Local Classical Solutions to the Equations of Relativistic Hydrodynamics

In this paper, we prove that the convexity of the negative thermodynamical entropy of the equations of relativistic hydrodynamics for ideal gas keeps its invariance under the Lorentz transformation if and only if the local sound speed is less than the light speed in vacuum. Then a symmetric form for the equations of relativistic hydrodynamics is presented and the local classical solution is obtained. Based on this, we prove that the nonrelativistic limit of the local classical solution to the relativistic hydrodynamics equations for relativistic gas is the local classical solution of the Euler equations for polytropic gas.     

Rough Burgers-like equations with multiplicative noise

We construct solutions to vector valued Burgers type equations perturbed by a multiplicative space–time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise.     

Navier-Stokes方程的精确解——Dirac-Pauli表象的复变函数理论及其在流体力学中的应用(Ⅱ)

本文是文[1]的继续。在文[1]中我们应用Dirac-Pauli表象的复变函数理论并引入Kaluza“鬼”坐标,将不可压缩粘流动力学的Navier-Stokes方程化成只有一对复未知函数的非线性方程。在本文中,我们将除时间之外的复自变量进行重新组合,从而成对地减少了复自变量的数目。最后,我们将Navier-Stokes方程化成经典的Burgers方程。联结Burgers方程与扩散方程的Cole-Hopf变换实际上是B?cklund变换,而扩散方程众所周知是具有通解的。于是,我们利用B?cklund变换求得了Navier-Stokes方程的精确解。     

一类Burgers方程的孤立波解

Burgers方程在工程上有着重要的应用,它可以用来描述湍流、车队的交通流、氏族的随机迁移、化学工程中的分离等现象,对Burgers方程求解方法的研究有着重要的现实意义.对Burgers方程求解主要是应用差分和微分两方面的方法来展开求解的,1/G展开法是近年来发展起来的求解非线性偏微分方程的一种较为有效的微分解法.采用微分方程方面的方法,利用1/G展开法对一类Burgers方程进行求解,得到了此方程的一类孤立波解和扭曲波解,同时描绘出解的图像并分析解的结构和变化趋势.     

Relativistic Burgers and nonlinear SchrÖdinger equations

We construct relativistic complex Burgers-Schrödinger and nonlinear Schrödinger equations. In the nonrelativistic limit, they reduce to the standard Burgers and nonlinear Schrödinger equations and are integrable through all orders of relativistic corrections.     

Efficient numerical schemes for the solution of generalized time fractional Burgers type equations

The main aim of this paper is to propose two semi-implicit Fourier pseudospectral schemes for the solution of generalized time fractional Burgers type equations, with an analysis of consistency, stability, and convergence. Under some assumptions, the unconditional stability of the schemes is shown. In implementation of these schemes, the fast Fourier transform (FFT) can be used efficiently to improve the computational cost. Various test problems are included to illustrate the results that we have obtained regarding the proposed schemes. The results of numerical experiments are compared with analytical solutions and other existing methods in the literature to show the efficiency of proposed schemes in both accuracy and CPU time. As numerical solution of fractional stochastic nonlinear partial differential equations driven by Brownian motions are among current related research interests, we report the performance of these schemes on stochastic time fractional Burgers equation as well.     

Burgers and Kadomtsev-Petviashvili hierarchies: A functional representation approach

Functional representations of (matrix) Burgers and potential Kadomtsev-Petviashvili (pKP) hierarchies (and others), as well as some corresponding Bäcklund transformations, can be obtained surprisingly simply from a “discrete” functional zero-curvature equation. We use these representations to show that any solution of a Burgers hierarchy is also a solution of the pKP hierarchy. Moreover, the pKP hierarchy can be expressed in the form of an inhomogeneous Burgers hierarchy. In particular, this leads to an extension of the Cole-Hopf transformation to the pKP hierarchy. Furthermore, these hierarchies are solved by the solutions of certain functional Riccati equations.     

On the one- and one-half-dimensional relativistic Vlasov-Fokker-Planck-Maxwell system

In this paper, we study the relativistic Vlasov-Fokker-Planck-Maxwell system in one space variable and two momentum variables. This non-linear system of equations consists of a transport equation for the phase space distribution function combined with Maxwell's equations for the electric and magnetic fields. It is important in modelling distribution of charged particles in the kinetic theory of plasma. We prove the existence of a classical solution when the initial density decays fast enough with respect to the momentum variables. The solution which shares this same decay condition along with its first derivatives in the momentum variables is unique.     

Distributed optimal control of the viscous Burgers equation via a Legendre pseudo‐spectral approach

This paper presents a computational technique based on the pseudo‐spectral method for the solution of distributed optimal control problem for the viscous Burgers equation. By using pseudo‐spectral method, the problem is converted to a classical optimal control problem governed by a system of ordinary differential equations, which can be solved by well‐developed direct or indirect methods. For solving the resulting optimal control problem, we present an indirect method by deriving and numerically solving the first‐order optimality conditions. Numerical tests involving both unconstrained and constrained control problems are considered. Copyright © 2015 John Wiley & Sons, Ltd.     

爱因斯坦场方程的一类新解

作者构造了真空爱因斯坦场方程的一类新解,并给出了两个具体的例子,其中一个例子是时间周期解,而时间周期解和循环宇宙有着密切的关系.作者证明了这一类新解不是Minkowski的,并且与其它已有的解有本质上的不同.作者期望这种解可以在现代宇宙学和广义相对论中有所应用.     

Asymptotic behavior of weak solutions for the relativistic Vlasov-Maxwell equations with large light speed

We study here the behavior of time periodic weak solutions for the relativistic Vlasov-Maxwell boundary value problem in a three-dimensional bounded domain with strictly star-shaped boundary when the light speed becomes infinite. We prove the convergence toward a time periodic weak solution for the classical Vlasov-Poisson equations.     

The theory of the Lanczos spinor

Summary The Lanczos tensor is studied by using the van der Waerden two component spinor calculus. The algebraic and invariant theory of the resulting Lanczos spinor is investigated, and the spinor form of the equation relating the Lanczos and Weyl tensor is derived. The Weyl-Lanczos equations are transcribed into spin coefficient form and the radial dependence of the Lanczos coefficients is determined for eight relativistic space-times. Entrata in Redazione il 12 luglio 1973.     

The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method

In this paper, a scheme is developed to study numerical solution of the space- and time-fractional Burgers equations with initial conditions by the variational iteration method (VIM). The exact and numerical solutions obtained by the variational iteration method are compared with that obtained by Adomian decomposition method (ADM). The results show that the variational iteration method is much easier, more convenient, and more stable and efficient than Adomian decomposition method. Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed.     

Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators

In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a Banach space with locally monotone operators, which is a generalization of the classical result for monotone operators. In particular, we show that local monotonicity implies pseudo-monotonicity. The main results are applied to PDE of various types such as porous medium equations, reaction–diffusion equations, the generalized Burgers equation, the Navier–Stokes equation, the 3D Leray-α model and the p-Laplace equation with non-monotone perturbations.     

Homotopy perturbation method for numerical solutions of KdV‐Burgers' and Lax's seventh‐order KdV equations

In this article, we applied homotopy perturbation method to obtain the solution of the Korteweg‐de Vries Burgers (for short, KdVB) and Lax's seventh‐order KdV (for short, LsKdV) equations. The numerical results show that homotopy perturbation method can be readily implemented to this type of nonlinear equations and excellent accuracy. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Multi-soliton solution,rational solution of the Boussinesq–Burgers equations

In this paper we consider the Boussinesq–Burgers equations and establish the transformation which turns the Boussinesq–Burgers equations into the single nonlinear partial differential equation, then we obtain an auto-Bäcklund transformation and abundant new exact solutions, including the multi-solitary wave solution and the rational series solutions. Besides the new trigonometric function periodic solutions are obtained by using the generalized tan h method.     

DYNAMICS OF RELATIVISTIC FLUID FOR COMPRESSIBLE GAS

In this paper the relativistic fluid dynamics for compressible gas is studied.We show that the strict convexity of the negative thermodynamical entropy preserves invariant under the Lorentz transformation if and only if the local speed of sound in this gas is strictly less than that of light in the vacuum.A symmetric form for the equations of relativistic hydrodynamics is presented,and thus the local classical solutions to these equations can be deduced.At last,the non-relativistic limits of these local cla...     

Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers‐Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi‐linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two‐dimensional inviscid, incompressible fluid flows. Thus, the Burgers‐Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers‐Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near‐identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers‐Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small‐amplitude smooth solutions of the Burgers‐Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.     

Jump conditions in stress relaxation and creep experiments of Burgers type fluids: a study in the application of Colombeau algebra of generalized functions

We discuss stress relaxation and creep experiments of fluids that are generalizations of the classical model due to Burgers by allowing the material moduli such as the viscosities and relaxation and retardation times to depend on the stress. The physical problem, which is cast within the context of one dimension, leads to an ordinary differential equation that involves nonlinear terms like product of a function with a jump discontinuity and the derivative of a function with a jump discontinuity. As the equations are nonlinear, standard techniques that are used to study problems concerning linear viscoelastic fluids such as Laplace transforms and the theory of distributions are not applicable. We find it necessary to seek the solution in a more general setting. We discuss the mathematical and physical issues concerning the jump discontinuities and nonlinearity of the governing equation, and we show that the solution to the governing equation can be found in the sense of the generalized functions introduced by Colombeau. In the framework of Colombeau algebra we, under certain assumptions, derive jump conditions that shall be used in stress relaxation and creep experiments of fluids of the Burgers type. We conclude the paper with a discussion of the physical relevance of these assumptions.     

Analytic approximate solutions to Burgers,Fisher, Huxley equations and two combined forms of these equations

In this paper, we are giving analytic approximate solutions to a class of nonlinear PDEs using the homotopy analysis method (HAM). The Burgers, Fisher, Huxley, Burgers–Fisher and Burgers–Huxley equations are considered. We aim two goals: one is to highlight the efficiency of HAM in solving this class of PDEs and the other is that, although the considered equations have different combinations of nonlinear terms, when applying HAM, we use the same initial guess, the same auxiliary linear operator and the same auxiliary function for all of them.