Approximate direct reduction method: infinite series reductions to the perturbed mKdV equation

The approximate direct reduction method is applied to the perturbed mKdV equation with weak fourth order dispersion and weak dissipation. The similarity reduction solutions of different orders conform to formal coherence, accounting for infinite series reduction solutions to the original equation and general formulas of similarity reduction equations. Painlevé II type equations, hyperbolic secant and Jacobi elliptic function solutions are obtained for zero-order similarity reduction equations. Higher order similarity reduction equations are linear variable coefficient ordinary differential equations.     

Evolution of Distribution of Dislocations in Ni3Ge Single Crystals During Deformation

The evolution of the distribution of dislocations in Ni₃Ge single crystals subjected to deformation in uniaxial compression is studied. The dislocation ensemble in the material under review is found to be of a chaotic homogeneous type. Contact interactions between dislocations prevail, and a linear relation of the spacing between dislocations to the length of dislocation segments is observed for stoppers of an arbitrary type. An equation is derived for the probability density function of the fraction of mobile dislocation segments. The solution to the equation is the normal distribution law. This solution can be extended to parameters that are functions of the dislocation density or spacing between dislocations. The experimental histograms of the spacing between dislocations and of that between arbitrary stoppers with a high significance level obey the lognormal law for all degrees of reduction studied.     

组合KdV方程的显式精确解

借助计算机代数系统Mathematica，利用双曲函数法找到了组合KdV方程(Combined KdV Equation)的精确孤立波解，包括钟型孤立波解和扭结型孤立波解.在此基础上又对双曲函数法的思想进行了推广，从而获得了其更多的显式精确解，包括间断型激波解和指数函数型解.这种方法也适用于求解其他非线性发展方程(组). 关键词： 组合KdV方程 双曲函数法 孤立波解 精确解     

Chapman-Kolmogorov equation and path integrals for discrete chaos in presence of noise

We derive an equation of the Chapman-Kolmogorov type for discrete multi-dimensional mappings under the action of additive and multiplicative noise with arbitrary distribution function. The resulting equation is reduced to a Fredholm integral equation. By iteration of the Chapman-Kolmogorov equation as usual, a path integral solution is found. Specializing the distribution function of the noise to a Gaussian distribution and taking the Fourier transform contant can be made with the path integral formulation used by Shraiman, Wayne and Martin.     

Kaon and τ-decay rates in QCD

The derivation of string type equations from QCD is reexamined in the framework of renormalized perturbation theory. Renormalizing the equation for the second functional derivative of the Wilson functional at different points one observes a short distance problem which is studied by the help of OPE and RG. On this line the reduction of the equation to a linear one of string-type can be understood.     

Distribution of the angle of rotation plane random walks

We study the probability distribution of the angle by which the tangent to the trajectory rotates in the course of a plane random walk. It is shown that the determination of this distribution function can be reduced to an integral equation, which can be rigorously transformed into a differential equation of Hill's type. We derive the asymptotic distribution for very long walks.     

Electromagnetic wave propagation in random media

The propagation of a narrow frequency band beam of electromagnetic waves in a medium with randomly varying index of refraction is considered. A novel formulation of the governing equation is proposed. An equation for the average Green function (or transition probability) can then be derived. A Fokker-Planck type equation is contained as a limiting case. The results are readily generalized to include the features of the random coupling model and it is argued that the present problem is particularly suited for an analysis of this type.     

Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving...     

The Benney hierarchy and the Dirichlet boundary problem in two dimensions

A theoretical connection between reductions of the Benney hierarchy and the Dirichlet problem for Laplace's equation in the plane is made. The connection is used to deduce general formulas for the uniformizations of two spectral functions associated with N-parameter reductions of the hierarchy. Two types of reduction are considered: one type has been considered by previous authors using alternative arguments, the second type is new. The formulas are general and are expressed in terms of the modified Green's function (for Laplace's equation) in arbitrary N-connected, reflectionally-symmetric, planar domains. The Benney moments are found to be purely geometrical quantities associated with these domains.     

A modified complex short pulse equation of defocusing type

In this paper, we are concerned with a modified complex short pulse (mCSP) equation of defocusing type. Firstly, we show that the mCSP equation is linked to a complex coupled dispersionless equation of defocusing type via a hodograph transformation, thus, its Lax pair can be deduced. Then the bilinearization of the defocusing mCSP equation is formulated via dependent variable and hodograph transformations. One- and two-dark soliton solutions are found by Hirota’s bilinear method and their properties are analyzed. It is shown that, depending on the parameters, the dark soliton solution can be either smoothed, cusponed or looped one. More specifically, the dark soliton tends to be evolved into a singular (cusponed or looped) one due to the increase of the spatial wave number in background plane waves and the increase of the depth of the trough. In the last part of the paper, we derive the defocusing mCSP equation from the single-component extended KP hierarchy by the reduction method. As a by-product, the N-dark soliton solution in the form of determinants for the defocusing mCSP is provided.     

Microscopic dielectric response functions in semiconductor quantum dots

We calculate and model the microscopic dielectric response function for quantum dots using first principle methods. We find that the response is bulklike inside the quantum dots, and the reduction of the macroscopic dielectric constants is a surface effect. We present a model for the microscopic dielectric function which reproduces well the directly calculated results and can be used to solve the Poisson equation in a nanosystem.     

A coupled pressure-based computational method for incompressible/compressible flows

Pressure-based flow solvers couple continuity and linearized truncated momentum equations to derive a Poisson type pressure correction equation and use the well known SIMPLE algorithm. Momentum equations and the pressure correction equation are typically solved sequentially. In many cases this method results in slow and often difficult convergence. The current paper proposes a novel computational algorithm, solving for pressure and velocity simultaneously within a pressure-correction coupled solution approach using finite volume method on structured and unstructured meshes. The method can be applied to both incompressible and subsonic compressible flows. For subsonic compressible flows, the energy equation is also coupled with flow field and the density of fluid is obtained by equation of state. The procedure eliminates the pressure correction step, the most expensive component of the SIMPLE-like algorithms. The proposed coupled continuity-momentum-energy equation method can be used to simulate steady state or transient flow problems. The method has been tested on several CFD benchmark cases with excellent results showing dramatically improved numerical convergence and significant reduction in computational time.     

Non-minimal gravitational coupling of phantom and big rip singularity

We consider a non-minimal coupling of a perfect fluid matter system with geometry, which the coupling function is taken to be an arbitrary function of the Ricci scalar. Due to such a coupling, the matter stress tensor is no longer conserved and there is an energy transfer between the two components. By solving the conservation equation and applying the second law of thermodynamics, we show that direction of the energy transfer depends on the equation of state of the matter fluid. In particular, a phantom fluid should loose energy with expansion of the universe. This energy reduction can avoid the universe to end with a cosmic doomsday.     

Exact travelling wave solutions for （1＋ 1）-dimensional dispersive long wave equation

A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.     

Thermodynamical and Green function many-body Wick theorems

The thermodynamical and Green function many-body reduction theorems of Wick type are proved for the arbitrary mixtures of the fermion, boson and spin systems. “Many-body” means that the operators used are the products of the arbitrary number of one-body standard basis operators [of the fermion or (and) spin types] with different site (wave vector) indices, but having the same “time” (in the interaction representation). The method of proving is based on: 1) the first-order differential equation of Schwinger type for: 1a) -product of operators; 1b) its average value; 2) KMS boundary conditions for this average. It is shown that the fermion, boson and spin systems can be unified in the many-body formulation (bosonification of the fermion systems). It is impossible in the one-body approach. Both of the many-body versions of the Wick theorem have the recurrent feature: nth order moment diagrams for the free energy or Green functions can be expressed by the (n −1)th order ones. This property corresponds to the automatic realization of: (i) summations over Bose-Einstein or (and) Fermi-Dirac frequencies; (ii) elimination of Bose-Einstein or (and) Fermi-Dirac distributions. The procedures (i) and (ii), being the results of using the Green function one-body reduction theorem, have constituted the significant difficulty up to now in the treatment of quantum systems.     

The time fractional diffusion-wave equation

The time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order 2β with 0<β≤1/2 or 1/2<β≤1, respectively. Using the method of the Laplace transform, it is shown that the fundamental solutions of the basic Cauchy and signalling problems can be expressed in terms of an auxiliary function M (z; β), where z is the similarity variable. Such function, which reduces to the well-known Gaussian function for β=1/2 (ordinary diffusion), is proved to be an entire function of Wright type.     

ANALYTICAL SOLUTION OF THE 2n-DIMENSIONAL FOKKER-PLANCK EQUATION WITH HARMONIC OSCILLATOR POTENTIAL

An analytical solution is obtained for the 2n-dimensiona Fokker-Planck equation (F-P equation for short) with the harmonic oscillator potential. A few steps are involved in the derivation. First,the Lagrangian subsidiary equation is solved; then with its integral constants as new variables of the F-P equation, the diffusion equation is obtained and solved; at last, expressed in the original phase space, the solution of the F-P equation .is finally obtained. The analysis for the solution is made. The solution is a Gaussian type function and a δ-function of time. If a particle moves in a well in ali directions, then as t→∞, the distribution function can reach a stationary nonzero distribution-Maxuwell-Boltzmann type distribution (M-B distribution for short).As an example, the 2-dimensional F-P equation is solved and discussed in detail.     

Analytic solution of Percus-Yevick equation for fluid of hard spheres

A new method of analytic solution of the Percus-Yevick equation for the radial distribution functiong(r) of hard-sphere fluid is proposed. The original non-linear integral equation is reduced to non-homogeneous linear integral equation of Volterra's type of the second order. The kernel of this new equation has a polynomial form which allows to find analytic expression forg(r) itself without using the Laplace transformation. In addition, the first three moments of the total correlation function can be found.     

Lie symmetry analysis and reduction of a new integrable coupled KdV system

In this paper, Lie symmetry is investigated for a new integrable coupled Korteweg--de Vries (KdV) equation system. Using some symmetry subalgebra of the equation system, we obtain five types of the significant similarity reductions. Abundant solutions of the coupled KdV equation system, such as the solitary wave solution, exponential solution, rational solution and polynomial solution, etc. are obtained from the reduced equations. Especially, one type of group-invariant solution of reduced equations can be acquired by means of the Painlev\'e I transcendent function.     

双曲函数型辅助方程构造具5次强非线性项的波方程的新精确孤波解

在辅助方程法的基础上利用两种函数变换和一种双曲函数型辅助方程，通过符号计算系统Mathematica构造了在力学当中一个重要的模型，有5次强非线性项的波方程的新三角函数型和双曲函数型精确孤波解.这种方法寻找其他具5次强非线性项的非线性发展方程的新精确解方面具有普遍意义. 关键词： 双曲函数型辅助方程 函数变换 具5次强非线性项的波方程 精确孤波解