1 Introduction

Electron-positron (e-p) pair plasmas are present in many astrophysical environments such as the solar wind [1,2,3,4,5,6], the Earth's magnetosphere [7,8], pulsars [9,10], and microquasars [11]. Moreover, e-p plasmas can be created by ultra-intense laser interaction with matter in the laboratory [12,13,14,15,16]. The long-lived runaway positrons can also be generated in post-disruption tokamak plasmas [17]. In dense astrophysical environments, ions usually exist in addition to electrons and positrons, for example, nearby hot white dwarfs and microquasar [18,19]. Energetic electrons, accelerated to high suprathermal energies, are also found to be produced in ultra-intense laser fields [12], tokamaks [20], the solar wind [21,22], and the Earth's magnetosphere [23]. In particular, the energy distribution of suprathermal electrons in solar flares was found to be well described by a power law with a maximum high-energy cutoff of 3 GeV [22]. Hence, studying e-p pair plasmas with suprathermal electrons are important for both laboratory and astrophysical plasmas.

Electrostatic waves usually occur in a plasma containing distinct electron populations with different temperatures [24,25,26,27], namely cool electrons, $T_{c}$, and hot electrons, $T_{h}$. The cool electron motion provides the inertia required to maintain electrostatic oscillations, while the hot electron pressure produces the restoring force for electrostatic waves propagating at a phase speed between the cool and hot electron thermal velocities. In such a plasma, the ions can be assumed to make a stationary background providing charge neutrality. It is found that Landau damping is minimized if the cool electron fraction of the total number density of electrons is in the range of $0.2\lesssim n_{c}/(n_{c}+n_{h})\lesssim0.8$ and the hot electron temperature is much higher than the cool electron temperature $T_{h}/T_{c} \gg10$  [26,27,28,29]. The dynamics of electron-acoustic waves in a two-electron-temperature plasma have been studied by many authors [24,25,26,27,28]. Moreover, linear and nonlinear studies of electron-acoustic waves in the presence of suprathermal (or non-thermal) electrons have received a great deal of interest in recent years, both in unmagnetized [29,30,31] and in magnetized plasmas [32,33]. Negative polarity electrostatic wave structures were found to exist in a two-electron-temperature plasma with excess suprathermal electrons [32,31], which are associated with the inertia of mobile cold electrons. However, positive polarity electrostatic waves moving at velocities comparable to electrons have been reported in the auroral magnetosphere [34,35]. Inclusion of a beam component [36,37] or finite inertia [38,39] may lead to a positive polarity electrostatic wave. Alternatively, a fraction of mobile positrons (or electron holes), which are created by the solar wind, may maintain the inertia for the propagation of positive polarity electrostatic waves. Interestingly, a considerable fraction of positrons has been recently observed at the solar wind: $\phi(\mathrm{e}^{+})/(\phi(\mathrm{e}^{+})+\phi(\mathrm{e}^{-}))\lesssim 0.1$ at energies 0.04-1GeV [1,2,3,4,5,6]. Moreover, a significant positron density has been measured in laser-plasma experiences: $n(\mathrm{e}^{+})/(n(\mathrm{e}^{+})+n(\mathrm{e}^{-}))\sim0.05$-0.1 in $\sim10$MeV [14] and $\sim0.01$-0.1 in $150$MeV [15]. The experimental temperature of positrons was measured to be roughly half of the effective electron temperature in ultra-intense laser fields [14] (In this paper, $T_{p} \sim T_{c}$, while $T_{p}/T_{h} \ll0.1$ to minimize Landau damping; see Ref. [40]). The positron effect may have important implications for the dynamics of positive polarity electrostatic waves of the auroral magnetosphere.

The propagation of electrostatic waves in e-p pair plasmas can also be supported by the inertia of mobile cool positrons, while background hot electrons act as the restoring force. Interestingly, the coherent microwave radiation has been reported in pulsars [41,42], which is assumed to be originated from electric fields of e-p pair plasma over the polar caps of neutron stars [43,44]. It was proposed that coherent pulsar radio emissions could be due to nonlinear electrostatic solitary oscillations generated by effective electron-positron streams on the polar caps in rotating magnetized neutron stars [45]. Studies of e-p pair plasmas demonstrated that electrostatic solitary waves can be generated [40], though a Maxwellian distribution was assumed. A number of papers have also been devoted to the linear and nonlinear dynamics of electron-acoustic waves [46,47], electrostatic waves [48,49,50,51,52,53,54,55], and in the presence of suprathermal (and non-thermal) electrons [50,46,52] in e-p pair plasmas. Moreover, the propagation of ion-acoustic waves [56,57,58,59], and dust-acoustic waves [60,61,62] have recently been studied in e-p plasmas. However, the nonlinear dynamics and the existence domains of electrostatic solitary waves have not fully been investigated in the presence of positrons. It is important to study the occurrence of electrostatic solitary wave structures in e-p pair plasmas with suprathermal electrons, which may lead to the (co-)existence of positive and negative polarity electrostatic waves similar to what observed in the Earth's magnetosphere [34,35], as well as a possible explanation for coherent pulsar radio emissions [45].

In this paper, we aim to explore the effect of mobile cool positrons (electron holes) on electrostatic solitary waves in an e-p pair plasma with suprathermal electrons. In Section 2, a two-fluid model is presented. In Section 3, a dispersion relation is derived. In Section 4, a nonlinear pseudopotential (Sagdeev) method is used to investigate the existence of large-amplitude electrostatic solitary waves. Section 5 is devoted to a parametric investigation of the nonlinear form and the characteristics of electrostatic solitary wave structures. Finally, our results are summarized in the concluding section 6.