On the stable hole solutions in the complex Ginzburg–Landau equation

We show numerically that the one-dimensional quintic complex Ginzburg–Landau equation admits four different types of stable hole solutions. We present a simple analytic method which permits to calculate the region of existence and approximate shape of stable hole solutions in this equation. The analytic results are in good agreement with numerical simulations.     

Numerical soliton-like solutions of the potential Kadomtsev–Petviashvili equation by the decomposition method

In this Letter we present an Adomian's decomposition method (shortly ADM) for obtaining the numerical soliton-like solutions of the potential Kadomtsev–Petviashvili (shortly PKP) equation. We will prove the convergence of the ADM. We obtain the exact and numerical solitary-wave solutions of the PKP equation for certain initial conditions. Then ADM yields the analytic approximate solution with fast convergence rate and high accuracy through previous works. The numerical solutions are compared with the known analytical solutions.     

Analytic Solutions of Linearized Lattice Boltzmann Equation for Simple Flows

A general procedure to obtain analytic solutions of the linearized lattice Boltzmann equation for simple flows is developed. As examples, the solutions for the Poiseuille and the plane Couette flows in two-dimensional space are obtained and studied in detail. The solutions not only have a component which is the solution of the Navier–Stokes equation, they also include a kinetic component which cannot be obtained by the Navier–Stokes equation. The kinetic component of the solutions is due to the finite-mean-free-path effect. Comparison between the analytic results and the numerical results of lattice-gas simulations is made, and they are found to be in accurate agreement.     

On an integral equation arising in the transport of radiation through a slab involving internal reflection

In an earlier contribution to this journal [M.M.R. Williams, Eur. Phys. J. B 47, 291 (2005)], we derived an integral equation for the transmission of radiation through a slab of finite thickness which incorporated internal reflection at the surfaces. Here we generalise the problem to the case when there is a source on each face and the reflection coefficients are different at each face. We also discuss numerical and analytic solutions of the equation discussed in [M.M.R. Williams, Eur. Phys. J. B 47, 291 (2005)] when the reflection is governed by the Fresnel conditions. We obtain numerical and graphical results for the reflection and transmission coefficients, the scalar intensity and current and the emergent angular distributions at each face. The incident source is either a mono-directional beam or a smoothly varying distribution which goes from isotropic to a normal beam. Of particular interest is the philosophy of the numerical solution and whether a direct numerical approach is more effective than one involving a more elegant analytical solution using replication and the Hilbert problem. We also develop the solution of this problem using diffusion theory and compare the results with the exact transport solution. An erratum to this article is available at .     

Multiple exp-function method for soliton solutions of nonlinear evolution equations

We applied the multiple exp-function scheme to the(2+1)-dimensional Sawada-Kotera(SK) equation and(3+1)-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota's perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.     

强阻尼广义sine-Gordon方程特征问题的变分迭代法

研究了在数学、力学中广泛出现的一类非线性强阻尼广义sine-Gordon扰动微分方程问题. 首先, 引入行波变换, 求出退化方程的精确解. 再构造一个泛函, 创建了一个变分迭代算法, 最后, 求出原非线性强阻尼广义sine-Gordon扰动微分方程问题的近似行波解析解. 用变分迭代法可得到的各次近似解, 具有便于求解、精度高等特点. 求得的近似解析解弥补了单纯用数值方法的模拟解的不足.     

拉曼增益系数可分离光纤中的受激拉曼散射

给出了拉曼增益系数分离光纤的定义,在考虑了N个单色光之间的两两受激拉曼散射,得到了这种光纤中单向传输的N个不同的单色光之间的受激拉曼散射（SRS）稳态耦合波方程的解析解。它适用于各单色光功率任意大小、光频率分布不均匀时的一般情况。这一结果可应用于光纤通信系统、拉曼光纤放大器和拉曼光纤激光器的设计与分析。最后把解析解与数值解进行了比较,两者取得了很好的一致。     

A review on the solution of Grad–Shafranov equation in the cylindrical coordinates based on the Chebyshev collocation technique

Equilibrium reconstruction consists of identifying, from experimental measurements, a distribution of the plasma current density that satisfies the pressure balance constraint. Numerous methods exist to solve the Grad–Shafranov equation, describing the equilibrium of plasma confined by an axisymmetric magnetic field. In this paper, we have proposed a new numerical solution to the Grad–Shafranov equation (an axisymmetric, magnetic field transformed in cylindrical coordinates solved with the Chebyshev collocation method) when the source term (current density function) on the right-hand side is linear. The Chebyshev collocation method is a method for computing highly accurate numerical solutions of differential equations. We describe a circular cross-section of the tokamak and present numerical result of magnetic surfaces on the IR-T1 tokamak and then compare the results with an analytical solution.     

A comparison of differential and integral equations of radiative transfer

The equations of radiative transfer and of statistical equilibrium of a two-level atom are solved by means of differential and integral equations for a one-dimensional medium. The numerical solutions are compared to the analytic solution. It is found that the integral equation for piecewise quadratic source functions gives more accurate results than does the differential equation.     

Numerical simulation of the regularized long wave equation by He's homotopy perturbation method

In this Letter, we present the homotopy perturbation method (shortly HPM) for obtaining the numerical solution of the RLW equation. We obtain the exact and numerical solutions of the Regularized Long Wave (RLW) equation for certain initial condition. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of other methods have led us to significant consequences. The numerical solutions are compared with the known analytical solutions.     

Foundations of quantum mechanics: The Langevin equations for QM

Stochastic derivations of the Schrödinger equation are always developed on very general and abstract grounds. Thus, one is never enlightened which specific stochastic process corresponds to some particular quantum mechanical system, that is, given the physical system—expressed by the potential function, which fluctuation structure one should impose on a Langevin equation in order to arrive at results identical to those comming from the solutions of the Schrödinger equation. We show, from first principles, how to write the Langevin stochastic equations for any particular quantum system. We also show the relation between these Langevin equations and those proposed by Bohm in 1952. We present numerical simulations of the Langevin equations for some quantum mechanical problems and compare them with the usual analytic solutions to show the adequacy of our approach. The model also allows us to address important topics on the interpretation of quantum mechanics.     

Scaling Solutions of Inelastic Boltzmann Equations with Over-Populated High Energy Tails

This paper deals with solutions of the nonlinear Boltzmann equation for spatially uniform freely cooling inelastic Maxwell models for large times and for large velocities, and the nonuniform convergence to these limits. We demonstrate how the velocity distribution approaches in the scaling limit to a similarity solution with a power law tail for general classes of initial conditions and derive a transcendental equation from which the exponents in the tails can be calculated. Moreover on the basis of the available analytic and numerical results for inelastic hard spheres and inelastic Maxwell models we formulate a conjecture on the approach of the velocity distribution function to a scaling form.     

A second-order accurate numerical method for the two-dimensional fractional diffusion equation

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow, finance and others. In this paper, we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation. This numerical method combines the alternating directions implicit (ADI) approach with a Crank–Nicolson discretization and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method. The stability and the consistency of the method are established. Numerical solutions for an example super-diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method.     

Optimization strategy to find shapes of soliton molecules

Frequently, a certain solution of a nonlinear wave equation is of interest, but no analytic form is known, and one must work with approximations. We introduce a search strategy to find solutions of the propagation of soliton molecules in a dispersion-managed optical fiber and to determine their shape with some precision. The strategy compares shapes before and after propagation and invokes an optimization routine to minimize the difference. The scheme is designed to be implemented in an experiment so that all fiber parameters are taken into account. Here, we present a full numerical study and a verification of convergence; we validate the method with cases of known solutions. We also compare the performance of two optimization procedures, the Nelder–Mead simplex method and a genetic algorithm.     

Numerical solution of the Fokker-Planck equation with variable coefficients

We study the numerical solution of the Fokker-Planck equation. This equation gives a good approximation to the radiative transport equation when scattering is peaked sharply in the forward direction which is the case for light propagation in tissues, for example. We derive first the numerical solution for the problem with constant coefficients. This numerical solution is constructed as an expansion in plane wave solutions. Then we extend that result to take into account coefficients that vary spatially. This extension leads to a coupled system of initial and final value problems. We solve this system iteratively. Numerical results show the utility of this method.     

谱方法用于非定常流动计算的隐式求解

本文对谱方法用于周期性非定常流动的隐式求解方法进行了探讨,分析了影响计算稳定性和收敛速度的因素.提出了结合多重网格的隐式求解方法并对算法进行了验证,初步计算表明本文算法具有良好的稳定性和收敛速度.对于周期性非定常流动,结合本文提出的隐式求解的时域谱方法可以达到很高的精度且具有良好的计算效率.     

A Lattice Boltzmann Model and Simulation of KdV-Burgers Equation

A lattice Boltzmann model of KdV-Burgers equation is derived by using the single-relaxation form of the lattice Boltzmann equation. With the present model, we simulate the traveling-wave solutions, the solitary-wave solutions, and the sock-wave solutions of KdV-Burgers equation, and calculate the decay factor and the wavelength of the sock-wave solution, which has exponential decay. The numerical results agree with the analytical solutions quite well.     

A Lattice Boltzmann Model and Simulation of KdV-Burgers Equation

A lattice Boltzmann model of KdV-Burgers equation is derived by using the single-relaxation form of the lattice Boltzmann equation. With the present model, we simulate the traveling-wave solutions, the solitary-wave solutions, and the sock-wave solutions of KdV-Burgers equation, and calculate the decay factor and the wavelength of the sock-wave solution, which has exponential decay. The numerical results agree with the analytical solutions quite well.     

大尺度浅水波方程中相互调制的非线性波

基于一般的浅水波方程, 根据大尺度正压大气的特点, 得到无量纲的控制大尺度大气的动力学非线性方程组. 利用多尺度法, 由无量纲的动力学方程组导出了扰动位势的非线性控制方程. 采用椭圆方程构造该扰动位势控制方程的解, 获得了扰动位势和速度的多周期波与冲击波(爆炸波) 并存的解析解. 扰动位势的解表明经向和纬向具有不同周期和波长的周期波, 且都受纬向孤波的调制; 速度的解表明大尺度大气流动存在气旋和反气旋周期性分布的现象. 关键词： 浅水波方程 大尺度正压大气 解析解 非线性波     

A general method for evaluating the current system and its magnetic field of a plane current sheet, uniform except for a certain area of different uniform conductivity, with results for a square area

Summary We derive a linear Fredholm integral equation of the second kind on an arbitrary closed plane contour which divides an infinite plane current sheet into two regions of different uniform integrated conductivities. This integral equation is satisfield along the above-described contour by a certain combination of the limiting values of the electric potential at both sides of the boundary. This electric potential is due to the currents created in the sheet when a uniform electric field is applied to it. The derived integral equation admits exact solutions in closed form for the cases of circular and elliptical insertions. These solutions are identical with those previously obtained, by other methods, for the same cases. A general method is given for the numerical solution of the integral equation. As an illustration, this method is applied to the case of a square insertion where we used the results of Ashour to obtain numerical estimation for the results of the additional magnetic field on and around the square insertion. To speed up publication, the proofs were not sent to the authors and were supervised by the Scientific Commettee.