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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 43, NO. 11, NOVEMBER 2015

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Application of Particle-in-Cell Simulation to the Description of Ion Acoustic Solitary Waves Xin Qi, Yan-Xia Xu, Xiao-Ying Zhao, Ling-Yu Zhang, Wen-Shan Duan, and Lei Yang

Abstract— 1-D particle-in-cell simulations are used to investigate the propagation and decomposition of the ion acoustic solitary waves (IASWs) in plasmas. Our results show that for small-amplitude conditions, IASWs are stable and the simulation results are consistent with the theoretical predictions of the reductive perturbation method. As the amplitudes of IASWs increase, the waves become unstable and trains of oscillating waves are emitted behind the main waves. When the amplitude is large enough, the wave cannot exist and will decay into a series of waves with a small amplitude. By comparing our simulations with the theoretical solutions of Kortewag–de Vries soliton, the upper limitation of the amplitude of IASWs in plasmas is found. Moreover, our results show that although the reductive perturbation method is valid only for small perturbations, the application scope of the reductive perturbation method can be expanded to describe the potential profiles of IASWs with any amplitude. Meanwhile, the application scope for the density profiles is still limited in the perturbation cases. Index Terms— Ion acoustic solitary wave (IASW), Korteweg–de Vries (KdV) soliton solution, particle-in-cell (PIC) simulation.

I. I NTRODUCTION

P

LASMA is a complex system that can support various kinds of nonlinear waves, such as Langmuir waves, Alfvén waves, and electron or ion acoustic waves [1]–[4]. In general, nonlinear waves in plasmas arise due to the collective interaction between particles and self-consistent

Manuscript received June 11, 2015; revised August 14, 2015 and August 21, 2015; accepted September 1, 2015. Date of publication September 25, 2015; date of current version November 6, 2015. This work was supported in part by the National Magnetic Confinement Fusion Science Program, China, under Grant 2014GB104002, in part by the Strategic Priority Research Program through the Chinese Academy of Sciences under Grant XDA03030100, and in part by the National Natural Science Foundation of China under Grant 11505261, Grant 11275156, and Grant 11047010. (Corresponding authors: Wen-Shan Duan and Lei Yang.) X. Qi, X.-Y. Zhao, and W.-S. Duan are with the Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China, and also with the Joint Laboratory of Atomic and Molecular Physics, Institute of Modern Physics, Chinese Academy of Sciences, Northwest Normal University, Lanzhou 730070, China (e-mail: [email protected]; zxy@impcas. ac.cn; [email protected]). Y.-X. Xu is with the State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China (e-mail: victoryrain@ 126.com). L.-Y. Zhang is with the Software Center for High Performance Numerical Simulation, China Academy of Engineering Physics, Beijing 100088, China (e-mail: [email protected]). L. Yang is with the Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China, also with the Joint Laboratory of Atomic and Molecular Physics of NWNU, Institute of Modern Physics, Chinese Academy of Sciences, Northwest Normal University, Lanzhou 730070, China, and also with the Department of Physics, Lanzhou University, Lanzhou 730000, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPS.2015.2477102

electromagnetic field. The research on nonlinear wave phenomena in plasmas constitutes an important component in the physics of plasmas for its wide application in astrophysics and space physics. It is always carried out via two different approaches, i.e., the fluid approximation and kinetic method. Due to the complexity and variety of the issues related to nonlinear waves, the treatments of the problem are also diverse. The most classic ones are the reductive perturbation method for weak nonlinearities [10] and the fully nonlinear Sagdeev (or pseudopotential) approach [13]–[15]. While the Sagdeev treatment can be successfully applied to many systems [16]–[18], some of the underlying dynamics that favor or inhibit the existence of solitary structures are not explicit in the treatment. In this respect, McKenzie [19] and McKenzie et al. [20] developed the gas-dynamic description, which is the ideal complement to the Sagdeev treatment of nonlinear waves. Ion acoustic solitary wave (IASW) is a very typical nonlinear electrostatic structure. The formation and propagation of IASWs were experimentally observed by Ikezi et al. [5] in an argon plasma with density 109 cm−3 , electron temperature 1.5–3 eV, and ion temperature 0.2 eV. After that, IASWs were frequently observed in laboratory and space plasmas [6]–[9]. Since its first study by Washimi and Taniuti [10] via the reductive perturbation method [21], [22], it has been a subject of long-standing interest because of its practical importance in many physical fields [24]–[26]. In recent years, studies of the search for new techniques accelerating charged particles [27], the progress in nonlinear dynamics, the inertial confinement fusion [28], and the complementary heating method in fusion devices [29] are closely related to the dynamics of IASWs. A wide range of research studies have been carried out for IASWs, ranging from excitation [30], [31], formation [12], [32], propagation [33], [34] to reflection [35], [36], scattering [37], [38], decay [39], [40], and energy properties, as well as wave instabilities [41], [42] and collisions [43] of IASWs. Especially, lots of numerical studies have been conducted through different approaches: Kakad et al. [44] have studied the formation of ion acoustic solitary chain through fluid simulation, Matsumoto et al. [46] and Omura et al. [47] have studied IAWSs via particle simulation, Nopoush and Abbasi [47] have employed hybrid [particle in cell (PIC)–fluid] simulation to study the generation of IASWs, the propagation of IASWs in an electron–positron–ion plasma was investigated by Ferdousi et al. [48], [49]. The head-on collision processes of IASWs have been studied by Qi et al. [50] via the PIC method. Sharma et al. [51] studied the propagation of

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small- and large-amplitude ion acoustic solitons. They found that the small-amplitude ion acoustic solitons will propagate without distortion, while large-amplitude solitons intensify initially as they propagate by readjusting and releasing energy till they reach the solution of the full set of nonlinear equations. Moreover, the overtaking of large-amplitude ion acoustic solitons are also discussed in [51]. Recently, it has been found that there exists upper limitation of the amplitude of dust acoustic waves [52]. Is this also the case of IASWs in plasmas without dust grains? Besides, work has seldom been done to treat with the application scope of the reductive perturbation method in IASWs. To address these issues, 1-D PIC simulation [53] is performed to study the properties of IASWs in plasmas in this paper. The existence of the maximum amplitude and decomposition of IAWSs during the wave propagation have been exhibited and analyzed. By comparing the results from the reductive perturbation method with those from the PIC simulation, the application scope of the reductive perturbation method in IASWs is discussed and determined. This paper is structured as follows. Section II gives a brief introduction to the reductive perturbation method and PIC simulation method used in this paper. The results and discussion, both analytical and numerical, are presented in Section III, and the summary of conclusions is exhibited in Section IV. II. B RIEF I NTRODUCTION TO THE R EDUCTIVE P ERTURBATION M ETHOD AND PIC M ETHOD We consider a collisionless plasma composed of hot electrons and cold ions. The analysis is carried out based on the 1-D continuity and momentum fluid equations for ions, Boltzmann’s distribution for electrons, and Poisson’s equation. The dimensionless dynamical equations are as follows [54]: ∂ ∂n i + (n i v i ) = 0 (1) ∂t ∂x ∂v i ∂φ ∂v i + vi = −Z (2) ∂t ∂x ∂x ∂ 2φ = μeφ − Z n i (3) ∂x2 where n i is the number density of ions, Z is the charge number of the ion, v i is the velocity of the ion fluid, and φ is the self-consistent electrostatic potential. Besides, μ = n e0 /n i0 , and n e0 and n i0 are the equilibrium number densities of electrons and ions, respectively. Charge neutrality at equilibrium requires that n e0 = Z n i0 , and thus, there will be μ = 1 for Z = 1, which has been chosen in our study. And the following normalization is used in (1)–(3): ωpi t ≡ t  ,

x ≡ x , λD

vi ≡ v i , ci

ni ≡ n i , n i0

φ ≡ φ φ0

(4)

of which λ D = (kTe /4πe2 n i0 )(1/2), ωpi = (4πn i0 e2 /m i )(1/2), −2 ), with e indicating ci = (kTe /m i )(1/2) , and φ0 = (m i λ2D /eωpi the magnitude of the electron charge, m i the mass of ion, and Te the electron temperature. t  , x  , v i , n i , and φ  are the normalized physical parameters. In the following text, these normalized physical parameters are represented by t, x, v i , n i , and φ, respectively, for simplicity.

Now we adopt a reductive perturbation analysis to (1)–(3) to obtain a Korteweg–de Vries (KdV) description. The independent variables are scaled as ξ = ε(x − c0 t) and τ = ε3 t [55], [56], where c0 is the velocity of linear IASWs and ε is a small parameter characterizing the strength of the nonlinearity. The related variables are expanded around the equilibrium values in powers of ε as n i = 1 + ε2 n 1 + ε4 n 2 + · · ·

(5)

vi = ε2 v1 + ε4 v2 + · · · φ = ε 2 φ1 + ε 4 φ2 + · · ·

(6) (7)

Introducing (5)–(7) into the basic equations (1)–(3), by applying the first-order approximation, we can obtain n 1 = v 1 = φ1 and c02 = 1. For the second-order approximation, there will be the standard KdV equation ∂φ1 1 ∂ 3 φ1 ∂φ1 + φ1 + = 0. (8) ∂τ ∂ξ 2 ∂ξ 3 The stationary solitary wave solution of (8) can be written as ξ − u0τ (9) w where u 0 is a positive constant indicating the normalized dimension of velocity, φ1m = (3u 0 /a) indicates the amplitude of solitary wave, w = 2(b/u 0 )1/2 indicates the width of the wave, and a = 1 and b = 1/2. On the order of ε 2 , we can obtain the distribution of the IASWs as follows: ξ − u0τ , n i = 1 + φ, v i = φ. (10) φ = 3ε 2 u 0 Sech2 w In this paper, we use the 1-D PIC method to simulate the transition of IASWs in infinite background plasmas. In simulations, ion particles are represented by a limited ensemble of so-called superparticles (SPs), while electrons are treated as Boltzmann fluids. Each SP has a weight factor S specifying the number of real particles contained. Instead of calculating the field quantities on every SPs itself, the properties of SPs are weighted on the x grid to obtain the electric density ρ. Once ρ at time t is known, (3) is numerically solved to obtain the electric potential φ and field E in plasmas. The electronic field moves the SPs according to Newton’s equations of motion. Therefore, Newton’s equation of motion governs the ion part φ1 = φ1m Sech2

dx j dv j = q j Ej , = vj (11) dt dt where m j , q j , and x j are the mass, charge, and position of the j th SP, respectively, and E j is the electric field at the position of the j th SP. In the PIC simulation, the normalizations exhibited in (4) have also been adopted. At the beginning of the simulation, the SPs are uniformly distributed in the whole simulation region. The weighting parameter S and velocity of each SP are carefully chosen to make sure that the density distribution n i (x) and velocity distribution v i (x) of ions are from the KdV solutions in (10) √ ε u 0 [x − x 0 ] (12) n i (x) = 1 + 3ε 2 u 0 Sech2 2 √ ε u 0 [x − x 0 ] v i (x) = 3ε 2 u 0 Sech2 . (13) 2 mj

QI et al.: APPLICATION OF PIC SIMULATION TO THE DESCRIPTION OF IASWs

Fig. 1. Evolution of soliton in an electrostatic potential profile with (a) ε2 = 0.01, (b) ε 2 = 0.1, (c) ε 2 = 0.2, and (d) ε 2 = 0.6. For IASWs with a small amplitude, waves propagate without distortion, as shown in (a). With the increase in ε 2 , waves become unstable and trains of oscillating waves are emitted behind the main waves.

Fig. 2. Evolution of soliton in an ion density profile with (a) ε 2 = 0.01, (b) ε2 = 0.1, (c) ε 2 = 0.2, and (d) ε 2 = 0.6. For IASWs with a small amplitude, waves propagate without distortion, as shown in (a). With the increase in ε 2 , the profile becomes sharp and narrow and trains of oscillating waves are emitted behind the main waves.

As is done in [53] and [58], u 0 is chosen to be 1.0 through the whole simulation. In the whole process of simulation, the regions of simulation are moved along with the solitary waves to make sure that the waves are in the considering region. III. R ESULTS AND D ISCUSSION The evolution procedures of IASWs are exhibited in Figs. 1 and 2 for different values of ε 2 during different times in simulations. Fig. 1 shows the results of potential profiles, while Fig. 2 shows the results of density profiles. In Fig. 1(a), it can be found that for small perturbations, the IASWs steadily propagate with their amplitude and width unchanged. As ε2 increases, the wave becomes unstable and a series of small-amplitude oscillating waves whose amplitudes grow larger as ε 2 increases is observed behind the IASWs. Similar results for density profiles can be observed in Fig. 2. It is noted that the shape of the density profiles becomes very narrow and sharp, as shown in Fig. 2(d).

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Fig. 3. Comparison of the electrostatic potential profiles obtained from PIC simulations with those from the reductive perturbation method, where (a) ε2 = 0.01, (b) ε 2 = 0.1, (c) ε 2 = 0.2, and (d) ε 2 = 0.6. In (a)–(c), when the amplitude of the waves is not large enough, the profiles from PIC simulations match well with the KdV solution. For larger IASWs, the wave will break down into a series of waves with a small amplitude, as shown in (d).

Fig. 4. Comparison of the ion density profiles obtained from PIC simulations with those from the reductive perturbation method, where (a) ε 2 = 0.01, (b) ε2 = 0.1, (c) ε 2 = 0.2, and (d) ε 2 = 0.6. In (a), when the amplitude of the waves is small, the profiles from PIC simulations match well with the KdV solution. For larger IASWs, the profiles will become much narrower and sharper than those from the KdV solutions, as shown in (c)–(d).

In order to get further understanding of the evolution of IASWs, the corresponding comparisons between PIC numerical results and the analytical ones of the reductive perturbation method (both with the same initial conditions) are given in Figs. 3 and 4. Fig. 3 shows the comparison between the evolution of electrostatic potential φ obtained from analytical solutions and numerical experiments for different values of ε 2 . From Fig. 3(a), with a small amplitude, it can be noted that the excited solitary wave agrees very well with the analytical solution of IASWs from the reductive perturbation method, which means that the excited solitary wave is an IASW. Thus, the accuracy of the PIC code in studying IASWs has been confirmed. With the increase in ε 2 , the wave decomposition appears. In Fig. 3(b) and (c), there is a train of small oscillating waves behind each prime wave. However, the prime wave can

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Fig. 5. Dependence of the maximum electrostatic potential on ε 2 . When ε2 > 0.44, the wave cannot exist and will break down into a series of small waves.

Fig. 6. Dependence of the maximum ion number density on ε 2 . When ε2 > 0.44, the wave cannot exist and will break down into a series of small waves.

still match the analytical solutions. In Fig. 3(d), when ε 2 is large enough, the wave will break into a series of waves with small amplitudes. Similarly, Fig. 4 shows the comparison between the evolution of the ion density profiles obtained from the reductive perturbation method and PIC simulation for ε 2 = 0.01, 0.1, 0.2, and 0.6, respectively. It shows that for a small amplitude, the numerical result is in good agreement with the analytical one from the reductive perturbation method. However, the larger the value of ε 2 is, the larger the difference between results from the two approaches will be. Besides, it is noted that the amplitude of ion density profile also changes as ε 2 increases, whereas it does not obey the same law with the analytical soliton solution. The difference between simulation results and the analytical solution appears even when ε 2 = 0.1, as shown in Fig. 4(b). When ε 2 increases, the wave becomes more and more sharp. It is observed in Figs. 2 and 4 that our results are in good agreement with Sharma et al.’s report [51]. For small IASWs, the simulation results closely match the KdV soliton solutions. As the amplitude of IASWs increases, the given profile initially evolves and then settles down to the exact solution of the full nonlinear Poisson equation. However, as the amplitude becomes larger and larger, the waves will break down, as shown in Figs. 3(d) and 4(d). Thus, it is easy to realize the existence of the maximum amplitude during the increasing process of ε 2 . In order to find the maximum amplitudes of electrostatic potential φ and number density n, the dependence of φmax and that of n max on ε2 have been shown in Figs. 5 and 6, with φmax and n max denoting the maximum electrostatic potential and ion density, respectively. According to (10), the analytical φmax satisfies φmax = φ1m = 3u 0 ε2 , which increases as ε 2 increases, so there exists no upper limit, as is shown in blue in Fig. 5. The results from PIC simulation show that the upper limit of amplitude is obtained when ε 2 = 0.44. For ε 2 < 0.44, φmax increases with ε 2 in a smaller rate than the analytical one. However, for any ε 2 larger than 0.44, the amplitude of φ becomes smaller. In Fig. 6, the analytical n max also increases linearly with the increase in ε 2 in the form of n max = 3u 0 ε2 , with no maximum value obtained. However, PIC simulation shows that after the initial perturbation is given, the maximum value of n

increases severely for ε 2 ≤ 0.44 and the largest value of n max is obtained at ε 2 = 0.44. For any ε 2 > 0.44, the value of n max becomes smaller. We carry out a large number of simulations with different initial conditions as in [52]. All the simulation results show that the limitation amplitude of the IASWs does exist. Any IASW with potential amplitude exceeding 1.233 or density amplitude exceeding 62.3 will break down into small waves. In Figs. 5 and 6, the difference between the simulations and analytical results is mainly due to the fact that the analytical solution is based on small-amplitude approximation and has neglected the effects of high-order components. As ε 2 grows larger, the nonlinearity grows stronger, and there will be inevitably nonlinear wave–particle interaction, which may lead to the variation of wave properties, such as wave damping. Therefore, it is reasonable to get a smaller amplitude of φ in PIC simulations. However, although the reductive perturbation method is valid only when ε 2  1, for any ε 2 lies in (0, 0.44], the relative difference between the results from two approaches is always less than 1.7%. Considering the maximum potential amplitude of the IASWs is on the condition ε 2 = 0.44, it means that the reductive perturbation method can be applied to describe the potential profiles of IASWs with any amplitude. In Fig. 6, the analytical n max also increases linearly with the increase in ε 2 in the form of n max = 3u 0 ε2 , with no maximum value obtained. However, the PIC simulation shows that after the initial perturbation is given, the maximum value of n increases severely for ε 2 ≤ 0.44. Compared with Fig. 5, the relative difference between the results obtained from two approaches has already reached about 6.7% at ε 2 = 0.1 and it rapidly increases for larger values of ε 2 . Therefore, the reductive perturbation method cannot describe the density profiles of IASWs for large amplitudes, and the application scope for the potential profiles is still limited within ε 2  0.1. IV. C ONCLUSION In this paper, 1-D PIC simulation has been applied to study the formation and propagation of IASWs in plasmas composed of cold ions and hot electrons. It is found that both the reductive perturbation method and PIC simulation can precisely describe IASWs for small-amplitude conditions.

QI et al.: APPLICATION OF PIC SIMULATION TO THE DESCRIPTION OF IASWs

However, as the value of ε 2 grows larger, the results of PIC simulations on this subject are more reliable. Because of ignoring the higher components, the wave properties in cases of strong nonlinearities, i.e., larger value of ε 2 , cannot be exactly predicted by the reductive perturbation method. Our results show that as the amplitudes of IASWs increase, the waves become unstable and trains of oscillating waves are emitted behind the main waves, which is consistent with the previous study. Moreover, when the amplitude is larger enough, the wave cannot exist and will decay into a series of waves with a small amplitude. The dependence of the maximum n and that of φ on ε 2 have been studied both analytically and numerically and the upper limitations of n and φ for IASWs have been obtained. Besides, the application scope of the reductive perturbation method in studying IASWs is discussed. Although the reductive perturbation method is valid only when ε 2  1, the application scope of the reductive perturbation method in describing the potential profiles of IASWs can be expanded to IASWs with any amplitude. However, the reductive perturbation method cannot describe the density profiles of IASWs with large amplitudes, and its application scope for the potential profiles is still limited within ε 2  0.1. R EFERENCES [1] S. Bardwell and M. V. Goldman, “Three-dimensional Langmuir wave instabilities in type III solar radio bursts,” Astrophys. J., vol. 209, pp. 912–926, Nov. 1976. [2] M. Bárta and M. Karlický, “Energy mode distribution at the very beginning of parametric instabilities of monochromatic Langmuir waves,” Astron. Astrophys., vol. 353, pp. 757–770, Jan. 2000. [3] N. Devi, R. Gogoi, G. C. Das, and R. Roychoudhury, “Studies on the formation of large amplitude kinetic Alfvén wave solitons and double layers in plasmas,” Phys. Plasmas, vol. 14, no. 1, p. 012107, 2007. [4] J. Castro, P. McQuillen, and T. C. Killian, “Ion acoustic waves in ultracold neutral plasmas,” Phys. Rev. Lett., vol. 105, p. 065004, Aug. 2010. [5] H. Ikezi, R. J. Taylor, and D. R. Baker, “Formation and interaction of ion-acoustic solitions,” Phys. Rev. Lett., vol. 25, no. 1, pp. 11–14, Jul. 1970. [6] K. E. Lonngren, “Soliton experiments in plasmas,” Plasma Phys., vol. 25, pp. 943–982, Sep. 1983. [7] Y. Nakamura, T. Ito, and K. Koga, “Excitation and reflection of ionacoustic waves by a gridded plate and a metal disk,” J. Plasma Phys., vol. 49, no. 2, pp. 331–339, 1993. [8] P. O. Dovner, A. I. Eriksson, R. Boström, and B. Holback, “Freja multiprobe observations of electrostatic solitary structures,” Geophys. Res. Lett., vol. 21, no. 17, pp. 1827–1830, 1994. [9] Y. Nakamura, H. Bailung, and P. K. Shukla, “Observation of ionacoustic shocks in a dusty plasma,” Phys. Rev. Lett., vol. 83, no. 8, pp. 1602–1605, Aug. 1999. [10] H. Washimi and T. Taniuti, “Propagation of ion-acoustic solitary waves of small amplitude,” Phys. Rev. Lett., vol. 17, pp. 996–998, Nov. 1966. [11] A. R. Osborne and T. L. Burch, “Internal solitons in the Andaman Sea,” Science, vol. 208, no. 4443, pp. 451–460, May 1980. [12] G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: Universality and diversity,” Science, vol. 286, no. 5444, pp. 1518–1523, Nov. 1999. [13] R. Z. Sagdeev, “Cooperative phenomena and shock waves in collisionless plasmas,” Rev. Plasma Phys., vol. 4, pp. 23–91, 1966. [14] R. Z. Sagdeev and A. Galeev, Nonlinear Plasma Theory. New York, NY, USA: W. A. Benjamin, 1969. [15] S. Watanabe, “Ion acoustic soliton in plasma with negative ion,” J. Phys. Soc. Jpn., vol. 53, no. 3, pp. 950–956, 1984. [16] D. W. Bullett, “Applications of localised pseudopotential theory,” in Proc. AIP Conf., vol. 20. 1974, p. 139. [17] T. F. G. Green and J. R. Yates, “Relativistic nuclear magnetic resonance J-coupling with ultrasoft pseudopotentials and the zeroth-order regular approximation,” J. Chem. Phys., vol. 140, no. 23, p. 234106, 2014.

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Xiao-Ying Zhao received the bachelor’s and master’s degrees from Lanzhou University, Lanzhou, China, in 2008 and 2011, respectively. She has been with the Institute of Modern Physics, Lanzhou, since 2011. Her expertise is in the field of beam propagation in plasmas by particle-in-cell simulation.

Ling-Yu Zhang received the Ph.D. degree from the Institute of Modern Physics, Chinese Academy of Science, Lanzhou, China, in 2010. She has been with the Software Center for High Performance Numerical Simulation, China Academy of Engineering Physics, Beijing, China. Her expertise is in the field of ion beam dynamics, molecular, and atom physics.

Wen-Shan Duan received the Ph.D. degree from Nanjing University, Nanjing, China, in 1997. He is currently a Professor with Northwest Normal University, Lanzhou, China. His expertise is in the field of plasma waves, dusty plasma, nonlinear equations, and material sciences.

Xin Qi received the Ph.D. degree from the Institute of Modern Physics (IMP), Chinese Academy of Science, Lanzhou, China, in 2010. He is currently an Associate Professor with IMP. His expertise is in the field of plasma waves, transport of ion beams, accelerator, and computer simulation.

Yan-Xia Xu received the B.S. and M.S. degrees from Northwest Normal University, Lanzhou, China, in 2011 and 2014, respectively. She is currently pursuing the Ph.D. degree with East China Normal University, Shanghai, China. Her current research interests include nonlinear waves in plasmas, particle-in-cell simulation on laser-plasma interaction, and the transmitting properties of electromagnetic waves in plasmas.

Lei Yang received the Ph.D. degree from Lanzhou University, Lanzhou, China, in 1999. He is currently a Professor with the Institute of Modern Physics, Chinese Academy of Science, Lanzhou. His expertise is in the field of nonlinear dynamics, material sciences, and nuclear physics.