尘埃颗粒大小分布对尘埃等离子体集体行为影响的研究
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摘要
随着科学技术的高速发展,非线性科学以其令人惊异的复杂性成为人们的研究热点。对混沌、分形和孤立波等非线性现象的深入研究,改变了人们对自然界的认识,同时也成为非线性科学的的热门研究课题之一。
本文运用约化摄动法分别研究了冷尘埃等离子体、热尘埃等离子体、磁化尘埃等离子体和尘埃等离子体晶体中的尘埃声孤波现象,着重考虑尘埃颗粒大小分布对尘埃声波的影响。
论文第一章为前言部分,简要介绍非线性科学的研究内容、等离子体物理基本概念和摄动方法的基本原理。
论文第二章研究未磁化的、无碰撞的、双温度离子的冷尘埃等离子体中的有限小振幅尘埃声波。通过使用约化摄动法,得到尘埃声波在不同情况下满足的Kadomtsev-Petviashvili (KP), Modified KP和Coupled KP方程。结果表明:1.将尘埃颗粒大小呈Power Law Distribution (PLD)分布的双温度离子冷尘埃等离子体中的尘埃声孤波与尘埃颗粒大小相同的尘埃等离子体中的尘埃声孤波相比,其传播速度满足v_0>(?),振幅满足φ_m<(?),宽度满足ω>(?)。当尘埃颗粒半径的最大值和最小值之比c=a_(max)/a_(min)增大时,尘埃声孤波的传播速度加快,振幅减小,而宽度增大。2.采用Hirota方法解析求解KP方程,得到KP方程多孤立子解的解析表达式,并用图形模拟多孤子的相互作用过程。
论文第三章研究在高阶横向扰动下,不同尘埃颗粒大小分布对未磁化的、无碰撞的含有两种不同温度离子的热尘埃等离子体中二维尘埃声波的影响。研究结果显示:不论是等温过程还是绝热过程,将尘埃颗粒大小服从PLD分布的热尘埃等离子体中的尘埃声孤波与尘埃颗粒大小相同的热尘埃等离子体中的尘埃声孤波作比较发现:当尘埃温度与有效温度之比σ=T_d/T_(eff)增大时,孤波传播速度之比v_0/(?)随着σ的增大而减小,振幅之比φ_m(?)随着σ的增大而增大,宽度之比ω/(?)随着σ的增大而减小。
论文第四章研究磁化的双温度离子尘埃等离子体中的有限小振幅尘埃声波。通过使用约化摄动法,我们得到描述磁化尘埃等离子体的Zakharov-Kuznetsov (ZK), Modified ZK和Coupled ZK方程。结果表明:1.尘埃颗粒的极性对磁化尘埃等离子体中的尘埃声波有重要影响:(ⅰ) 对于含有同种极性的尘埃等离子体,当所有的尘埃颗粒均带负电时,无碰撞的三维磁化尘埃等离子体中存在稀疏的尘埃声孤波;当所有尘埃颗粒均带正电,即Z_(dj)<0时,此系统中存在压缩尘埃声孤波;(ⅱ)对于含有不同极性的尘埃等离子体,若sum from j=1 to N n_(d0j)Z_(dj)~3/m_(dj)~2>0,则系统中的孤立波为稀疏尘埃声孤波;若sum from j=1 to N n_(d0j)Z_(dj)~3/m_(dj)~2>0,则系统中存在压缩尘埃声孤波。另一方面,用数值模拟的方法研究高阶横向扰动下Coupled ZK方程的孤立波解的稳定性问题。研究发现:对于尘埃颗粒大小呈一定分布的磁化尘埃等离子体中的尘埃声波来说,如果在横向方向存在高阶扰动,且横向扰动的波数K和L满足条件0<(K~2+L~2)~(1/2)Nowadays, with the rapid development of science and technology, the nonlinear science has been the hot topic of investigation because of it's amazing complexity. The study of chaos, fractals and solitons has changed the comprehension of nature for the people. Meanwhile, it has been the pop problem of the nonlinear science.In this paper, the effect of dust size distribution for the dust acoustic solitary wave (DASW) in the cold dusty plasma, the hot dusty plasma, the magnetized dusty plasmas and the dusty plasmas crystals have been studied respectively.In chapter one, the primary content of nonlinear science, the basic definition of plasma physics and the essential theory of perturbation method have been introduced briefly.In chapter two, the small but infinite amplitude dust acoustic wave (DAW) in a collisionless, unmagnetized two-ion-temperature cold dusty plasma has been studied. By using the reductive perturbation method, the Kadomtsev-Petviashvili (KP), Modified KP and Coupled KP equations have been obtained for different cases. It seems that: 1. For the power law distribution (PLD) cases, the DAW in the cold dusty plasma propagate quicker than that of the mono-sized dusty plasma. It is found that the DASW for PLD dusty plasmas is lower than that of the mono-sized dusty plasmas. Moreover, the DASW of PLD dusty plasmas is wider than that of the mono-sized dusty plasmas. For the DASW in PLD dusty plasmas, the amplitude decreases as c increases, but the width and the velocity increases as c increases, where c = a_max/a_min (a_max and a_min refer to the maximum and minimum radii of dust particles respectively). 2. The N-soliton solution of KP equation has been obtained by the Hirota method. Moreover, the interaction process of N-soliton is described by numerical results.In chapter three, the effect of dust size distribution for the two-dimensional DAW in the collisionless, unmagnetized two-ion-temperature hot dusty plasmas has been studied by considering the higher order transverse perturbation. It can be concluded that: For the isothermal and adiabatic cases, the velocity for DAW in the PLD hot dusty plasmas is larger than that of the mono-sized dusty plasmas. Meanwhile, for DASW in the PLD and mono-sized dusty plasmas, the ratio of amplitude increases as σ increases, but the ratio of velocity and the ratio of width decreases as σ increases, where σ = T_d/T_eff (T_d is the temperature of dust particles and T_eff is the effective temperature).In chapter four, the small but infinite amplitude DAW in the magnetized two-ion-temperature dusty plasmas have been investigated. The Zakharov-Kuznetsov (ZK), Modified ZK and Coupled ZK equations have been obtained by using the reductive perturbation method. It indicate that: 1. The polarity of dust particles has important effect on DAW for
the magnetized dusty plasmas: (i) There is a rarefactive DASW when all the dust particles are negatively charged. On the other hand, a compressive DASW exist in this system while the dust particles are positively charged, (ii) For a dusty plasmas with opposite polarity particles, it is noted that the DASW is rarefactive if Otherwise, the DASW is compressive. On the other hand, the stability of the solitary wave solution for the Coupled ZK equation has been discussed by numerical method. It is found that, for the DAW in a magnetized dusty plasmas with dust size distribution, there is a instability to the transverse perturbations with the wave numbers K and L satisfies 0 < (K~2 + L~2)~1/2 < s_c. 2. The N-soliton solution of ZK equation has been obtained by the Hirota method, meanwhile, the different process of two-soliton has been simulated numerically.In chapter five, the characteristic of dust lattice wave (DLW) in a dusty plasma crystal has been studied. It can be concluded that: 1. For the dust lattice solitary wave (DLSW) of weakly inhomogeneous one-dimensional dust lattice, it is found that the amplitude and velocity decreases as time r increases, but the width increases as time r increases: On the other hand, the frequency to, for the envelope wave in dusty plasmas crystal, increases as the wave number k and the charge of dust particles Q increases. 2. It has been investigated that the nonlinear envelope wave in different directions for a two-dimensional hexagonal crystal. It is found that the frequencies in different directions are all increases as the wave number k and the charge of dust particles Q increases. Moreover, the frequency and the velocity in m-direction is larger than that of n-direction.In chapter six, the main results have been summarized and the further works have been introduced.
本文运用约化摄动法分别研究了冷尘埃等离子体、热尘埃等离子体、磁化尘埃等离子体和尘埃等离子体晶体中的尘埃声孤波现象,着重考虑尘埃颗粒大小分布对尘埃声波的影响。
论文第一章为前言部分,简要介绍非线性科学的研究内容、等离子体物理基本概念和摄动方法的基本原理。
论文第二章研究未磁化的、无碰撞的、双温度离子的冷尘埃等离子体中的有限小振幅尘埃声波。通过使用约化摄动法,得到尘埃声波在不同情况下满足的Kadomtsev-Petviashvili (KP), Modified KP和Coupled KP方程。结果表明:1.将尘埃颗粒大小呈Power Law Distribution (PLD)分布的双温度离子冷尘埃等离子体中的尘埃声孤波与尘埃颗粒大小相同的尘埃等离子体中的尘埃声孤波相比,其传播速度满足v_0>(?),振幅满足φ_m<(?),宽度满足ω>(?)。当尘埃颗粒半径的最大值和最小值之比c=a_(max)/a_(min)增大时,尘埃声孤波的传播速度加快,振幅减小,而宽度增大。2.采用Hirota方法解析求解KP方程,得到KP方程多孤立子解的解析表达式,并用图形模拟多孤子的相互作用过程。
论文第三章研究在高阶横向扰动下,不同尘埃颗粒大小分布对未磁化的、无碰撞的含有两种不同温度离子的热尘埃等离子体中二维尘埃声波的影响。研究结果显示:不论是等温过程还是绝热过程,将尘埃颗粒大小服从PLD分布的热尘埃等离子体中的尘埃声孤波与尘埃颗粒大小相同的热尘埃等离子体中的尘埃声孤波作比较发现:当尘埃温度与有效温度之比σ=T_d/T_(eff)增大时,孤波传播速度之比v_0/(?)随着σ的增大而减小,振幅之比φ_m(?)随着σ的增大而增大,宽度之比ω/(?)随着σ的增大而减小。
论文第四章研究磁化的双温度离子尘埃等离子体中的有限小振幅尘埃声波。通过使用约化摄动法,我们得到描述磁化尘埃等离子体的Zakharov-Kuznetsov (ZK), Modified ZK和Coupled ZK方程。结果表明:1.尘埃颗粒的极性对磁化尘埃等离子体中的尘埃声波有重要影响:(ⅰ) 对于含有同种极性的尘埃等离子体,当所有的尘埃颗粒均带负电时,无碰撞的三维磁化尘埃等离子体中存在稀疏的尘埃声孤波;当所有尘埃颗粒均带正电,即Z_(dj)<0时,此系统中存在压缩尘埃声孤波;(ⅱ)对于含有不同极性的尘埃等离子体,若sum from j=1 to N n_(d0j)Z_(dj)~3/m_(dj)~2>0,则系统中的孤立波为稀疏尘埃声孤波;若sum from j=1 to N n_(d0j)Z_(dj)~3/m_(dj)~2>0,则系统中存在压缩尘埃声孤波。另一方面,用数值模拟的方法研究高阶横向扰动下Coupled ZK方程的孤立波解的稳定性问题。研究发现:对于尘埃颗粒大小呈一定分布的磁化尘埃等离子体中的尘埃声波来说,如果在横向方向存在高阶扰动,且横向扰动的波数K和L满足条件0<(K~2+L~2)~(1/2)
the magnetized dusty plasmas: (i) There is a rarefactive DASW when all the dust particles are negatively charged. On the other hand, a compressive DASW exist in this system while the dust particles are positively charged, (ii) For a dusty plasmas with opposite polarity particles, it is noted that the DASW is rarefactive if Otherwise, the DASW is compressive. On the other hand, the stability of the solitary wave solution for the Coupled ZK equation has been discussed by numerical method. It is found that, for the DAW in a magnetized dusty plasmas with dust size distribution, there is a instability to the transverse perturbations with the wave numbers K and L satisfies 0 < (K~2 + L~2)~1/2 < s_c. 2. The N-soliton solution of ZK equation has been obtained by the Hirota method, meanwhile, the different process of two-soliton has been simulated numerically.In chapter five, the characteristic of dust lattice wave (DLW) in a dusty plasma crystal has been studied. It can be concluded that: 1. For the dust lattice solitary wave (DLSW) of weakly inhomogeneous one-dimensional dust lattice, it is found that the amplitude and velocity decreases as time r increases, but the width increases as time r increases: On the other hand, the frequency to, for the envelope wave in dusty plasmas crystal, increases as the wave number k and the charge of dust particles Q increases. 2. It has been investigated that the nonlinear envelope wave in different directions for a two-dimensional hexagonal crystal. It is found that the frequencies in different directions are all increases as the wave number k and the charge of dust particles Q increases. Moreover, the frequency and the velocity in m-direction is larger than that of n-direction.In chapter six, the main results have been summarized and the further works have been introduced.
引文
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