Startseite Ion acoustic quasi-soliton in an electron-positron-ion plasma with superthermal electrons and positrons
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Ion acoustic quasi-soliton in an electron-positron-ion plasma with superthermal electrons and positrons

  • Jianyong Wang , Ying Zeng , Zufeng Liang EMAIL logo , Yani Xu und Yuanxiang Zhang
Veröffentlicht/Copyright: 1. September 2018
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Aus der Zeitschrift Open Physics Band 16 Heft 1

Abstract

In this work, we are concerned with the ion acoustic quasi-soliton in an electron-positron-ion plasma with superthermal electrons and positrons. By using the reductive perturbation method, the Korteweg-de Vries equation is derived from the governing equations of ion acoustic waves. An interesting soliton-cnoidal wave solution of the Korteweg-de Vries equation and its quasi-soliton behaviour are presented. The influence of electron superthermality, positron superthermality and positron concentration ratio on characteristics of the quasi-soliton is confirmed to be significant.

Keywords: Ion acoustic wave; electron-positron-ion plasma; KdV equation; quasi-soliton behaviour
PACS: 52.35.Fp; 52.35.Mw; 52.35.Sb; 52.35.Tc

1 Introduction

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 [1]. 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 [6]. 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 [12]. 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. [20] 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 [26] or topological soliton on a cnoidal wave background [27].

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

∂ni∂t+∂(niui)∂x=0,(1)
∂ui∂t+ui∂ui∂x=−∂ϕ∂x,(2)
∂2ϕ∂x2=ne−pnp−(1−p)ni,(3)

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

ne=1−ϕκe−32−κe+12,np=(1+σϕκp−32)−κp+12,(4)

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 ωpi−1=mi/(4πne0e2) , and space variable x is normalized by the Debye length λDe=Te/(4πne0e2).

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)

∂2ϕ∂x2≃(1−p)(1−ni)+aκϕ+12bκϕ2,(5)

where

aκ=2κe−12κe−3+(2κp−1)pσ2κp−3,(6)
bκ=4κe2−1(2κe−3)2−(4κp2−1)pσ2(2κp−3)2.(7)

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

ξ=ϵ1/2(x−Vt),τ=ϵ3/2t,(8)

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

ni=1+ϵn1+ϵ2n2+⋯,(9)
ϕ=ϵϕ1+ϵ2ϕ2+⋯,(10)
ui=ϵu1+ϵ2u2+⋯.(11)

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

n1=aκ1−pϕ1, u1=1Vϕ1, V=1−paκ.(12)

The second order terms in ϵ yield a further set of compatibility conditons. Together with the known results (12), we have

n2=bκ2(1−p)ϕ12−11−p∂2ϕ1∂ξ2+aκ1−pϕ2,(13)

and

u2=14bκaκV−aκ(1−p)Vϕ12−12aκV∂2ϕ1∂ξ2+1Vϕ2.(14)

Eliminating the second order quantities, we obtain the KdV equation

∂ϕ1∂τ+Aϕ1∂ϕ1∂ξ+B∂3ϕ1∂ξ3=0,(15)

where the nonlinear and dispersive coefficients A and B are given by

A=32V−bκV32(1−p), B=V32(1−p).(16)

The soliton-cnoidal wave solution of the KdV equation as well as its quasi-soliton behaviour has been studied in detail [26], which reads

ϕ1=3(V1−V2)2AG2[(m2−1+2G)2(m2−1)tanh⁡(w)2−2δmS(m2−1+2G)tanh⁡(w)+m2−1]−(m2+7)V1−(3m2+5)V22A(m2−1),(17)

with

w=ξ−V1τW1+c1arctanh(c2S(η,m)),G=1−m2S2+δmCD,η=ξ−V2τW2S≡sn(η,m),C≡cn(η,m),D≡dn(η,m).

and the wave parameters are determined as

W1=8B(1−m2)V1−V2,W2=2B(1−m2)V1−V2,c1=δ2,c2=m,δ2=1.(18)

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

u=3VAsech2x−VtW,W=4BV.(19)

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 [26]

Figure 1 Density plot of the collisions between a soliton and a cnoidal wave, given by equations (17) and (18), with the plasma parameters are κe = 2, κp = 5, p = 0.2, σ = 0.5, and the wave parameters are (a) m = 1/3, V1 = 0.5, V2 = 0, δ = 1; (b) m = 1/3, V1 = 0.5, V2 = 0.25, δ = 1; and (c) m = 1/3, V1 = 0.5, V2 = −0.25, δ = 1, respectively.
Figure 1

Density plot of the collisions between a soliton and a cnoidal wave, given by equations (17) and (18), with the plasma parameters are κe = 2, κp = 5, p = 0.2, σ = 0.5, and the wave parameters are (a) m = 1/3, V1 = 0.5, V2 = 0, δ = 1; (b) m = 1/3, V1 = 0.5, V2 = 0.25, δ = 1; and (c) m = 1/3, V1 = 0.5, V2 = −0.25, δ = 1, respectively.

Δcn=2W2K(m)=λc2.(20)

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.

Figure 2 Variations of the ion acoustic quasi-soliton with respect to the electron superthermal parameter κe. Other plasma parameters are taken as κp = 50, p = 0.1, σ = 0.9, and the wave parameters are m = 0.04, V1 = −V2 = 0.05 and δ = −1.
Figure 2

Variations of the ion acoustic quasi-soliton with respect to the electron superthermal parameter κe. Other plasma parameters are taken as κp = 50, p = 0.1, σ = 0.9, and the wave parameters are m = 0.04, V1 = −V2 = 0.05 and δ = −1.

Figure 3 Variations of the ion acoustic quasi-soliton with respect to the positron superthermal parameter κp. Other parameters are κe = 3, p = 0.1, σ = 0.9, m = 0.025, V1 = −V2 = 0.05 and δ = −1.
Figure 3

Variations of the ion acoustic quasi-soliton with respect to the positron superthermal parameter κp. Other parameters are κe = 3, p = 0.1, σ = 0.9, m = 0.025, V1 = −V2 = 0.05 and δ = −1.

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.

Figure 4 Variations of the ion acoustic quasi-soliton with respect to positron-to-electron concentration ratio p. Other plasma parameters are selected as κe = 3, p = 0.1, σ = 0.9, and wave parameters are m = 0.025, V1 = −V2 = 0.05 and δ = 1.
Figure 4

Variations of the ion acoustic quasi-soliton with respect to positron-to-electron concentration ratio p. Other plasma parameters are selected as κe = 3, p = 0.1, σ = 0.9, and wave parameters are m = 0.025, V1 = −V2 = 0.05 and δ = 1.

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.

Acknowledgement

The work is sponsored by National Natural Science Foundation of China (Nos. 11475052, 11605102, 11675055, 51605252).

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Received: 2018-02-23
Accepted: 2018-05-21
Published Online: 2018-09-01

© 2018 Jianyong Wang et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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  95. Special Issue Applications of Nonlinear Dynamics
  96. Study on dynamic characteristics of silo-stock-foundation interaction system under seismic load
  97. Microblog topic evolution computing based on LDA algorithm
  98. Modeling the creep damage effect on the creep crack growth behavior of rotor steel
  99. Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs
  100. Chinese open information extraction based on DBMCSS in the field of national information resources
  101. 10.1515/phys-2018-0079
  102. CPW-fed circularly-polarized antenna array with high front-to-back ratio and low-profile
  103. Intelligent Monitoring Network Construction based on the utilization of the Internet of things (IoT) in the Metallurgical Coking Process
  104. Temperature detection technology of power equipment based on Fiber Bragg Grating
  105. Research on a rotational speed control strategy of the mandrel in a rotary steering system
  106. Dynamic load balancing algorithm for large data flow in distributed complex networks
  107. Super-structured photonic crystal fiber Bragg grating biosensor image model based on sparse matrix
  108. Fractal-based techniques for physiological time series: An updated approach
  109. Analysis of the Imaging Characteristics of the KB and KBA X-ray Microscopes at Non-coaxial Grazing Incidence
  110. Application of modified culture Kalman filter in bearing fault diagnosis
  111. Exact solutions and conservation laws for the modified equal width-Burgers equation
  112. On topological properties of block shift and hierarchical hypercube networks
  113. Elastic properties and plane acoustic velocity of cubic Sr2CaMoO6 and Sr2CaWO6 from first-principles calculations
  114. A note on the transmission feasibility problem in networks
  115. Ontology learning algorithm using weak functions
  116. Diagnosis of the power frequency vacuum arc shape based on 2D-PIV
  117. Parametric simulation analysis and reliability of escalator truss
  118. A new algorithm for real economy benefit evaluation based on big data analysis
  119. Synergy analysis of agricultural economic cycle fluctuation based on ant colony algorithm
  120. Multi-level encryption algorithm for user-related information across social networks
  121. Multi-target tracking algorithm in intelligent transportation based on wireless sensor network
  122. Fast recognition method of moving video images based on BP neural networks
  123. Compressed sensing image restoration algorithm based on improved SURF operator
  124. Design of load optimal control algorithm for smart grid based on demand response in different scenarios
  125. Face recognition method based on GA-BP neural network algorithm
  126. Optimal path selection algorithm for mobile beacons in sensor network under non-dense distribution
  127. Localization and recognition algorithm for fuzzy anomaly data in big data networks
  128. Urban road traffic flow control under incidental congestion as a function of accident duration
  129. Optimization design of reconfiguration algorithm for high voltage power distribution network based on ant colony algorithm
  130. Feasibility simulation of aseismic structure design for long-span bridges
  131. Construction of renewable energy supply chain model based on LCA
  132. The tribological properties study of carbon fabric/ epoxy composites reinforced by nano-TiO2 and MWNTs
  133. A text-Image feature mapping algorithm based on transfer learning
  134. Fast recognition algorithm for static traffic sign information
  135. Topical Issue: Clean Energy: Materials, Processes and Energy Generation
  136. An investigation of the melting process of RT-35 filled circular thermal energy storage system
  137. Numerical analysis on the dynamic response of a plate-and-frame membrane humidifier for PEMFC vehicles under various operating conditions
  138. Energy converting layers for thin-film flexible photovoltaic structures
  139. Effect of convection heat transfer on thermal energy storage unit
https://doi.org/10.1515/phys-2018-0073
Schlagwörter für diesen Artikel
Ion acoustic wave; electron-positron-ion plasma; KdV equation; quasi-soliton behaviour
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  93. A method to reduce partial discharges in motor windings fed by PWM inverter
  94. Reluctance network lumped mechanical & thermal models for the modeling and predesign of concentrated flux synchronous machine
  95. Special Issue Applications of Nonlinear Dynamics
  96. Study on dynamic characteristics of silo-stock-foundation interaction system under seismic load
  97. Microblog topic evolution computing based on LDA algorithm
  98. Modeling the creep damage effect on the creep crack growth behavior of rotor steel
  99. Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs
  100. Chinese open information extraction based on DBMCSS in the field of national information resources
  101. 10.1515/phys-2018-0079
  102. CPW-fed circularly-polarized antenna array with high front-to-back ratio and low-profile
  103. Intelligent Monitoring Network Construction based on the utilization of the Internet of things (IoT) in the Metallurgical Coking Process
  104. Temperature detection technology of power equipment based on Fiber Bragg Grating
  105. Research on a rotational speed control strategy of the mandrel in a rotary steering system
  106. Dynamic load balancing algorithm for large data flow in distributed complex networks
  107. Super-structured photonic crystal fiber Bragg grating biosensor image model based on sparse matrix
  108. Fractal-based techniques for physiological time series: An updated approach
  109. Analysis of the Imaging Characteristics of the KB and KBA X-ray Microscopes at Non-coaxial Grazing Incidence
  110. Application of modified culture Kalman filter in bearing fault diagnosis
  111. Exact solutions and conservation laws for the modified equal width-Burgers equation
  112. On topological properties of block shift and hierarchical hypercube networks
  113. Elastic properties and plane acoustic velocity of cubic Sr2CaMoO6 and Sr2CaWO6 from first-principles calculations
  114. A note on the transmission feasibility problem in networks
  115. Ontology learning algorithm using weak functions
  116. Diagnosis of the power frequency vacuum arc shape based on 2D-PIV
  117. Parametric simulation analysis and reliability of escalator truss
  118. A new algorithm for real economy benefit evaluation based on big data analysis
  119. Synergy analysis of agricultural economic cycle fluctuation based on ant colony algorithm
  120. Multi-level encryption algorithm for user-related information across social networks
  121. Multi-target tracking algorithm in intelligent transportation based on wireless sensor network
  122. Fast recognition method of moving video images based on BP neural networks
  123. Compressed sensing image restoration algorithm based on improved SURF operator
  124. Design of load optimal control algorithm for smart grid based on demand response in different scenarios
  125. Face recognition method based on GA-BP neural network algorithm
  126. Optimal path selection algorithm for mobile beacons in sensor network under non-dense distribution
  127. Localization and recognition algorithm for fuzzy anomaly data in big data networks
  128. Urban road traffic flow control under incidental congestion as a function of accident duration
  129. Optimization design of reconfiguration algorithm for high voltage power distribution network based on ant colony algorithm
  130. Feasibility simulation of aseismic structure design for long-span bridges
  131. Construction of renewable energy supply chain model based on LCA
  132. The tribological properties study of carbon fabric/ epoxy composites reinforced by nano-TiO2 and MWNTs
  133. A text-Image feature mapping algorithm based on transfer learning
  134. Fast recognition algorithm for static traffic sign information
  135. Topical Issue: Clean Energy: Materials, Processes and Energy Generation
  136. An investigation of the melting process of RT-35 filled circular thermal energy storage system
  137. Numerical analysis on the dynamic response of a plate-and-frame membrane humidifier for PEMFC vehicles under various operating conditions
  138. Energy converting layers for thin-film flexible photovoltaic structures
  139. Effect of convection heat transfer on thermal energy storage unit
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