A maple package for the Painleve Test of Nonlinear Partial Differential Equations

A Maple package,named PLtest,is presented to study whether or not nonlinear partial differential equations (PDEs) pass the Painleve test.This package is based on the so-called WTC-Kruskal algorithm,which combines the standard WTC algorithm and the Kruskal simplification algorithm.Therefore,we not only study whether the given PDEs pass the test or not,but also obtain its truncated expansion form related to some integrability properties.Several well-known nonlinear models with physical interests illustrate the effectiveness of this package.     

A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations

For the Noyes-Fields equations, two-dimenslonal hyperbolic equations of conversation laww and the Burgers-KdV equation, a class of travellng wave solutions has been obtained by constructhag appropriate function transformations. The main idea of solving the equations is that nonlinear partial differential equations are changed into solving algebraic equations. This method has a wide-ranging practicability.     

Rare Events in Stochastic Partial Differential Equations on Large Spatial Domains

A methodology is proposed for studying rare events in stochastic partial differential equations in systems that are so large that standard large deviation theory does not apply. The idea is to deduce the behavior of the original model by breaking the system into appropriately scaled subsystems that are sufficiently small for large deviation theory to apply but sufficiently large to be asymptotically independent from one another. The methodology is illustrated in the context of a simple one-dimensional stochastic partial differential equation. The application reveals a connection between the dynamics of the partial differential equation and the classical Johnson–Mehl–Avrami–Kolmogorov nucleation and growth model. It also illustrates that rare events are much more likely and predictable in large systems than in small ones due to the extra entropy provided by space.     

Fluctuations of Quantum Currents and Unravelings of Master Equations

The very notion of a current fluctuation is problematic in the quantum context. We study that problem in the context of nonequilibrium statistical mechanics, both in a microscopic setup and in a Markovian model. Our answer is based on a rigorous result that relates the weak coupling limit of fluctuations of reservoir observables under a global unitary evolution with the statistics of the so-called quantum trajectories. These quantum trajectories are frequently considered in the context of quantum optics, but they remain useful for more general nonequilibrium systems. In contrast with the approaches found in the literature, we do not assume that the system is continuously monitored. Instead, our starting point is a relatively realistic unitary dynamics of the full system     

Unified approach to Miura,Bäcklund and Darboux Transformations for Nonlinear Partial Differential Equations

Abstract

This paper is an attempt to present and discuss at some length the Singular Manifold Method. This Method is based upon the Painlevé Property systematically used as a tool for obtaining clear cut answers to almost all the questions related with Nonlinear Partial Differential Equations: Lax pairs, Miura, Bäcklund or Darboux Transformations as well as τ -functions, in a unified way. Besides to present the basics of the Method we exemplify this approach by applying it to four equations in (1+1)-dimensions. Two of them are related with the other two through Miura transformations that are also derived by using the Singular Manifold Method.     

Supersymmetric Content of the Dirac and Duffin-Kemmer-Petiau Equations

We study subsolutions of the Dirac and Duffin-Kemmer-Petiau equations described in our earlier papers. It is shown that subsolutions of the Duffin-Kemmer-Petiau equations and those of the Dirac equation obey the same Dirac equation with some built-in projection operator. This covariant equation can be referred to as supersymmetric since it has bosonic as well as fermionic degrees of freedom.     

Construction of Generalized Synchronization for a Kind of Array Differential Equations and Applications

Chaos synchronization, as a special complex phenomenon, has been studied for about a decade. Only recently, have generalized chaotic synchronization phenomena been realized to be general in the real world and to have potential applications. We present two theorems for constructing a kind of array differential equations with generalized synchronization (GS) with respect to linear transformations. Two array differential equation systems with GS are introduced based on our theorems. Numerical simulations show that the two systems display periodic GS and chaotic GS, respectively.     

Sub-ODE＇s New Solutions and Their Applications to Two Nonlinear Partial Differential Equations with Higher-Order Nonlinear Terms

In the present paper, with the aid of symbolic computation, families of new nontrivial solutions of the first-order sub-ODE F12 = AF2 ＋ BF2＋p ＋ CF2＋2p （where F1= dF/dε, p 〉 0） are obtained. To our best knowledge, these nontrivial solutions have not been found in [X.Z. Li and M.L. Wang, Phys. Lett. A 361 （2007） 115] and IS. Zhang, W. Wang, and J.L. Tong, Phys. Lett. A 372 （2008） 3808] and other existent papers until now. Using these nontrivial solutions, the sub-ODE method is described to construct several kinds of exact travelling wave solutions for the generalized KdV-mKdV equation with higher-order nonlinear terms and the generalized ZK equation with higher-order nonlinear terms. By means of this method, many other physically important nonlinear partial differential equations with nonlinear terms of any order can be investigated and new nontrivial solutions can be explicitly obtained with the help of symbolic computation system Maple or Mathematics.     

Discrete Spectrum of the Deficit Angle and the Differential Structure of a Cosmic String

Differential properties of Klein-Gordon and electromagnetic fields on the space-time of a straight cosmic string are studied with the help of methods of the differential space theory. It is shown that these fields are smooth in the interior of the cosmic string space-time and that they loose this property at the singular boundary except for the cosmic string space-times with the following deficit angles: Δ=2π(1−1/n), n=1,2,… . A connection between smoothness of fields at the conical singularity and the scalar and electromagnetic conical bremsstrahlung is discussed. It is also argued that the smoothness assumption of fields at the singularity is equivalent to the Aliev and Gal’tsov “quantization” condition leading to the above mentioned discrete spectrum of the deficit angle.     

荧光法同时测定饲料及食品中V.A和V.E

用石油醚(30°～60℃)作抽提剂和萃取剂,借乙醇介质中的抗坏血酸使V.E避免氧化;用60%的硫酸抑制V.A对V.E的干扰作用。用荧光法分別在λ_(EX)295nm/λ_(EM)340nm和λ_(EX)340nm/λ_(EM)480nm处简便、快速地测定饲料和食品中V.A和V.E含量。样品不需经充氮皂化和柱分离,即可得到满意的结果。测定V.A和V.E的相对标准偏差分别为3.59%和1.76%,V.A和V.E的平均回收率分别为95.74%和95.05%。     

A Hierarchy of Differential-Difference Equations and Their Integrable Couplings

Starting from a discrete spectral problem, the corresponding hierarchy of nonlinear differential-difference equation is proposed. It is shown that the hierarchy possesses the bi-Hamiltionian structures. Further, two integrable coupling systems for the hierarchy are constructed through enlarged Lax pair method.     

(G′/G)-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics

In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.     

A Re-interpretation of the Concept of Mass and of the Relativistic Mass-Energy Relation

For over a century the definitions of mass and derivations of its relation with energy continue to be elaborated, demonstrating that the concept of mass is still not satisfactorily understood. The aim of this study is to show that, starting from the properties of Minkowski spacetime and from the principle of least action, energy expresses the property of inertia of a body. This implies that inertial mass can only be the object of a definition—the so called mass-energy relation—aimed at measuring energy in different units, more suitable to describe the huge amount of it enclosed in what we call the “rest-energy” of a body. Likewise, the concept of gravitational mass becomes unnecessary, being replaceable by energy, thus making the weak equivalence principle intrinsically verified. In dealing with mass, a new unit of measurement is foretold for it, which relies on the de Broglie frequency of atoms, the value of which can today be measured with an accuracy of a few parts in 10⁹.     

David Malament and the Conventionality of Simultaneity: A Reply

In 1977, David Malament proved the valuable technical result that the simultaneity relation of standard synchrony $\epsilon=\frac{1}{2}$ with respect to an inertial observer O is uniquely definable in terms of the relation κ of causal connectibility. And he claimed that this definability undermines my own version of the conventionality of metrical simultaneity within an inertial frame. But Malament’s proof depends on the imposition of several supposedly “innocuous” constraints on any candidate for the simultaneity relation relative to O. Relying on Allen I. Janis’s 1983 challenge to one of these constraints, I argue that Malament’s technical result did not undermine my philosophical construal of the ontological status of relative metrical simultaneity. Furthermore, I show that (a) Michael Friedman’s peremptorily substantivalist critique of my conception, which Malament endorses, is ill-founded, and (b) if Malament had succeeded in discrediting my own conventionalist version of metrical simultaneity, he would likewise have invalidated Einstein’s pioneering version of it.     

A Stochastic Differential Equation Model with Jumps for Fractional Advection and Dispersion

The path of a tracer particle through a porous medium is typically modeled by a stochastic differential equation (SDE) driven by Brownian noise. This model may not be adequate for highly heterogeneous media. This paper extends the model to a general SDE driven by a Lévy noise. Particle paths follow a Markov process with long jumps. Their transition probability density solves a forward equation derived here via pseudo-differential operator theory and Fourier analysis. In particular, the SDE with stable driving noise has a space-fractional advection-dispersion equation (fADE) with variable coefficients as the forward equation. This result provides a stochastic solution to anomalous diffusion models, and a solid mathematical foundation for particle tracking codes already in use for fractional advection equations.     

Nonequilibrium Thermodynamics of the First and Second Kind: Averages and Fluctuations

We elaborate and compare two approaches to nonequilibrium thermodynamics, the two-generator bracket formulation of time-evolution equations for averages and the macroscopic fluctuation theory, for a purely dissipative isothermal driven diffusive system under steady state conditions. The fluctuation dissipation relations of both approaches play an important role for a detailed comparison. The nonequilibrium Helmholtz free energies introduced in these two approaches differ as a result of boundary conditions. A Fokker-Planck equation derived by projection operator techniques properly reproduces long range fluctuations in nonequilibrium steady states and offers the most promising possibility to describe the physically relevant fluctuations around macroscopic averages for time-dependent nonequilibrium systems.