentration \(({\mu }_{ph})\) and hot electron concentration \(({\mu }_{e})\), etc.

The auxiliary function for the two-soliton solution is presented as:

$$f\left( {\xi ,\tau } \right) = 1 + e^{{\eta_{1} }} + e^{{\eta_{2} }} + a_{12} e^{{\eta_{1} + \eta_{2} }} ,$$
(4.5)

where \({\eta }_{i}={k}_{i} \xi -B{k}_{i}^{3}\tau\) (\(i=1\), 2). Using \(f\left(\xi ,\tau \right)\) and \(R=\frac{12 B}{A}\) in (4.2), and then in (3.13), we get the phase shift factor \({a}_{12}\) as

$$a_{12} = \frac{{\left( {k_{1} - k_{2} } \right)^{2} }}{{\left( {k_{1} + k_{2} } \right)^{2} }}.$$
(4.6)

Thus, the two-soliton solution is obtained as

$$\psi_{2}^{\left( 1 \right)} \left( {\xi ,\tau } \right) = \frac{12 B}{A}\left[ {\frac{{k_{1}^{2} e^{{\eta_{1} }} + k_{2}^{2} e^{{\eta_{2} }} + a_{12} e^{{\eta_{1} + \eta_{2} }} \left( {k_{2}^{2} e^{{\eta_{1} }} + k_{1}^{2} e^{{\eta_{2} }} } \right) + 2\left( {k_{1} - k_{2} } \right)^{2} e^{{\eta_{1} + \eta_{2} }} }}{{\left( {1 + e^{{\eta_{1} }} + e^{{\eta_{2} }} + a_{12} e^{{\eta_{1} + \eta_{2} }} } \right)^{2} }}} \right],$$
(4.7)

where \({k}_{i}=\sqrt{\frac{{U}_{i}}{B}}\) (\(i=1\), 2).

When \(\tau \gg 1\), the non-dominant terms \({e}^{-({\eta }_{1}+{\eta }_{2})}\), \({e}^{-({2\eta }_{1}+{\eta }_{2})}\), \({e}^{-({\eta }_{1}+{2\eta }_{2})}\) can be neglected and (4.7) is asymptotically transformed into a superposition of two single-soliton solutions (Ismael et al. 2020). Thus

$$\psi_{2}^{\left( 1 \right)} \left( {\xi ,\tau } \right)\sim \mathop \sum \limits_{i = 1}^{2} \frac{{3 Bk_{i}^{2} }}{A}{\text{sech}}^{2} \frac{{k_{i} }}{2}\left( { \xi - Bk_{i}^{2} \tau + \Delta_{i} } \right),$$
(4.8)

where the amplitude is \({A}_{i}=\frac{3 B{k}_{i}^{2}}{A}\), \({k}_{i}=\sqrt{\frac{{U}_{i}}{B}}\) (\(i=1\), 2) and the phase shifts of the solitons due to interaction are \({\Delta }_{i}=\frac{2B}{{k}_{i}} lo{\text{g}}\left|\sqrt{{a}_{12}}\right|\), \({a}_{12}\) is related to phase shifts of overtaking collisions.

For three-soliton solutions, the auxiliary function (where \({\eta }_{i}={k}_{i} \xi -B{k}_{i}^{3}\tau\), (\(i=1\), 2, 3) are:

$$f\left( {\xi ,\tau } \right) = 1 + e^{{\eta_{1} }} + e^{{\eta_{2} }} + e^{{\eta_{3} }} + a_{12} e^{{\eta_{1} + \eta_{2} }} + a_{13} e^{{\eta_{1} + \eta_{3} }} + a_{23} e^{{\eta_{2} + \eta_{3} }} + A_{123} e^{{\eta_{1} + \eta_{2} + \eta_{3} }}$$
(4.9)

Use of (4.9) and (4.2) in (3.13) yields \({a}_{ij}\), and it can be generalized to

$$a_{ij} = \frac{{\left( {k_{i} - k_{j} } \right)^{2} }}{{\left( {k_{i} + k_{j} } \right)^{2} }}, 1 \le i < j \le 3$$
(4.10)

and

$$A_{123} = a_{12} a_{23} a_{13} = \left[ {\frac{{\left( {k_{1} - k_{2} } \right)\left( {k_{2} - k_{3} } \right)\left( {k_{1} - k_{3} } \right)}}{{\left( {k_{1} + k_{2} } \right)\left( {k_{2} + k_{3} } \right)\left( {k_{1} + k_{3} } \right)}}} \right]^{2} .$$
(4.11)

Here, the Eq. (4.10) demonstrates that the KdV equation does not exhibits resonant phenomenon as \({a}_{ij}\) is not zero or infinity for \({k}_{i}\ne {k}_{j}\) (Wazwaz 2009a; b; 2016).

Thus, the three-soliton solution can be found by putting the values of \({a}_{ij}\) and \({A}_{123}\) in (4.2) as

$$\psi_{3}^{\left( 1 \right)} \left( {\xi , \tau } \right) = \frac{12 B}{A}\left[ {\frac{{L_{1} }}{{L_{3} }} - \left( {\frac{{L_{2} }}{{L_{3} }}} \right)^{2} } \right],$$
(4.12)

where, \({L}_{1}=\sum_{i=1}^{3}{k}_{i}^{2}{e}^{{\eta }_{i}}+{\sum }_{1\le i<j\le 3}{\left({k}_{i}-{k}_{j}\right)}^{2} {e}^{{\eta }_{i}+{\eta }_{j}} +{A}_{123} {\left(\sum_{i=1}^{3}{k}_{i}\right)}^{2}{e}^{\sum_{i=1}^{3}{\eta }_{i}}\),

$$L_{2} = \mathop \sum \limits_{i = 1}^{3} k_{i} { }e^{{\eta_{i} }} + \mathop \sum \limits_{1 \le i < j \le 3} \frac{{\left( {k_{i} - k_{j} } \right)^{2} }}{{\left( {k_{i} + k_{j} } \right)}}{ }e^{{\eta_{i} + \eta_{j} }} + A_{123} \left( {\mathop \sum \limits_{i = 1}^{3} k_{i} } \right)e^{{\mathop \sum \limits_{i = 1}^{3} \eta_{i} }} ,$$
$$L_{3} = 1 + { }\mathop \sum \limits_{i = 1}^{3} e^{{\eta_{i} }} + \mathop \sum \limits_{1 \le i < j \le 3} \frac{{\left( {k_{i} - k_{j} } \right)^{2} }}{{\left( {k_{i} + k_{j} } \right)^{2} }}{ }e^{{\eta_{i} + \eta_{j} }} + A_{123} { }e^{{\mathop \sum \limits_{i = 1}^{3} \eta_{i} }} .$$

Thus, for \(\tau \gg 1\), the three-soliton solution is asymptotically transformed into a superposition of three single-soliton solutions as

$$\psi_{3}^{\left( 1 \right)} \left( {\xi ,\tau } \right)\sim \mathop \sum \limits_{i = 1}^{3} \frac{{3 Bk_{i}^{2} }}{A}{\text{sech}}^{2} \frac{{k_{i} }}{2}\left( {\xi - Bk_{i}^{2} \tau + \Delta_{i}^{*} } \right),$$
(4.13)

where the amplitude is \({A}_{i}=\frac{3 B{k}_{i}^{2}}{A}\), \({k}_{i}=\sqrt{\frac{{U}_{i}}{B}} (i=1\), 2, 3) and the phase shifts of the solitons due to interaction are \({\Delta }_{1}^{*}=\frac{2B}{{k}_{1}} lo{\text{g}}\left|\sqrt{\frac{{A}_{123}}{{a}_{23}}}\right|\), \({\Delta }_{2}^{*}=\frac{2B}{{k}_{2}} lo{\text{g}}\left|\sqrt{\frac{{A}_{123}}{{a}_{13}}}\right|\) and \({\Delta }_{3}^{*}=\frac{2B}{{k}_{3}} lo{\text{g}}\left|\sqrt{\frac{{A}_{123}}{{a}_{12}}}\right|\).

The solitary waves will be associated with positive (negative) potential if \(A>0 \left(<0\right).\) That is, compressive (hump) and rarefactive (dip) solitons will be produced if \(A>0 \left(<0\right)\). But if there exist some parametric values for which \(A=0\), the waves amplitudes approach to infinity. To overcome the complexity, we will use the following set of solutions of the mKdV equation.

The single-, two and three-soliton solutions of the mKdV Eq. (3.14) as

$$\psi_{{1\left( {mKdV} \right)}}^{\left( 1 \right)} \left( {\xi ,\tau } \right) = \sqrt {\frac{24 B}{C} } \frac{{k_{1} e^{{k_{1} \xi - Bk_{1}^{3} \tau }} }}{{1 + \left( {e^{{k_{1} \xi - Bk_{1}^{3} \tau }} } \right)^{2} }} = \sqrt {\frac{{6 U_{1} }}{C}} {\text{sech}}\sqrt {\frac{{U_{1} }}{B}} \left( {\xi - U_{1} \tau } \right),$$
(4.14)
$$\psi_{{2\left( {mKdV} \right)}}^{\left( 1 \right)} \left( {\xi ,\tau } \right) = \sqrt{\frac{24}{C}} \left[ { \frac{{T_{1} M_{1} + N_{1} R_{1} }}{{N_{1}^{2} + M_{1}^{2} }}} \right],$$
(4.15)
$$\psi_{{3\left( {mKdV} \right)}}^{\left( 1 \right)} \left( {\xi ,\tau } \right) = \sqrt{\frac{24}{C}} \left[ { \frac{{T_{2} M_{2} + N_{2} R_{2} }}{{N_{2}^{2} + M_{2}^{2} }}} \right],$$
(4.16)

where \({T}_{1}={\sum }_{i=1}^{2}\sqrt{{U}_{i}}{e}^{{\eta }_{i}}\), \({M}_{1}=1-\sum_{1\le i<j\le 2}\frac{{\left(\sqrt{{U}_{i}} -\sqrt{{U}_{j}}\right)}^{2}}{{\left(\sqrt{{U}_{i}} +\sqrt{{U}_{j}}\right)}^{2}} {e}^{{\eta }_{i}+{\eta }_{j}}\), \({N}_{1}={\sum }_{i=1}^{2}{e}^{{\eta }_{i}}\),

$$\begin{gathered} R_{1} = \mathop \sum \limits_{1 \le i < j \le 2} \frac{{\left( {\sqrt {U_{i} } - \sqrt {U_{j} } } \right)^{2} }}{{\left( {\sqrt {U_{i} } + \sqrt {U_{j} } } \right)}} e^{{\eta_{i} + \eta_{j} }} , T_{2} = \mathop \sum \limits_{i = 1}^{3} \sqrt {U_{i} } e^{{\eta_{i} }} {-\!\!-}\mathop \sum \limits_{i = 1}^{3} k_{i} \left( {\mathop \prod \limits_{1 \le i < j \le 3} \frac{{\left( {\sqrt {U_{i} } - \sqrt {U_{j} } } \right)^{2} }}{{\left( {\sqrt {U_{i} } + \sqrt {U_{j} } } \right)^{2} }}} \right)e^{{\mathop \sum \limits_{i = 1}^{3} \eta_{i} }} , \hfill \\ M_{2} = 1 - \mathop \sum \limits_{1 \le i < j \le 3} \frac{{\left( {\sqrt {U_{i} } - \sqrt {U_{j} } } \right)^{2} }}{{\left( {\sqrt {U_{i} } + \sqrt {U_{j} } } \right)^{2} }} e^{{\eta_{i} + \eta_{j} }} , N_{2} = \mathop \sum \limits_{i = 1}^{3} e^{{\eta_{i} }} - \left( {\mathop \prod \limits_{1 \le i < j \le 3} \frac{{\left( {\sqrt {U_{i} } - \sqrt {U_{j} } } \right)^{2} }}{{\left( {\sqrt {U_{i} } + \sqrt {U_{j} } } \right)^{2} }}} \right)e^{{\mathop \sum \limits_{i = 1}^{3} \eta_{i} }} , \hfill \\ R_{2} = \mathop \sum \limits_{1 \le i < j \le 3} \frac{{\left( {\sqrt {U_{i} } - \sqrt {U_{j} } } \right)^{2} }}{{\left( {\sqrt {U_{i} } + \sqrt {U_{j} } } \right)}} e^{{\eta_{i} + \eta_{j} }} . \hfill \\ \end{gathered}$$

Thus, for \(\tau \gg 1\), the soliton solutions are asymptotically transformed into a superposition of multiple single-soliton solution as

$$\psi_{{1\left( {mKdV} \right)}}^{\left( 1 \right)} \left( {\xi ,\tau } \right) = \sqrt {\frac{24 B}{C} } \frac{{k_{1} e^{{k_{1} \xi - Bk_{1}^{3} \tau }} }}{{1 + \left( {e^{{k_{1} \xi - Bk_{1}^{3} \tau }} } \right)^{2} }} = \sqrt {\frac{{6B k_{1}^{2} }}{C}} {\text{sech}}k_{1} \left( {\xi - Bk_{1}^{2} \tau } \right).$$
(4.17)
$$\psi_{{2\left( {mKdV} \right)}}^{\left( 1 \right)} \left( {\xi ,\tau } \right) \sim \mathop \sum \limits_{i = 1}^{2} \sqrt {\frac{{6B k_{i}^{2} }}{C}} {\text{sech}}k_{i} \left( {\xi - Bk_{i}^{2} \tau + \Delta_{i} } \right), k_{i} = \sqrt {\frac{{U_{i} }}{B}} \left( {i = 1,2} \right).$$
(4.18)
$$\psi_{{3\left( {mKdV} \right)}}^{\left( 1 \right)} \left( {\xi ,\tau } \right) \sim \mathop \sum \limits_{i = 1}^{3} \sqrt {\frac{{6B k_{i}^{2} }}{C}} {\text{sech}}k_{i} \left( {\xi - Bk_{i}^{2} \tau + \Delta_{i}^{*} } \right), k_{i} = \sqrt {\frac{{U_{i} }}{B}} \left( {i = 1,2, 3} \right).$$
(4.19)

Similarly, we can determine \(N\)-soliton solutions when \(N\ge 3.\) In this case, the wave propagates with speed \({U}_{i}\left(=B{k}_{i}^{2}\right)\), and moves with an amplitude \(\sqrt{\frac{6 {U}_{i}}{C}}\) and width \(\sqrt{\frac{B}{{U}_{i}}}\).

5 Results and discussion

This segment intends to investigate the parametric influences on the soliton structure, while the values of the parameters (viz. \({n}_{i0} \sim 1.5-3\) cm−3, \({n}_{ph0} \sim 1.5-3\) cm−3, \({n}_{pc0} \sim 5-10\) cm−3, \({T}_{e} \sim (200-1000) eV\), \({T}_{ph} \sim (200-1000) eV\), 3/2 < \({\kappa }_{e}\), \({\kappa }_{p}\)<\(\infty\)) have been chosen from experimental work of Pottelette et al. (1999) and different plasma assumptions (Alam et al. 2014; Akter and Hafez 2022). The properties of solitons (amplitude, width, phase velocity, etc.) vary significantly with the parameters (such as hot positron concentration, the effects of index parameters of electrons and positrons, etc.), which are discussed in detail here.

Figure 1 describe the wave propagation of the single-soliton when the parameters are assumed as \({\mu }_{ph}=0.15-0.6,{\mu }_{i}=0.15-0.6,{\sigma }_{1}=0.7,{\sigma }_{2}=0.8, {\kappa }_{e}=1.8, {\kappa }_{p}=3, {U}_{1}=0.04\). Here, 1(a) depicts the 3D wave profile of single-soliton of the KdV equation, whereas 1(b) depicts its 2D view of the solitons at time \(\tau =-100, -25, 0, 100\). According to the graphs, the single-soliton solution of the KdV equation results a bell-shaped wave pattern with a single global peak that vanishes at \(\xi \to \pm \infty\).

Fig. 1
figure 1

Single-soliton profile described by the solution (4.4) while the parametric values are: \({\mu }_{ph}=0.25,{\mu }_{i}=0.3,{\sigma }_{1}=0.7,{\sigma }_{2}=0.8, {\kappa }_{e}=1.8, {\kappa }_{p}=3,{U}_{1}=0.04.\) a Depicts Spatial plots in the interval \(\xi \in \left[-20, 20\right], \tau \in [-\mathrm{100,100}]\) and b shows 2D profiles of the single-solitons at time \(\tau =-100, -25, 0, 100\)

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Figure 2 depicts the wave-structure of the two-solitons drawn from (4.7). Here, the two-soliton wave propagation is studied by considering the parametric values as \({\mu }_{ph}=0.25,{\mu }_{i}=0.3,{\sigma }_{1}=0.7,{\sigma }_{2}=0.8, {\kappa }_{e}=1.8, {\kappa }_{p}=3, {U}_{1}=0.04,{U}_{2}=0.14\). The temporal evaluation of the two-solitons interaction is presented for a range of values of τ in Fig. 2. As illustrated in Fig. 2b, the bigger amplitude soliton is behind the smaller amplitude soliton at τ = − 100. Subsequently, as depicted in Fig. 2c, two solitons merge at τ = −25, and then in Fig. 2d they completely merge to form a single soliton at τ = 0. Figure 2e shows how the solitons depart from one another at τ = 100. Figure 2f displays the combined profile of the two solitons It is evident that the merging soliton’s amplitude is less than that of the taller soliton but larger than that of the shorter soliton.

Fig. 2
figure 2

Profile of two-soliton described by (4.7) while the parametric values are: \({\mu }_{ph}=0.25,{\mu }_{i}=0.3,{\sigma }_{1}=0.7,{\sigma }_{2}=0.8, {\kappa }_{e}=1.8, {\kappa }_{p}=3, {U}_{1}=0.04,{U}_{2}=0.14.\) a Depicts spatial plots in the interval \(\xi \in \left[-\mathrm{20,20}\right], \tau \in [-\mathrm{100,100}]\) and b–f give 2D profiles of the two-solitons at time \(\tau =-100, -25, 0, 100\)

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Similarly, Fig. 3 represents the wave-structure of three-solitons. Here, 3(a) and 3(b)-3(f) represent the wave profiles’ 3D and 2D views, respectively, where \({\mu }_{ph}=0.25,{\mu }_{i}=0.3,{\sigma }_{1}=0.7,{\sigma }_{2}=0.8, {\kappa }_{e}=1.8, {\kappa }_{p}=3, {U}_{1}=0.04,{U}_{2}=0.14,{U}_{3}=0.33\). The figures exemplify how the wave moves with three humps in the + ive direction of \(\xi\). At time \(\tau =-100\), the soliton of higher amplitude is behind the soliton of lower amplitude. The solitons start to merge at later time \(\tau =-25\), say. At τ = 0, the solitons fully merge to form a single soliton. Again, they start to diverge from one another and eventually, at last, each soliton reverts to its initial shape and speed. As illustrated in Fig. 3e, the smaller amplitude solitary wave is behind the greater amplitude soliton at τ = 100. However, the fluctuation of the combined compressive three-soliton profiles for various values of τ is displayed in Fig. 3f.

Fig. 3
figure 3

Profile of three-soliton described by (4.12) while the parametric values are: \({\mu }_{ph}=0.25,{\mu }_{i}=0.3,{\sigma }_{1}=0.7,{\sigma }_{2}=0.8, {\kappa }_{e}=1.8, {\kappa }_{p}=3, {U}_{1}=0.04,{U}_{2}=0.14, {U}_{3}=0.33.\) 2(a) shows spatial plots in the interval \(\xi \in \left[-40, 40\right], \tau \in [-100, 100]\) and 2(b)-2(f) give 2D profile of the three-solitons at time \(\tau =-100, -25, 0, 100\)

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Since \(A\) and \(B\) are always positive for the considered ranges of parametric values, the solitons are associated with positive potential only. Figures 1, 2 and 3 give the validity of the observation.

Figure 4 demonstrates the impact of hot positron concentration on the salient features of the single-, two- and three-soliton profiles. The figures are taken against different hot positron concentration \({\mu }_{ph}=0.2, 0.4, 0.6\) while the other parameter are: \({\mu }_{i}=0.3\), \({\sigma }_{1}=0.7\), \({\sigma }_{2}=0.8\), \({\kappa }_{e}=1.8\), \({\kappa }_{p}=3\),\({U}_{1}=0.04\), \({U}_{2}=0.14\), \({U}_{3}=0.33\). The figures show that hot positron concentration has a reducing effect on the solitons’ amplitudes and widths.

Fig. 4
figure 4

Variations of the 2D profiles of multiple-solitons for different hot positron concentration \({\mu }_{ph}=0.2, 0.4, 0.6\), when \({\mu }_{i}=0.3\), \({\sigma }_{1}=0.7\), \({\sigma }_{2}=0.8\), \({\kappa }_{e}=1.8\), \({\kappa }_{p}=3\),\({U}_{1}=0.04,{U}_{2}=0.14, {U}_{3}=0.33\)

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Figures 5a–c show how the single-, two-, and three-soliton natures are affected by the index parameter of electrons while the parameters are: \({\mu }_{ph}=0.25,{\mu }_{i}=0.3,{\sigma }_{1}=0.7,{\sigma }_{2}=0.8, {\kappa }_{p}=3, {U}_{1}=0.11,{U}_{2}=0.12,{U}_{3}=0.13.\) It is observed from the figures that the index parameter of electrons has increasing impact on the soliton’s amplitude and width.

Fig. 5
figure 5

Variations of the 2D profiles of multiple-solitons for different the index parameter of electrons \({\kappa }_{e}=1.8, 2.0, 2.2\), when\({\mu }_{i}=0.3\),\({\sigma }_{1}=0.7\),\({\sigma }_{2}=0.8\),\({\kappa }_{e}=1.8\),\({\kappa }_{p}=3\),\({U}_{1}=0.04,{U}_{2}=0.14, {U}_{3}=0.33\)

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Figure 6 implies that the hot positron concentration (index parameter of electrons) reduces (increases) the phase velocity drastically. Besides, we observe from the dispersion relation that the index parameter of positrons \(({\kappa }_{p})\) has an increasing effect on the phase speed for \({\kappa }_{p}\ge \frac{3}{2}\) only.

Fig. 6
figure 6

a Variation of the phase speed against \({\mu }_{ph}\) \(\in \left[0.2, 0.6\right]\) and b Variation of the phase speed against \({k}_{e}\) \(\in \left[1.5, 6\right]\) when \({\mu }_{i}=0.3,{\sigma }_{1}=0.7,{\sigma }_{2}=0.8,{\kappa }_{p}=3\)

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In Fig. 7a, we have shown the phase shift variation for two-soliton against \({\kappa }_{e}\) with the fixed values of the other parameters (viz. \({\mu }_{ph}=0.25,\) \({\mu }_{i}=0.3,{\sigma }_{1}=0.7,{\sigma }_{2}=0.8,{\kappa }_{p}=3\)). Similarly, in Fig. 7b, we have presented the variation of phase shifts for three-soliton against \({\kappa }_{e}\) with fixed values of the other parameters (stated as above).

Fig. 7
figure 7

a Variation of the phase shift against \({\kappa }_{e}\) \(\in \left[2, 10\right]\) for two solitons and b Variation of the phase shift against \({\kappa }_{e}\) \(\in \left[2, 10\right]\) for three solitons when \({\mu }_{ph}=0.25,\) \({\mu }_{i}=0.3,{\sigma }_{1}=0.7,{\sigma }_{2}=0.8,{\kappa }_{p}=3\)

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If there exists a physical phenomena for which \(A=0\) (though \(A\) is always positive for our considered plasma problem and compressive solitons exists only), multiple-solitons cannot be described by the KdV equation. In this case, the mKdV equation will be suitable for explanation of the physical situation.

6 Conclusion

We have effectively built the multiple-soliton solutions to the KdV and mKdV equations. Although Hirota’s bilinear method can be used to discover the soliton solutions, simplified Hirota method is more user-friendly and employs no bilinear operators. According to the result, it is observed that the solitons’ amplitudes and widths are reduced by the hot positron concentration. Furthermore, the amplitudes and widths are increasingly influenced by the density of the index parameter of electrons (similar observations are found by Alam et al. 2014; Ullah et al. 2020; Ismael et al. 2020). Besides, it is evident that the merging soliton’s amplitude is less than that of the taller soliton but larger than that of the shorter soliton, and the similarly result is observed by Drazin and Johnson (1989). The aforementioned findings lead us to conclude that the plasma parameters significantly influence the fundamental characteristics of the multiple-solitons. Since the results are consistent with the fundamentals of plasma and previous works, we may anticipate that the results will be useful in illustrating the positron-acoustic multiple-solitons propagating in various unmagnetized plasma systems that exist in astrophysical and laboratory plasmas, such as the pulsar magnetosphere, the active galactic nuclei, the quasar’s relativistic jet, the inner region of the accretion disk surrounding a black hole, ultra-intense laser matter interaction, positron injection into the electron–ion system, or relativistic heavy-ion collisions, etc.