A Model for Periodic Nonlinear Electric Field Structures in Space Plasmas

In this study, we present a physical model to explain the generation mechanism of nonlinear periodic waves with a large amplitude electric field structures propagating obliquely and exactly parallel to the magnetic field. The ＂Sagdeev potential＂ from the MHD equations is derived and the nonlinear electric field waveforms are obtained when the Mach number, direction of propagation, and the initial electric field satisfy certain plasma conditions. For the parallel propagation, the amplitude of the electric field waves with ion-acoustic mode increases with the increase of initial electric field and Mach number but its frequency decreases with the increase of Mach number. The amplitude and frequency of the electric field waves with ion-cyclotron mode decrease with the increase of Mach number and become less spiky, and its amplitude increases with the increase of initial electric field. For the oblique propagation, only periodic electric field wave with an ion-cyclotron mode obtained, its amplitude and frequency increase with the increase of Mach number and become spiky. From our model the electric field structures show periodic, spiky, and saw-tooth behaviours corresponding to different plasma conditions.     

A Model for Periodic Nonlinear Electric Field Structures in Space Plasmas

In this study, we present a physical model to explain the generation mechanism of nonlinear periodic waves with a large amplitude electric field structures propagating obliquely and exactly parallel to the magnetic field. The ``Sagdeev potential' fromthe MHD equations is derived and the nonlinear electric field waveforms are obtained when the Mach number, direction of propagation, and the initial electric field satisfy certain plasma conditions. For the parallel propagation, the amplitude of theelectric field waves with ion-acoustic mode increases with the increase of initial electric field and Mach number but its frequency decreases with the increase of Mach number. The amplitude and frequency of the electric field waves with ion-cyclotron mode decrease with the increase of Mach number and become less spiky, andits amplitude increases with the increase of initial electric field. For the oblique propagation, only periodic electric field wave with an ion-cyclotron mode obtained, its amplitude and frequency increase with the increase of Mach number and become spiky. From our model the electric field structures show periodic, spiky, and saw-tooth behaviours corresponding to different plasma conditions.     

A Laplace Decomposition Method for Nonlinear Partial Diferential Equations with Nonlinear Term of Any Order

A Laplace decomposition algorithm is adopted to investigate numerical solutions of a class of nonlinear partial diferential equations with nonlinear term of any order,utt+auxx+bu+cup+du2p 1=0,which contains some important equations of mathematical physics.Three distinct initial conditions are constructed and generalized numerical solutions are thereby obtained,including numerical hyperbolic function solutions and doubly periodic ones.Illustrative figures and comparisons between the numerical and exact solutions with diferent values of p are used to test the efciency of the proposed method,which shows good results are achieved.     

A Quick Numerical Method for Analyzing Nonlinear Effects of Excited State Absorption

A semi-analytical difference method for analyzing nonlinear effects of excited state optical absorption is presented. In comparison with Hughes' method and traditional difference method, the advantages of the semi-analytical difference method are speed-up in computation time, high accuracy and good stabilization.     

A Quick Numerical Method for Analyzing Nonlinear Effects of Excited State Absorption

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Anomalous Scaling of Structure Functions and Dynamic Constraints on Turbulence Simulations

The connection between anomalous scaling of structure functions (intermittency) and numerical methods for turbulence simulations is discussed. It is argued that the computational work for direct numerical simulations (DNS) of fully developed turbulence increases as Re ⁴, and not as Re ³ expected from Kolmogorov’s theory, where Re is a large-scale Reynolds number. Various relations for the moments of acceleration and velocity derivatives are derived. An infinite set of exact constraints on dynamically consistent subgrid models for Large Eddy Simulations (LES) is derived from the Navier–Stokes equations, and some problems of principle associated with existing LES models are highlighted     

On the Fully Implicit Solution of a Phase-Field Model for Binary Alloy Solidification in Three Dimensions

A fully implicit numerical method, based upon a combination of adaptively refined hierarchical meshes and geometric multigrid, is presented for the simulation of binary alloy solidification in three space dimensions. The computational techniques are presented for a particular mathematical model, based upon the phase-field approach, however, their applicability is of greater generality than for the specific phase-field model used here. In particular, an implicit second order time discretization is combined with the use of second order spatial differences to yield a large nonlinear system of algebraic equations as each time step. It is demonstrated that these equations may be solved reliably and efficiently through the use of a nonlinear multigrid scheme for locally refined grids. In effect, this paper presents an extension of earlier research in two space dimensions (J. Comput. Phys., 225 (2007), pp. 1271-1287) to fully three-dimensional problems. This extension is validated against earlier two-dimensional results and against some of the limited results available in three dimensions, obtained using an explicit scheme. The efficiency of the implicit approach and the multigrid solver are then demonstrated and some sample computational results for the simulation of the growth of dendrite structures are presented.     

An Extension of Mapping Deformation Method and New Exact Solution for Three Coupled Nonlinear Partial Differential Equations

In this paper, we extend the mapping deformation method proposed by Lou. It is used to find new exacttravelling wave solutions of nonlinear partial differential equation or coupled nonlinear partial differential equations(PDEs). Based on the idea of the homogeneous balance method, we construct the general mapping relation betweenthe solutions of the PDEs and those of the cubic nonlinear Klein-Gordon (NKG) equation. By using this relation andthe abundant solutions of the cubic NKG equation, many explicit and exact travelling wave solutions of three systemsof coupled PDEs, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic functionsolutions, and rational solutions, are obtained.     

Weak Power-Counting Theorem for the Renormalization of the Nonlinear Sigma Model in Four Dimensions

The formulation of the non linear σ-model in terms of flat connection allows the construction of a perturbative solution of a local functional equation by means of cohomological techniques which are implemented in gauge theories. In this paper we discuss some properties of the solution at the one-loop level in D = 4. We prove the validity of a weak power-counting theorem in the following form: although the number of divergent amplitudes is infinite only a finite number of counterterms parameters have to be introduced in the effective action in order to make the theory finite at one loop, while respecting the functional equation (fully symmetric subtraction in the cohomological sense). The proof uses the linearized functional equation of which we provide the general solution in terms of local functionals. The counterterms are expressed in terms of linear combinations of these invariants and the coefficients are fixed by a finite number of divergent amplitudes. These latter amplitudes contain only insertions of the composite operators φ₀ (the constraint of the non linear σ-model) and F _μ (the flat connection). The structure of the functional equation suggests a hierarchy of the Green functions. In particular once the amplitudes for the composite operators φ₀ and F _μ are given all the others can be derived by functional derivatives. In this paper we show that at one loop the renormalization of the theory is achieved by the subtraction of divergences of the amplitudes at the top of the hierarchy. As an example we derive the counterterms for the four-point amplitudes. PACS numbers: 11.10.Gh, 11.30.Rd     

Similarity Reduction and Integrability for the Nonlinear WaveEquations from EPM Model

Four types of similarity reductions are obtained for the nonlinear wave equation arising in the elasto-plastic-microstructure model by using both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou.As a result,the nonlinear wave equation is not integrable.     

Statistical analysis of cellular detonation dynamics from numerical simulations: one-step chemistry

In this paper, two methods are developed for statistically analysing the nonlinear cellular dynamics from numerical simulations of gaseous detonations, one use of which is the systematic determination of detonation cell sizes from such simulations. Both these methods rely on signed vorticity records in which the individual families of transverse waves are captured independently. The first method involves an automated extraction of the main triple-point tracks from the vorticity records, allowing statistical analysis of the spacings between neighbouring tracks. The second method uses the autocorrelation function to spectrally analyse the vorticity records. These methods are then employed for a preliminary analysis of the cellular dynamics of the standard, idealized one-step chemistry model. Evidence is found for ‘cell size doubling’ bifurcations in the one-step model as the cellular dynamics become more irregular (e.g. as the activation is increased). It is also shown that the statistical models converge slowly due to systematic ‘shot-to-shot’ variation in the cellular dynamics for fixed parameters with different initial perturbations. Instead, it appears that a range of equally probable cell sizes can be obtained for given parameters.     

Numerical Solutions of a Class of Nonlinear Evolution Equations with Nonlinear Term of Any Order

In this paper, the Adomian decomposition method is developed for the numerical solutions of a class of nonlinear evolution equations with nonlinear term of any order, utt＋auxx ＋ bu ＋ cu^p＋ du^2p-1=0, which contains some important famous equations. When setting the initial conditions in different forms, some new generalized numerical solutions： numerical hyperbolic solutions, numerical doubly periodic solutions are obtained. The numerical solutions are compared with exact solutions. The scheme is tested by choosing different values of p, positive and negative, integer and fraction, to illustrate the efficiency of the ADM method and the generalization of the solutions.     

封闭圆内开缝圆自然对流的非线性特性研究

本文通过数值计算对封闭圆内开缝圆自然对流的非线性特性进行了研究。数值计算以整个流场为计算区域,采用非稳态数学模型和具有QUICK差分格式的SIMPLE算法。计算结果表明,在不同参数下流动和换热存在稳态定常解、周期性振荡解、拟周期性振荡解和非周期性振荡解。稳态定常解的相图是一个点;周期性振荡解的相图是一个极限环,对应功率谱含一个基频及其谐波;拟周期性振荡解的相图为环面,对应功率谱含两个不相关的频率及它们的线性组合频率;非周期性振荡解的功率谱为无规则的宽带连续谱。     

Collisions of Two Soatial Solitons in Inhomogeneous Nonlinear Media

Collisions of spatial solitons occurring in the nonlinear Schroeinger equation with harmonic potential are studied, using conservation laws and the split-step Fourier method. We find an analytical solution for the separation distance between the spatial solitons in an inhomogeneous nonlinear medium when the light beam is self-trapped in the transverse dimension. In the self-focusing nonlinear media the spatial solitons can be transmitted stably, and the interaction between spatial solitons is enhanced due to the linear focusing effect （and also diminished for the linear defocusing effect）. In the self-defocusing nonlinear media, in the absence of self-trapping or in the presence of linear self-defocusing, no transmission of stable spatial solitons is possible. However, in such media the linear focusing effect can be exactly compensated, and the spatial solitons can propagate through.     

底部加热长方体腔内自然对流的非线性特性

采用SIMPLE算法,QUICK差分格式,对底部加热三维长方体腔内空气的自然对流进行了数值模拟。根据模拟结果,探讨了方腔内流体流动与换热的静态分岔与振荡等非线性现象。数值结果显示,在固定的几何尺寸和不同Ra的情况下,当初始场不同时,会出现若干不同的解,即存在解的静态分岔;在固定的几何尺寸和相同的初始场情况下,低Ra时流动和换热处于稳态,当Ra超过某一临界值时,流动和换热就会随时间振荡,并通过倍周期分岔过渡到混沌;当方腔的几何尺寸不同时,分岔点的特征值Ra也发生变化。     

单轴晶体三阶有效非线性系数的解析表达式

应用压缩角标方法所得到的三阶非线性系数的矩阵形式，首次给出了在单轴晶体事进行三次谐波实验时，各种相位匹配方式下，其三阶有效非线性系数的解析表达式。这些表达式对三次谐波实验具有指导意义。     

弱非线性热声理论模型的建立及简化

本文指出非零的周期平均热声效应在本质上是非线性的,应保留到二阶精度。在小振幅条件下,利用摄动方法,以无限大平板流道为例,建立了二阶精度的弱非线性热声理论模型,并在不同条件下对模型做了进一步简化。这一理论为理解热声系统的工作机制以及设计优化热声系统提供了强有力的理论工具。     

Collisions of Two Spatial Solitons in Inhomogeneous Nonlinear Media

Collisions of spatial solitons occurring in the nonlinear Schröinger equation with harmonic potential are studied, using conservation laws and the split-step Fourier method. We find an analytical solution for the separation distance between the spatial solitons in an inhomogeneous nonlinear medium when the light beam is self-trapped in the transverse dimension. In the self-focusing nonlinear media the spatial solitons can be transmitted stably, and the interaction between spatial solitons is enhanced due to the linear focusing effect (and also diminished for the linear defocusing effect). In the self-defocusing nonlinear media, in the absence of self-trapping or in the presence of linear self-defocusing, no transmission of stable spatial solitons is possible. However, in such media the linear focusing effect can be exactly compensated, and the spatial solitons can propagate through.     

Propagation Dynamics of Ultrashort Optical Pulses in Nonlinear Coupled-Cavity-Type Multilayered Structures: a Numerical Study

A numerical study of the propagation dynamics of ultrashort pulses in a coupled-cavity-type multilayered structure doped with Kerr-type nonlinearity showed that such a structure improves the switching contrast of recently proposed optical-limiting and switching devices consisting of a nonlinear Bragg reflector.