Nonlinear ion-acoustic structures in a nonextensive electron–positron–ion–dust plasma: Modulational instability and rogue waves

Abstract

The nonlinear propagation of planar and nonplanar (cylindrical and spherical) ion-acoustic waves in an unmagnetized electron–positron–ion–dust plasma with two-electron temperature distributions is investigated in the context of the nonextensive statistics. Using the reductive perturbation method, a modified nonlinear Schrödinger equation is derived for the potential wave amplitude. The effects of plasma parameters on the modulational instability of ion-acoustic waves are discussed in detail for planar as well as for cylindrical and spherical geometries. In addition, for the planar case, we analyze how the plasma parameters influence the nonlinear structures of the first- and second-order ion-acoustic rogue waves within the modulational instability region. The present results may be helpful in providing a good fit between the theoretical analysis and real applications in future spatial observations and laboratory plasma experiments.

Introduction

The electron–positron–ion plasma, characterized as a fully ionized gas containing electrons and positrons having equal masses and charges with opposite polarity, is believed to exist in many astrophysical objects such as active galactic nuclei [1], magnetospheres of pulsar [2], our early universe [3], and the inner regions of the accretion disks surrounding the black hole [4]. There is a possibility of the existence of astrophysical plasmas with a high concentration of positrons. From the view of large-scale structures in the cosmological environments, the positrons may have a high concentration and play an important role in the astrophysical plasmas. In the laboratory conditions, an electron–positron beam–plasma experiment has been performed with a large number of positrons [5] and the instability leading to a large amplitude has been also observed in this experiment. In 2008, physicists at the Lawrence Livermore National Laboratory in California produced more than 100 billion positrons [6], which could be valuable for the further laboratory plasma experiments with a long time existence and high concentration of positrons.

However, the dust grains ($\mu m$ to sub-$\mu m$ sizes) are ubiquitous in the laboratory and space environments such as cometary surroundings, interstellar clouds and planetary rings [7], [8]. When the dust grains are immersed into a plasma, they are usually charged due to the absorption of the charged particles. These charged dust grains could either modify the behavior of the normal waves and instabilities, or introduce new eigenmodes [9], [10], [11]. Therefore, the plasma may become an admixture of electrons, positrons, ions and dust grains. In the electron–positron–ion–dust (EPID) plasmas, the charge of the dust grains can be either negative or positive when the number of positrons depositing on them is either smaller or larger than the number of electrons. Accordingly, it is interesting and important to study ion-acoustic waves (IAWs) in the EPID plasmas [12], [13].

It is well known that the properties of wave motions in plasma depend on the velocity distribution of the plasma particles. In the past few decades, the most commonly used distribution was the well-known Maxwellian particle distribution. However, a number of space observations indicates the presence of particles which depart from the Maxwellian distribution. For example, the nonthermal and superthermal electrons have been observed in the Earth’s bow-shock, in the upper ionosphere of Mars and in the vicinity of the Moon [14], [15], [16], [17]. Also the non-Maxwellian electron velocity distributions have been observed and measured in the laboratory experiments [18]. Motivated by these findings, a nonextensive generalization of the Boltzmann–Gibbs–Shannon (BGS) entropy, first recognized by Renyi [19] and subsequently proposed by Tsallis [20], can suitably extend the standard additivity of the entropies to the nonextensive cases. In this context, the entropic index $q$ characterizes the degree of nonextensivity of the considered system. The $q$-nonextensive statistics can provide a powerful and convenient frame for the analysis of many astrophysical phenomena, such as the head-on collision of black holes, the dynamics of inflationary cosmologies and gravitational wave emission [21], [22], [23]. This distribution also presents a good fit to the experimental results. For electrostatic plane-wave propagation in a collisionless thermal plasma, the dispersion relation in Tsallis formalism fits the experimental data very well when $q<1$ [24], while the standard Bohm–Gross relation based on the classical Maxwellian distribution ($q=1$) only provides a crude description. In Ref. [25], the anomalous diffusion and non-Gaussian statistics fitting the Tsallis distribution were detected experimentally in a two-dimensional driven-dissipative plasma system. It is necessary to point out that the transformation $\kappa =1/(q-1)$ which was first provided by Leubner in Ref. [26] links $q$-statistic and $\kappa$-distribution [27] (note that for $q\to 1,\kappa \to \infty$).

Two-electron temperature plasmas are very common in the space [28], as well as in laboratory experiments [29]. For example, when out flows of the electron–positron plasma from pulsars enters an interstellar cold, low-density electron–ion plasma, the two-electron temperature plasma could be formed [30]. R. Sabry et al. studied the ion-acoustic envelope solitons in electron–positron–ion plasma with two-electron temperature distributions [31]. Also Mishra et al. investigated ion-acoustic double layers in electron–positron–ion plasma with two-electron temperature distributions [32].

Due to the carrier wave self-interaction or intrinsic medium nonlinearity, the propagation of nonlinear waves in a dispersive media is generally subject to amplitude modulation, which can be studied via the nonlinear Schrödinger equation (NLSE) derived by the reductive perturbation method (RPM) [33]. This method reduces the very complicated systems of equations modeling such complex behavior as wave propagation in the plasmas to one simple canonical forms and seeks to extract the effect of dominant nonlinearities.

The aim of this paper is to investigate the modulational instability (MI) of the planar and nonplanar IAWs in an EPID plasma with two-electron temperature distributions, whose components are either negatively or positively charged dust grains, positive ions, $q$-nonextensive positrons and electrons. For the planar case, the nonlinear evolution of the first- and second-order ion-acoustic rogue waves are presented in the paper. Rogue waves (also known as freak waves, killer waves, giant waves, or extreme waves), the singular, rare and high-energy phenomena with amplitude much higher than the average wave crests around it, have appeared in many physical systems such as oceans, Bose–Einstein condensates, optics, and super-fluids [34], [35], [36], [37]. The rogue waves have been successively reported by numerous news media, Nature News, BBC News, Science Daily, Physics World, Scientific American, for instance. Due to the MI, the rogue waves may arise from the instabilities of the initial conditions that tend to grow exponentially and thus have the possibility of increasing up to very high amplitudes. It has been shown that the experimental results on the study of first- and second-order rogue waves [38], [39], [40] in a water tank which can be modeled by the NLSE are in good agreement with the theory. It is necessary to point out that the rational solutions of NLSE play an important role in the study of rogue waves. The first-order rational solution was given by Peregrine [41] as early as 1983. Based on [42], the next-order one was presented in Ref. [43] as a possible explanation for rogue waves with higher amplitude. In addition, this kind of rational solutions could resolve the mystery of rogue waves observed in optical fibers [44], [45], [46] and multicomponent plasma [47].

The paper is organized as follows: In Section 2, we present the basic equations of our theoretical model. Then, the modified NLSE is derived via the RPM. In Section 3, the effects of the plasma parameters on the MI and amplitudes of the first- and second-order ion-acoustic rogue waves are discussed. Finally, conclusions are given in Section 4.