Ion acoustic solitary waves, a fundamental mode in nonlinear plasma environments, have been a subject of extensive research for several decades [1,2,3,4,5]. There are two main approaches to study ion-acoustic solitary waves, namely, the reductive perturbation method and the pseudopotential theory. For small-amplitude excitation, the reductive perturbation technique is applicable. As is widely known, the Korteweg-de Vries (KdV) equation and its generalizations are very useful approximation to the full set of fluid equations describing ion acoustic waves . Recently, Saha et al. studied bifurcation and quasi-periodic behaviours of ion acoustic waves in a magnetized electron-ion (e-i) plasma with nonthermal electrons featuring Cairns-Tsallis distribution . The Kadomtsev-Petviashvili equation equation is derived by employing the reductive perturbation method, and used to analyze the existence of solitary wave solutions and periodic traveling wave solutions. In addition to solitary waves, the study of cnoidal waves in field free or magnetized plasmas is another topic of great interest due to its importance in the nonlinear transport processes in plasmas [7,8,9,10,11,12]. Recently, Saha and Chatterjee studied ion acoustic periodic waves in an unmagnetized plasma with superthermal electrons . By applying a non-perturbative approach, they obtained a planar dynamical system and showed that plasma parameters significantly affect the characteristics of periodic wave structures. Besides, the interaction between solitons, such as the head-on collisions of ion acoustic waves, have been intensively studied by means of the extended Poincaré-Lighthill-Kuo perturbation method [13,14,15,16,17,18].
However, interactions among different types of nonlinear waves in e-i plasmas as well as electron-positron-ion (e-p-i) plasmas are rarely considered. Recently, Keane et al.  considered the propagation of Alfvén envelope solitons in a quantum plasma. It is found that the time-independent Alfvén density soliton possesses a quasi-soliton structure with the Gauss peak surrounded by small amplitude sinusoidal waves. Furthermore, it was numerically confirmed that the shape of the Gauss peak remains the same in space-time evolution. This interesting numerical phenomena reveals that the interaction between the soliton core and its surrounded oscillations is elastic. Consequently, it seems significant if one can establish an analytical solution to explain such type of wave interactions. Fortunately, analytic solutions, describing interactions between solitons and other nonlinear excitations, have been obtained by combining the symmetry reduction method with the Darboux transformation or Bäcklund transformation related nonlocal symmetries [21,22]. Then, the consistent Riccati expansion method has been proposed to find the interacting wave solutions for many integrable systems [23,24,25]. It turns out that the KdV, nonlinear Schrödinger, and sine-Gordon equations possess different types of interacting wave solutions, such as nanopteron solution  or topological soliton on a cnoidal wave background .
In this paper, we are concerned with the propagation of ion acoustic quasi-soliton in an e-p-i plasma with superthermal electrons and positrons. In the next section, the KdV equation, describing the propagation of nonlinear ion acoustic waves in an e-p-i plasma, is derived by applying the reductive perturbation method. In addition, exact soliton-cnoidal wave solution of the KdV equation and its quasi-soliton behaviour are presented. The influence of plasma parameters on characteristics of the quasi-soliton is presented in the third section. The last section is a discussion.
2 KdV equation and its quasi-soliton solution
We are interested in studying ion acoustic quasi-soliton in a fully ionized, collisionless unmagnetized plasma, whose components are cold ions, superthermal electrons and superthermal positrons. Adopting a one dimensional fluid formulation, the dynamics of the cold inertial ion component, ignoring the thermal pressure effect, is governed by the following dimensionless set of fluid equations
where ui is the dimensionless ion hydrodynamic velocity normalized by the ion acoustic speed Cs = (Te/mi)(1/2) with the electron temperature Te and the rest ion mass mi. ni, ne, and np are ion, positron and electron number densities normalized by their equilibrium values ni0, ne0, and np0, respectively. Under the slow ion acoustic wave time scale, annihilation between electrons and positrons is negligible. Thus, we assume that the number density of electrons and positrons satisfies the following distribution functions
with σ = Te/Tp being the electron to positron temperature ratio, ϕ is the ion acoustic wave potential normalized by Te/e, and p = np0/ne0 is the fractional concentration of positrons with respect to electrons in the equilibrium state. It is noted that time variable t is normalized ion plasma period , and space variable x is normalized by the Debye length .
Under the weak perturbation assumption with eϕ/ Te ≪ 1, we obtain the following approximation of the Poisson’s equation by substituting the distribution functions of electrons and positrons (4) into equation (3)
In order to investigate the nonlinear excitations of small-amplitude ion acoustic waves, we employ the reductive perturbation technique. A set of stretched variables are introduced as
with ϵ is a small parameter lies in the range 0 < ϵ < 1, and V is the normalized phase speed to be determined later. The perturbed quantities are expanded about their equilibrium states as
Substituting equations (8)-(11) into equations (1)-(3) and setting the coefficients of different powers of ϵ to zero, one finds a sequence of differential equations. From the order of ϵ3/2, we obtain the compatibility conditions about n1, u1 and ϕ1, which can be solved as
The second order terms in ϵ yield a further set of compatibility conditons. Together with the known results (12), we have
Eliminating the second order quantities, we obtain the KdV equation
where the nonlinear and dispersive coefficients A and B are given by
The soliton-cnoidal wave solution of the KdV equation as well as its quasi-soliton behaviour has been studied in detail , which reads
and the wave parameters are determined as
Obviously, under the ultralimit conditon m = 0 (G = 1), V1 = V and V2 = −V, the soliton-cnoidal wave solution (17) degenerates to the classical soliton solution of the KdV equation
Consequently, the solution (17) has an interesting quasi-soliton behaviour under the asymptotic condition V1 = V, V2 = −V, and m → 0. It is found that the soliton core profile tends to the classical soliton of the KdV equation while the surrounded conidal wave becomes small amplitude sinusoidal vibrations around zero. Thus, the solution (17) can be viewed as a quasi-soliton solution for its quasi-soliton behaviour.
3 Parametric investigation
In this section, we illustrate the space-time evolution of the soliton-cnoidal solution (17), and subsequently discuss the influence of relevant plasma parameters on characteristics of the quasi-soliton.
Since V1 and V2 are taken as the velocities of the soliton and the cnoidal wave, respectively, remain arbitrary, the soliton-cnoidal wave solution (17) can display different types of interactions. For instance, if setting V2 = 0, then the cnoidal wave degenerates into a standing wave, and one can view a scenario in which a soliton moves on a standing cnoidal wave background. Fig. 1 (a) depicts a right-moving soliton colliding with a standing cnoidal wave. Fig. 1 (b) shows an elastic overtaking interaction. Fig. 1 (c) displays an elastic head-on collision between a right-going soliton and a left-going cnoidal wave. It is evident from Fig. 1 that the interaction between the soliton and every peak of the cnoidal wave is elastic with phase shifts. Interestingly, it has been shown that the collision-induced phase shift of the cnoidal wave is half wavelength 
To demonstrate the effect of electron superthermal parameter κe, we have presented the quasi-soliton structure for fixed values of the plasma parameters, and κe = 2, 5, 50 denoting the red, green and blue curves, respectively, in Fig. 2. From the figure, one observes that the amplitude and width of the quasi-soliton flourish with increasing κe. Physically, the electron pressure provides the needed restoring force of the ion acoustic wave. Consequently, the increase in κe, which can be interpreted as an equivalent process of increasing electron pressure, makes the amplitude higher of the soliton core as well as the surrounded vibrations. On the other hand, the increase of κe causes a larger value of the dispersion term in the KdV equation, and leads to wider quasi-solitons. So increasing superthermality (smaller κe) leads to lower amplitude and narrower quai-solitons. It also should be noted that the amplitude and the width of the quasi-soliton will reach a maximum in the Maxwellian limit κe → ∞. In Fig. 3, a similar effect can be observed with varying positron superthermal parameter κp.
The effect of positron-to-electron concentration ratio p = np0/ne0 on characteristics of the ion acoustic quasi-soliton is also investigated. It is quite evident from Fig. 4 that the amplitude as well as the width of the quasi-soliton diminishes with increasing positron concentration. As is widely known, the driving force of the ion acoustic wave is offered by the ion inertia and the increasing positron concentration p implies the depopulation of ions. So increasing positron concentration ratio p leads to lower amplitude and narrower quai-solitons.
4 Summary and discussion
In this work, we are concerned with the propagation of ion acoustic quasi-soliton in an e-p-i plasma with superthermal electrons and positrons. Quasi-soliton behaviour of the soliton-cnoidal wave solution has been presented from its connection with the classical soliton solution. Numerical study shows that plasma parameters affect characteristics of the quasi-soliton significantly. It is hoped that the quasi-soliton can be observed in real nonlinear plasma environments.
The work is sponsored by National Natural Science Foundation of China (Nos. 11475052, 11605102, 11675055, 51605252).
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About the article
Published Online: 2018-09-01
Citation Information: Open Physics, Volume 16, Issue 1, Pages 563–567, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0073.
© 2018 Jianyong Wang et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0