Symmetry study of the coupled Burgers system

The inverse of the recursion operator of a coupled Burgers equation is given explicitly. Three sets of infinitely many symmetries of the considered model are obtained by acting the recursion operator and it’s inverse on the trivial symmetries, space translation, identity transformation and the scaling transformation respectively. These symmetries constitute an infinite dimensional Lie algebra.     

Order by Disorder,without Order,in a Two-Dimensional Spin System with O(2) Symmetry

We present a rigorous proof of an ordering transition for a two-component two-dimensional antiferromagnet with nearest and next-nearest neighbor interactions. The low-temperature phase contains two states distinguished by local order among columns or, respectively, rows. Overall, there is no magnetic order in accord with the classic Mermin-Wagner theorem. The method of proof employs a rigorous version of order by disorder, whereby a high degeneracy among the ground states is lifted according to the differences in their associated spin-wave spectra.Communicated by Jennifer Chayessubmitted 07/10/03, accepted 28/04/04     

Large Time Asymptotic Behaviors for Periodic Solutions of Generalized Burgers Equations With Spherical Symmetry or Linear Damping

Using the method of balancing arguments, large time asymptotic behaviors for the periodic solutions of generalized Burgers equations   u_t  +  u ³ u_x  +  ju /2 t  =δ/2 u_xx   and   u_t  +  u ³ u_x  +λ u  =δ/2 u_xx   subject to the periodic initial condition     and the vanishing boundary conditions   u (0,  t ) =  u ( l ,  t ) = 0,   t  ≥ 0   or    t ₀,  where   A ,  A ₁, δ, λ,  l ,  t ₀, ∈ R ⁺  and   j  = 1, 2  , are obtained.     

Analysis and Numerical Simulation of Viscous Burgers Equation

This work is concerned with the viscous Burgers equation inside a time dependent domain. We establish theoretical and numerical analysis such as existence and uniqueness of solutions, asymptotic behavior to the energy, and numerical discretization by Finite difference and Finite element methods.     

应力波放大器二维数值分析

运用动态有限元方法,对应力波在锥形应力波放大器中传播特性进行了二维数值分析,对影响透射波放大系数和波形的几何因素、输入脉冲形状、脉冲升时等进行了讨论,可供应力波铆接器设计之参考.计算结果与在Hopkinson杆上实验测得值以及特征线计算值进行了比较,其与实验结果吻合程度是令人相当满意的,与一维特征线方法相比.其计算精度也得到较大改善,特别是在透射波峰值压力附近.     

Symmetry Analysis of Abel's Equation

A solution algorithm for Abel's equation and some generalizations based on a nontrivial Lie symmetry of a particular kind, i.e., so-called structure-preserving symmetry, is described. For the existence of such a symmetry a criterion in terms of the coefficients of the so-called rational normal form of the given equation is derived. If it is affirmative, solving Abel's equation is reduced to a well-defined integration problem. It is shown that almost all known ad hoc methods for obtaining closed form solutions are consequences of this type of symmetry. Possible extensions of this scheme to more general classes of first-order ordinary differential equations are pointed out.     

Soliton solutions of Burgers equations and perturbed Burgers equation

This paper carries out the integration of Burgers equation by the aid of tanh method. This leads to the complex solutions for the Burgers equation, KdV-Burgers equation, coupled Burgers equation and the generalized time-delayed Burgers equation. Finally, the perturbed Burgers equation in (1+1) dimensions is integrated by the ansatz method.     

Symmetry Studies for Data Analysis

This paper describes some of the basic applications of the algebraic theory of canonical decomposition to the analysis of data. The notions of structured data and symmetry studies are discussed and applied to demonstrate their role in well known principles of analysis of variance and their applicability in more general experimental settings.      

二维非静力旋转流体方程组的稳定性

应用分层理论所提供的方法,证明了二维非静力Boussinesq近似的旋转流体方程组在二阶连续函数类中是一个不稳定性的方程．并给出方程组的形式解解空间构造和求解方法．对某些典型的初边值问题,给出了判断其是否存在形式解的充分必要条件以及计算形式解的具体的计算公式．     

Symbolic Differential Elimination for Symmetry Analysis

Differential problems are ubiquitous in mathematical modeling of physical and scientific problems. Algebraic analysis of differential systems can help in determining qualitative and quantitative properties of solutions of such systems. In this tutorial paper we describe several algebraic methods for investigating differential systems.     

Symmetry Analysis of Differential Equations Using MATHLIE

A general procedure for solving ordinary differential equations of arbitrary order is discussed. The method used is based on symmetries of a differential equation. The known symmetries allow the derivation of first integrals of the equation. The knowledge of at least r symmetries of an ordinary differential equation of order n with r n is the basis for deriving the solution. Our aim is to show that Lie's theory is a useful tool for solving ordinary differential equations of higher orders. Bibliography: 12 titles.     

Principal Component Analysis of Two-Dimensional Functional Data

This article presents and compares two approaches of principal component (PC) analysis for two-dimensional functional data on a possibly irregular domain. The first approach applies the singular value decomposition of the data matrix obtained from a fine discretization of the two-dimensional functions. When the functions are only observed at discrete points that are possibly sparse and may differ from function to function, this approach incorporates an initial smoothing step prior to the singular value decomposition. The second approach employs a mixed effects model that specifies the PC functions as bivariate splines on triangulations and the PC scores as random effects. We apply the thin-plate penalty for regularizing the function estimation and develop an effective expectation–maximization algorithm for calculating the penalized likelihood estimates of the parameters. The mixed effects model-based approach integrates scatterplot smoothing and functional PC analysis in a unified framework and is shown in a simulation study to be more efficient than the two-step approach that separately performs smoothing and PC analysis. The proposed methods are applied to analyze the temperature variation in Texas using 100 years of temperature data recorded by Texas weather stations. Supplementary materials for this article are available online.     

Symmetry and Uniqueness of Solutions of an Integral System

In this paper, we study the positive solutions for a class of integral systems and prove that all the solutions are radially symmetric and monotonically decreasing about some point. Moreover, we also obtain the uniqueness result for a special case. We use a new type of moving plane method introduced by Chen-Li-Ou [1]. Our new ingredient is the use of Hardy-Littlewood-Sobolev inequality instead of Maximum Principle.     

Estimates for the Maximum Modulus of Solutions to the Burgers System of Equations

We establish some nonlinear estimates on using the particular features of convective bounds.     

Two-Dimensional Riemann Problems for the Compressible Euler System

Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult, but a viable alternative remains missing. The author lists merits of one-dimensional Riemann problems and compares them with those for the current two-dimensional Riemann problems, to illustrate their worthiness. Two-dimensional Riemann problems are approached via the methodology promoted by Andy Majda in the spirits of modern applied mathematics; that is, simplified model is built via asymptotic analysis, numerical simulation and theoretical analysis. A simplified model called the pressure gradient system is derived from the full Euler system via an asymptotic process. State-of-the-art numerical methods in numerical simulations are used to discern smallscale structures of the solutions, e.g., semi-hyperbolic patches. Analytical methods are used to establish the validity of the structure revealed in the numerical simulation. The entire process, used in many of Majda＇s programs, is shown here for the two-dimensional Riemann problems for the compressible Euler systems of conservation laws     

二维拟定常可压流Euler方程组的简单波

简单波是这样的流动,它在像空间中的像是一条曲线．“简单波理论是除基本流动结构以外构造流动问题的解的基础”,见Courant和Friedrichs的经典著作《超声速流与冲击波》．该文主要研究二维拟定常可压流Euler方程组的简单波的几何结构.根据这些几何诠释,还构造了绕一拟流线弯曲部的疏散和压缩的简单波流动结构．这种流动结构将作为一个局部流动结构出现在4个接触间断的Riemann问题的整体解中．     

Stochastic stability of Burgers equation

The current paper is devoted to stochastic Burgers equation with driving forcing given by white noise type in time and periodic in space. Motivated by the numerical results of Hairer and Voss, we prove that the Burgers equation is stochastic stable in the sense that statistically steady regimes of fluid flows of stochastic Burgers equation converge to that of determinstic Burgers equation as noise tends to zero.