非线性演化方程孤立波的同伦分析法求解

利用同伦分析法求解了Burgers方程，得到了其扭结形孤立波的近似解析解，该解非常接近于相应的精确解.结果表明，同伦分析法可用来求解非线性演化方程的孤立波解.同时，也对所用方法进行了一定扩展，得到了Kadomtsev-Petviashvili(KP)方程的钟形孤立子解.经过扩展后的方法能够更方便地用于求解更多非线性演化方程的高精度近似解析解. 关键词： Burgers方程 同伦分析法 KP方程 孤立波解     

Homotopic mapping method of solitary wave solutions for generalized complex Burgers equation

A class of generalized complex Burgers equation is considered. First, a set of equations of the complex value functions are solved by using the homotopic mapping method. The approximate solution for the original generalized complex Burgers equation is obtained. This method can find the approximation of arbitrary order of precision simply and reliably.     

Nonlinear acoustic propagation and focusing

Two model equations of nonlinear acoustics are considered. The implications of a point transformation between forms of the generalised Burgers equation (GBE) is discussed. New exact and asymptotic solutions are obtained. The dissipative Zabolotskaya-Khokhlov (DZK) equation describing acoustic wave propagation with allowance for transverse amplitude variation is studied. By considering a transformation onto the GBE, solutions exhibiting caustic behaviour are presented. A mechanism for the control of such singularities is presented along with a comparison with shock formation time.Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, England. Published in Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 8, pp. 783–787, August, 1993.     

Sound beams with shockwave pulses

The beam equation for a sound beam in a diffusive medium, called the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, has a class of solutions, which are power series in the transverse variable with the terms given by a solution of a generalized Burgers’ equation. A free parameter in this generalized Burgers’ equation can be chosen so that the equation describes an N-wave which does not decay. If the beam source has the form of a spherical cap, then a beam with a preserved shock can be prepared. This is done by satisfying an inequality containing the spherical radius, the N-wave pulse duration, the N-wave pulse amplitude, and the sound velocity in the fluid.     

Painlevé Analysis and Some Solutions of(2+1)-Dimensional Generalized Burgers Equations

Burgers equation ut = 2uux + uxx describes a lot of phenomena in physics fields, and it has attracted much attention.In this paper,the Burgers equation is generalized to (2+1) dimensions.By means of the Painlev(e') analysis,the most generalized Painlev(e') integrable(2+1)-dimensional integrable Burgers systems are obtained.Some exact solutions of the generalized Burgers system are obtained via variable separation approach.     

An Invariance for (2+1)-Extension of Burgers Equation and Formulae to Obtain Solutions of KP Equation

In this article, we study the (2+1)-extension of Burgers equation and the KP equation. At first, based on a known Bäcklund transformation and corresponding Lax pair, an invariance which depends on two arbitrary functions for (2+1)-extension of Burgers equation is worked out. Given a known solution and using the invariance, we can find solutions of the (2+1)-extension of Burgers equation repeatedly. Secondly, we put forward an invariance of Burgers equation which cannot be directly obtained by constraining the invariance of the (2+1)-extension of Burgers equation. Furthermore, we reveal that the invariance for finding the solutions of Burgers equation can help us find the solutions of KP equation. At last, based on the invariance of Burgers equation, the corresponding recursion formulae for finding solutions of KP equation are digged out. As the application of our theory, some examples have been put forward in this article and some solutions of the (2+1)-extension of Burgers equation, Burgers equation and KP equation are obtained.     

一类广义扰动KdV-Burgers方程的同伦近似解

通过构造一个同伦映射, 研究了一类广义扰动KdV-Burgers方程. 在引入典型无扰动任意次广义KdV-Burgers方程扭状孤立波解的基础上, 研究了扰动方程的具有任意精度的近似解,指出了近似解级数的收敛性, 最后利用不动点定理,进一步说明近似解的有效性,并对精度进行了讨论. 关键词： 广义扰动KdV-Burgers方程 同伦映射 渐近方法 近似解     

An application of the K-integrals for solving the radiative transfer equation with Fresnel boundary conditions

We introduce the reader to an approximate method of solving the transport equation which was developed in the context of neutron thermalisation by Kladnik and Kuscer in 1962 [Kladnik R, Kuscer I. Velocity dependent Milne's problem. Nucl Sci Eng 1962;13:149]. Essentially the method is based upon two special weighted integrals of the one-dimensional transport equation which are valid regardless of the boundary conditions, and any solution must satisfy these integral relationships which are called the K-integrals. To obtain an approximate solution to the transport equation we turn the argument around and insist that any approximate solution must also satisfy the K-integrals. These integrals are particularly useful when the problem under consideration cannot be solved easily by analytic methods. It also has the marked advantage of being applicable to problems where there is energy exchange in a collision and anisotropy of scattering. To establish the feasibility of the method we obtain a number of approximate solutions using the K-integral method for problems to which we have exact analytical solutions. This enables us to validate the method. It is then applied to a new problem that has not yet been solved; namely the calculation of the discontinuity in the scalar intensity at the boundary between two optically dissimilar materials.     

Reduction operators and exact solutions of generalized Burgers equations

Reduction operators of generalized Burgers equations are studied. A connection between these equations and potential fast diffusion equations with power nonlinearity of degree −1 via reduction operators is established. Exact solutions of generalized Burgers equations are constructed using this connection and known solutions of the constant-coefficient potential fast diffusion equation.     

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.     

Approximation of the soliton solution for the generalized Vakhnenko equation

A class of generalized Vakhnemko equation is considered. First, we solve the nonlinear differential equation by the homotopic mapping method. Then, an approximate soliton solution for the original generalized Vakhnemko equation is obtained. By this method an arbitrary order approximation can be easily obtained and, similarly, approximate soliton solutions of other nonlinear equations can be acquired.     

New Multiple Soliton-like Solutions to (3+1)-Dimensional Burgers Equation with Variable Coefficients

A new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation, which has more new solutions. More new multiple soliton-like solutions are obtained for the (3 1 )-dimensional Burgers equation with variable coefficients.     

New Multiple Soliton-like Solutions to （3＋1）-Dimensional Burgers Equation with Variable Coefficients

A new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation, which has more new solutions. More new multiple soliton-like solutions are obtained for the (3 1)-dimensional Burgers equation with variable coefficients.     

A uniform approximation for the eigenmodes of a planar dielectric waveguide

This paper derives a new approximation for the eigenmodes of a planar waveguide. The approximation is uniformly valid at both the high and low frequency regions of the dispersion relation. It is shown that a Pade approximation of the frequency equation leads to very accurate solutions. The new approximate solution is used to compute the frequency spectrum and the results compared with the exact analytical solution. The solutions presented here are ideal for analytically studying transient wave fields by means of modal summation.     

On the Approximation of the Stochastic Burgers Equation

 We prove mathematical approximation results for the (hyperviscous) Burgers equation driven by additive Gaussian noise. In particular we show that solutions of ``approximating equations' driven by a discretized noise converge towards the solution of the original equation when the discretization parameter gets small. The convergence takes place in the expected value of arbitrary powers of certain norms; i.e., all moments of the difference of the solutions tend to zero in certain function spaces. For the hyperviscous Burgers equation, these results are applied to justify the approximation of certain correlation functions that play a major role in statistical turbulence theory. Received: 10 October 2001 / Accepted: 21 May 2002 Published online: 6 August 2002     

A new approximate analytical approach for dispersion relation of the nonlinear Klein-Gordon equation

A novel approach is presented for obtaining approximate analytical expressions for the dispersion relation of periodic wavetrains in the nonlinear Klein-Gordon equation with even potential function. By coupling linearization of the governing equation with the method of harmonic balance, we establish two general analytical approximate formulas for the dispersion relation, which depends on the amplitude of the periodic wavetrain. These formulas are valid for small as well as large amplitude of the wavetrain. They are also applicable to the large amplitude regime, which the conventional perturbation method fails to provide any solution, of the nonlinear system under study. Three examples are demonstrated to illustrate the excellent approximate solutions of the proposed formulas with respect to the exact solutions of the dispersion relation. (c) 2001 American Institute of Physics.     

A(2+1)-dimensional nonlinear model for Rossby waves in stratified fluids and its solitary solution

In this paper,we investigate a(2+1)-dimensional nonlinear equation model for Rossby waves in stratified fluids.We derive a forced Zakharov–Kuznetsov(ZK)–Burgers equation from the quasigeostrophic potential vorticity equation with dissipation and topography under the generalized beta effect,and by utilizing temporal and spatial multiple scale transform and the perturbation expansion method.Through the analysis of this model,it is found that the generalized beta effect and basic topography can induce nonlinear waves,and slowly varying topography is an external impact factor for Rossby waves.Additionally,the conservation laws for the mass and energy of solitary waves are analyzed.Eventually,the solitary wave solutions of the forced ZK–Burgers equation are obtained by the simplest equation method as well as the new modified ansatz method.Based on the solitary wave solutions obtained,we discuss the effects of dissipation and slowly varying topography on Rossby solitary waves.     

多前车速度差的车辆跟驰模型的稳定性与孤波

通过线性稳定性分析,得到了多前车速度差模型的稳定性条件, 并发现通过调节多前车信息,使交通流的稳定区域明显扩大. 通过约化摄动方法 研究了该模型的非线性动力学特性:在稳定流区域,得到了描述密度波的Burgers方程;在交 通流的不稳定区域内,在临界点附近获得了描述车头间距的修正的Korteweg-de Vries (modified Korteweg-de Vries, mKdV)方程; 在亚稳态区域内,在中性稳定曲线附近获得了描述车头间距 的KdV方程. Burgers的孤波解、mKdV方程的扭结-反扭结波解及KdV方程的 孤波解描述了交通流堵塞现象.     

Boundary conditions in the radiative relaxation problem

The problem of the dissipation of temperature perturbations in a finite homogeneous atmosphere is solved for the situation in which the temperature at one boundary is maintained constant (that is, the temperature perturbation is zero for all times) while energy can be freely radiated to space through the other boundary. Exact solutions are shown for the exponential-sum fit to the kernel of the basic integral equation. These solutions constitute the set of radiative eigenfunctions. Also, approximate solutions in terms of the radiative eigenfunctions in the diffusion approximation (one exponential term in the expansion of the kernel) are obtained. These, in turn, are used in the solution of an initial value problem. The constant temperature boundary condition simulates the interface between two regions in one of which the relaxation processes are much more rapid than the purely radiative relaxation of the other.