非热离子对非均匀碰撞热尘埃等离子体中三维孤波的影响

从理论上研究了非热离子、外部磁场、碰撞对非均匀热尘埃等离子体中三维非线性尘埃声孤波的影响。运用约化摄动法得到描述三维非线性尘埃声孤波的非标准的变系数Korteweg-de Vries(KdV)方程。然后把非标准KdV方程变为标准的变系数KdV方程,并且得到了标准的变系数KdV方程的近似解析解。由此解析解可以看出,非热离子的数目、碰撞、非均匀性、波的斜向传播、尘埃颗粒和非热离子的温度对三维非线性尘埃声孤波的振幅和宽度有很大的影响。外部磁场对三维非线性尘埃声孤波的宽度有影响,而对其振幅没有影响。此外,波的相速度与非热离子、波的斜向传播、尘埃颗粒的温度和非均匀性有关。     

The theoretical analysis of the lattice hydrodynamic models for traffic flow theory

The lattice hydrodynamic model is not only a simplified version of the macroscopic hydrodynamic model, but also connected with the microscopic car following model closely. The modified Korteweg-de Vries (mKdV) equation related to the density wave in a congested traffic region has been derived near the critical point since Nagatani first proposed it. But the Korteweg-de Vries (KdV) equation near the neutral stability line has not been studied, which has been investigated in detail for the car following model. We devote ourselves to obtaining the KdV equation from the original lattice hydrodynamic models and the KdV soliton solution to describe the traffic jam. Especially, we obtain the general soliton solution of the KdV equation and the mKdV equation. We review several lattice hydrodynamic models, which were proposed recently. We compare the modified models and carry out some analysis. Numerical simulations are conducted to demonstrate the nonlinear analysis results.     

第二种椭圆方程构造变系数非线性发展方程的无穷序列新精确解

本文为了获得非线性发展方程的无穷序列新精确解,进一步研究获得了第二种椭圆方程的几类新型解和Bäcklund变换.在此基础上,借助符号计算系统Mathematica,用带强迫项变系数组合KdV方程、(2+1)维和(3+1)维变系数Zakharov-Kuznetsov 方程为应用实例,构造了无穷序列新精确解.这里包括无穷序列Jacobi 椭圆函数光滑孤立子解、无穷序列Jacobi椭圆函数紧孤立子解、无穷序列三角函数紧孤立子解和无穷序列尖峰孤立子解. 关键词： 第二种椭圆方程 Bä cklund 变换 变系数非线性发展方程 无穷序列新精确解     

The Korteweg-de Vries soliton in the lattice hydrodynamic model

The lattice hydrodynamic model is not only a simplified version of the macroscopic hydrodynamic model, but is also closely connected with the microscopic car following model. The modified Korteweg-de Vries (mKdV) equation about the density wave in congested traffic has been derived near the critical point since Nagatani first proposed it. But the Korteweg-de Vries (KdV) equation near the neutral stability line has not been studied, which has been investigated in detail in the car following model. So we devote ourselves to obtaining the KdV equation from the lattice hydrodynamic model and obtaining the KdV soliton solution describing the traffic jam. Numerical simulation is conducted, to demonstrate the nonlinear analysis result.     

Modified Korteweg-De Vries equation for spatially inhomogeneous plasmas

The problem of propagation of ion-acoustic KdV solitons in weakly, spatially inhomogeneous plasmas is considered taking into account self-consistently the zeroth-order velocity and the potential existing in the system due to the presence of the inhomogeneity. An explicit soliton solution of the modified KdV equation is obtained.     

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利用数值方法研究了双温离子、磁场、非均匀性和波的斜向传播对三维非线性尘埃声孤波振幅和宽度的影响。运用约化摄动法得到描述三维非线性尘埃声孤波的非标准变系数Korteweg-de Vries(KdV)方程。然后把非标准变系数KdV方程变为标准变系数KdV方程,并且得到了此标准变系数KdV方程的近似解析解。研究结果表明,此系统中存在着两种形式的孤波,即压缩型孤波和稀疏型孤波,外部磁场对三维非线性尘埃声孤波的宽度有影响,而对其振幅没有影响。此外,波的相速度与波的斜向传播和非均匀性有着非常紧密的联系。     

Kadomstev-Petviashvili (KP) equation in warm dusty plasma with variable dust charge,two-temperature ion and nonthermal electron

In this work, the propagation of nonlinear waves in warm dusty plasmas with variable dust charge, two-temperature ion and nonthermal electron is studied. By using the reductive perturbation theory, the Kadomstev-Petviashvili (KP) equation is derived. The energy of the soliton and the linear dispersion relation are obtained. The effects of variable dust charge on the energy of soliton and the angular frequency of linear wave are also discussed.     

第二类变系数KdV方程的新类型无穷序列精确解

为了构造变系数非线性发展方程的无穷序列新精确解, 发掘第一种椭圆辅助方程的构造性和机械化性特点, 获得了该方程的 新类型解和相应的 Bäcklund 变换. 在符号计算系统 Mathematica 的帮助下, 以第二类变系数 KdV 方程为应用实例, 构造了三种类型的无穷序列新精确解. 这里包括无穷序列光滑类孤子解、无穷序列尖峰孤立子解和无穷序列紧孤立子解. 这种方法也可以获得其他变系数非线性发展方程的无穷序列新精确解.     

A Variable Separation Approach to Solve the Integrable and Nonintegrable Models:Coherent Structures of the (2 + 1)-Dimensional KdV Eqnation

We study the localized coherent structures ofa generally nonintegrable (2 1 )-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads to abundant coherent structures. However, in the general nonintegrable case, an additional condition has to be introduced for these arbitrary functions. Although the additional condition has been introduced into the solutions of the nonintegrable KdV equation, there still exist many interesting solitary wave structures. Especially, the nonintegrable KdV equation possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the integrable case. Furthermor,in the integrable case, the interaction between two travelling ring solitons is elastic, while in the nonintegrable case we cannot find even the single travelling ring soliton solution.     

The dynamics of ring dark soliton in inhomogeneous Bose-Einstein condensates

The dynamics of the ring dark soliton in an inhomogeneous Bose-Einstein condensates (BEC) with thin disk-shaped potential trapping is investigated analytically and numerically. Analytical result shows that the ring dark soliton is governed by a variable coefficients Korteweg-de Vries (KdV) equation. The effect of the ring curvature (nonplanar geometry) and the inhomogeneous of the background on soliton amplitude and the emitted radiation profiles are obtained analytically. The theoretical results are confirmed by the direct numerical results.     

A Series of Variable Separation Solutions and New Soliton Structures of （2＋1）-Dimensional Korteweg-de Vries Equation

Variable separation approach is introduced to solve the （2＋1）-dimensional KdV equation. A series of variable separation solutions is derived with arbitrary functions in system. We present a new soliton excitation model （24）. Based on this excitation, new soliton structures such as the multi-lump soliton and periodic soliton are revealed by selecting the arbitrary function appropriately.     

A Series of Variable Separation Solutions and New Soliton Structures of (2+1)-Dimensional Korteweg-de Vries Equation

Variable separation approach is introduced to solve the (2 1)-dimensional KdV equation. A series of variable separation solutions is derived with arbitrary functions in system. We present a new soliton excitation model (24). Based on this excitation, new soliton structures such as the multi-lump soliton and periodic soliton are revealed by selecting the arbitrary function appropriately.     

Soliton and kink jams in traffic flow with open boundaries

Soliton density wave is investigated numerically and analytically in the optimal velocity model (a car-following model) of a one-dimensional traffic flow with open boundaries. Soliton density wave is distinguished from the kink density wave. It is shown that the soliton density wave appears only at the threshold of occurrence of traffic jams. The Korteweg-de Vries (KdV) equation is derived from the optimal velocity model by the use of the nonlinear analysis. It is found that the traffic soliton appears only near the neutral stability line. The soliton solution is analytically obtained from the perturbed KdV equation. It is shown that the soliton solution obtained from the nonlinear analysis is consistent with that of the numerical simulation.     

Nonlinear waves in gas-fluidized beds

A mathematical model for gas-fluidized beds is examined which treats both the particles and gas as continua by volume averaging. The system is then considered as two interlocking one-phase fluids. For small perturbations to the uniform state, these equations have been shown by Crighton (1991) to reduce to the Burgers-KdV equation and under certain criteria, we have instability. We consider the unstable situation when the amplification effects are a perturbation to the KdV equation and take an initial condition of a single KdV soliton. The growth of this soliton is followed through several regions in which the unstable Burgers-KdV equation is no longer appropriate, but KdV remains the leading order equation. Eventually, there is a fundamental change in the solution and the new governing equations are fully nonlinear and O(1). These admit a solitary wave solution which matches back onto the KdV soliton. Thus, we can follow the formation of a bubble from a small amplitude perturbation to the uniform state.University of Cambridge, Cambridge, United Kingdom. Published in Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 8, pp. 797–800, August, 1993.     

Higher-Dimensional KdV Equations and Their Soliton Solutions

A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of bell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.     

Korteweg-de Vries方程的准孤立子解及其在离子声波中的应用

应用推广的tanh函数展开法,给出了Korteweg-de Vries方程具有准孤立子行为的两组孤子-椭圆周期波解,其中一组为新解.推导了均匀磁化等离子体中描述离子声波动力学行为的Korteweg-de Vries方程,发现电子分布、离子电子温度比、磁场大小、磁场方向对离子声准孤立子的波形具有显著影响.     

The effect of the negative particle velocity in a soliton gas within Korteweg–de Vries-type equations

The effect of changing the direction of motion of a defect (a soliton of small amplitude) in soliton lattices described by the Korteweg–de Vries and modified Korteweg–de Vries integrable equations (KdV and mKdV) was studied. Manifestation of this effect is possible as a result of the negative phase shift of a small soliton at the moment of nonlinear interaction with large solitons, as noted in [1], within the KdV equation. In the recent paper [2], an expression for the mean soliton velocity in a “cold” KdV-soliton gas has been found using kinetic theory, from which this effect also follows, but this fact has not been mentioned. In the present paper, we will show that the criterion of negative velocity is the same for both the KdV and mKdV equations and it can be obtained using simple kinematic considerations without applying kinetic theory. The averaged dynamics of the “smallest” soliton (defect) in a soliton gas consisting of solitons with random amplitudes has been investigated and the average criterion of changing the sign of the velocity has been derived and confirmed by numerical solutions of the KdV and mKdV equations.     

Effects of positron density and temperature on ion acoustic dressed solitons in an electron–positron–ion plasma

Ion acoustic dressed solitons in a three component plasma consisting of cold ions, hot electrons and positrons are studied. Using reductive perturbation method, Korteweg–de Vries (KdV) equation and a linear inhomogeneous equation, governing respectively the evolution of first and second order potentials are derived for the system. Renormalization procedure of Kodama and Taniuti is used to obtain nonsecular solutions of these coupled equations. It is found that electron–positron–ion plasma system supports only compressive solitons. For a given amplitude of soliton on increasing the positron concentration, velocity of the KdV as well as dressed soliton increases. For any arbitrary values of soliton's amplitude and positron concentration, velocity of the dressed soliton is found to be larger than that of the KdV soliton. For small amplitude of solitons, the width of KdV as well as dressed soliton decreases as positron concentration increases and width of dressed soliton is found to be larger than that of the KdV soliton. However, for a large value of soliton's amplitude as concentration of positrons increases, instead of decreasing width of dressed soliton starts to increase.     

Formation of solitons and shocks in toroidal ion temperature gradient mode in the presence of non-Maxwellian electrons

Linear and nonlinear phenomena are investigated in toroidal ion temperature gradient (TITG)-driven pure drift mode. The model includes inhomogeneity in background magnetic field, ion temperature, and density. Finite Larmor radius effect is incorporated to understand the effect of low-frequency wave on ion dynamics. Electrons are assumed to follow nonthermal distribution, that is, kappa and Cairns distributions. Dispersion relation is obtained to analyse the linear behaviour of the TITG mode in the presence of non-Maxwellian electron distribution. In the nonlinear regime, exact solutions (soliton and shocks) are obtained (in dispersive and dissipative medium respectively) by using functional variable method to solve the nonlinear partial differential equation obtained for the system under consideration. Graphical illustrations are used to exhibit the characteristics of linear and nonlinear structures and their dependence on different physical parameters. It is observed that for TITG-driven pure drift mode, rarefactive solitons are formed for both thermal and nonthermal electron distributions. It is also observed that variation of electrons from standard thermal distribution affects the propagation characteristics of linear and nonlinear structures in TITG-driven modes. Results of our investigations will be helpful to understand the low-frequency waves in inhomogeneous plasmas in the presence of nonthermal electron distributions which are frequently observed by satellite missions and are also observed in laboratory plasmas.     

Two velocity difference model for a car following theory

In the light of the optimal velocity model, a two velocity difference model for a car-following theory is put forward considering navigation in modern traffic. To our knowledge, the model is an improvement over the previous ones theoretically, because it considers more aspects in the car-following process than others. Then we investigate the property of the model using linear and nonlinear analyses. The Korteweg-de Vries equation (for short, the KdV equation) near the neutral stability line and the modified Korteweg-de Vries equation (for short, the mKdV equation) around the critical point are derived by applying the reductive perturbation method. The traffic jam could be thus described by the KdV soliton and the kink-anti-kink soliton for the KdV equation and mKdV equation, respectively. Numerical simulations are made to verify the model, and good results are obtained with the new model.