高阶色散效应常系数Ginzburg-Landau方程自相似脉冲演化的解析分析

采用自相似分析方法,基于常系数高阶色散的Ginzburg-Landau方程,通过分离变量法得出了高阶色散效应自相似脉冲演化的解析解,给出了自相似脉冲的振幅、相位、啁啾以及脉冲宽度的一般表达式.研究表明,在增益光纤的二阶正常色散区域,同时考虑高阶色散和增益色散双重效应影响下演化的自相似孤子脉冲仍然保持线性啁啾;振幅解析解的三阶色散效应显著.这与数值计算的结果非常一致. 关键词： 三阶色散 Ginzburg-Landau方程 自相似脉冲 二阶正常色散     

Analytic solutions of self-similar pulse based on Ginzburg-Landau equation with constant coefficients

Based on the constant coefficients of Ginzburg-Landau equation that considers the influence of the doped fiber retarded time on the evolution of self-similar pulse, the parabolic asymptotic self-similar solutions were obtained by the symmetry reduction algorithm. The parabolic asymptotic amplitude function, phase function, strict linear chirp function and the effective temporal pulse width of self-similar pulse are given in this paper. And these theoretical results are consistent with the numerical simulations. Supported by the Natural Science Foundation of Guangdong Province of China (Grant No. 04010397)     

自相似Euler方程的数值方法

Euler方程某些问题的解具有自相似特点，可以使用更准确的方法求解.提出了两种数值方法，分别称为自相似和准自相似方法，新方法可以使用现有守恒律方程的数值格式，无须设计特殊方法.对一维激波管问题、二维Riemann问题、激波反射以及激波折射问题进行了数值计算.对自相似Euler方程，一维计算结果显示数值解基本等同于精确解，二维结果也比现有文献计算的结果有更高的分辨率.对准自相似Euler方程，新方法可以求解不具有自相似性但接近自相似的问题，并在计算时间足够长时可以取得自相似Euler方程的效果.数值求解自相似Euler方程对自相似问题的研究，高分辨率、高精度格式的设计乃至Euler方程的精确解都有重要启示.      

Warm cascades and anomalous scaling in a diffusion model of turbulence

A phenomenological turbulence model in which the energy spectrum obeys a nonlinear diffusion equation is analyzed. The general steady state contains a nonlinear mixture of the constant-flux Kolmogorov and fluxless thermodynamic components. Such "warm cascade" solutions describe a bottleneck phenomenon of spectrum stagnation near the dissipative scale. Transient self-similar solutions describing a finite-time formation of steady cascades are analyzed and found to exhibit nontrivial scaling behavior.     

Kinetic Approach to Long time Behavior of Linearized Fast Diffusion Equations

We show that the rate of convergence towards the self-similar solution of certain linearized versions of the fast diffusion equation can be related to the number of moments of the initial datum that are equal to the moments of the self-similar solution at a fixed time. As a consequence, we find an improved rate of convergence to self-similarity in terms of a Fourier based distance between two solutions. The results are based on the asymptotic equivalence of a collisional kinetic model of Boltzmann type with a linear Fokker-Planck equation with nonconstant coefficients, and make use of methods first applied to the reckoning of the rate of convergence towards equilibrium for the spatially homogeneous Boltzmann equation for Maxwell molecules.     

Several Types of Similarity Solutions for the Hunter-Saxton Equation

We study separable and self-similar solutions to the HunterSaxton equation,a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal(among other applications).Essentially,we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the HunterSaxton equation.For each type of solution,we are able to obtain some simple exact solutions in closed-form,and more complicated solutions through an analytical approach.We find that there is a whole family of self-similar solutions,each of which depends on an arbitrary parameter.This parameter essentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data.The simpler solutions found constitute exact solutions to a nonlinear partial differential equation,and hence are also useful in a mathematical sense.Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions.Both types of solutions cast light on self-similar phenomenon arising in the HunterSaxton equation.     

The LSW Model for Domain Coarsening: Asymptotic Behavior for Conserved Total Mass

In the classical theory of domain coarsening the particles of the coarsening phase evolve by diffusional mass transfer with a mean field. We study the long-time behavior of measure-valued solutions with compact support to this model coupled with the constraint of conserved total mass, including mean-field mass. Unlike the case of conserved volume fraction, this system has no precisely self-similar solutions, and sufficiently low supersaturation can lead to the finite-time extinction of all particles. We find a new explicit family of asymptotically self-similar solutions, and in case that the largest particle size is unbounded we establish results similar to the volume-conserved case. These include necessary criteria for asymptotic self-similarity, and sensitive dependence of long-time behavior on the distribution of largest particles in the system.     

色散渐减光纤中Ginzburg-Landau方程的自相似脉冲演化的解析解

采用对称约简的分析方法，得出了变系数Ginzburg-Landau方程的抛物渐近自相似脉冲解析解的一般表达式.给出了二阶色散系数纵向双曲型变化和纵向指数型变化的色散渐减光纤中自相似脉冲的振幅、啁啾以及脉冲宽度的具体形式，并与数值解进行了对比，其结果符合得很好.从而证实了稀土元素掺杂的色散渐减光纤中，在增益色散因子的影响下，脉冲的演化具有抛物型自相似特性.     

Simple solutions of relativistic hydrodynamics for systems with ellipsoidal symmetry

Simple, self-similar, analytic solutions of (1+3)-dimensional relativistic hydrodynamics are presented for ellipsoidally symmetric finite fireballs corresponding to non-central collisions of heavy ions at relativistic bombarding energies. The hydrodynamical solutions are obtained for a new, general family of equations of state with the possibility of describing phase transitions.     

A class of blowup and global analytical solutions of the viscoelastic Burgersʼ equations

In this Letter, by employing the perturbational method, we obtain a class of analytical self-similar solutions of the viscoelastic Burgers? equations. These solutions are of polynomial-type whose forms, remarkably, coincide with that given by Yuen for the other physical models, such as the compressible Euler or Navier–Stokes equations and two-component Camassa–Holm equations. Furthermore, we classify the initial conditions into several groups and then discuss the properties on blowup and global existence of the corresponding solutions, which may be readily seen from the phase diagram.     

Exact self-similar wave solutions for the generalized (3 + 1)-dimensional cubic–quintic nonlinear Schröinger equation with distributed coefficients

In this paper, a similarity transformation is presented to reduce the generalized (3 + 1)-dimensional cubic–quintic nonlinear Schrödinger equation with distributed coefficients to the related constant-coefficients one. Then a number of spatiotemporal self-similar wave solutions are constructed. Under the specific choice of the dispersion, cubic and quintic nonlinearities, phase modulation and the gain/loss, we investigate the dynamical behaviors of those spatiotemporal self-similar waves in an inhomogeneous optical fiber media.     

Self-Similar Solutions of Variable-Coefficient Cubic-Quintic Nonlinear Schrödinger Equation with an External Potential

An improved homogeneous balance principle and an F-expansiontechnique are used to construct exact self-similar solutions to the cubic-quintic nonlinear Schrödinger equation. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and the external potential. Some simple self-similar waves are presented.     

Scalings for a Ballistic Aggregation Equation

We consider a mean field type equation for ballistic aggregation of particles whose density function depends both on the mass and momentum of the particles. For the case of a constant aggregation rate we prove the existence of self-similar solutions and the convergence of more general solutions to them. We are able to estimate the large time decay of some moments of general solutions or to build some new classes of self-similar solutions for several classes of mass and/or momentum dependent rates.     

中空圆柱形边界条件下非线性介质中的晶格孤子

本文采用自相似方法求解中空圆柱形边界贝塞尔晶格中变系数非线性薛定谔方程, 得到了与数值模拟解一致的解析解, 表明由非衍射贝塞尔光束诱导的光子晶格可支持稳定的自相似孤子簇. 精确解ψ_mn是n+2层, 2m极的多极孤立波, 其形状及大小在传播过程中保持不变. 关键词： 空间光孤子 贝塞尔晶格 边界 自相似     

Directional fractional kinetics

Kinetic equations used to describe systems with dynamical chaos may contain fractional derivatives of an order alpha in space and beta in time in order to represent processes of stickiness, intermittency, and so on. We demonstrate for a simple example that the kinetics is anisotropic not only in the angular dependence of the diffusion constant, but also in the angular dependence of the exponents alpha and beta. A theory of such kinetic processes has been developed on the basis of integral representation and asymptotic solutions for different cases have been obtained. The results show the existence of self-similar solutions as well as possible logarithmic deviations. (c) 2001 American Institute of Physics.     

Exponentially self-similar impact ionization waves

The existence of generalized self-similar solutions to the system of continuity and Poisson equations is analyzed for the problem of evolution of impact ionization waves (IIWs). It is shown that, for any physically reasonable electric-field dependence of the impact ionization coefficients, there exist only exponentially self-similar (“limiting”) asymptotic solutions. These solutions describe IIWs whose spatial scales and propagation velocities increase exponentially with time. Conditions are found for the existence of plane, cylindrical, and spherical waves of this type; their structure is described; analytical relations between the key parameters are derived; and effects of recombination (or attachment) and tunnel ionization are analyzed. It is shown that these IIWs are intermediate asymptotics of numerical solutions to the corresponding Cauchy problems. The most important and interesting type of exponentially self-similar IIWs are streamers in a uniform electric field. The simplest comprehensive and explicit model describing their evolution is a spherical IIW.     

Interaction of Four Rarefaction Waves in the Bi-Symmetric Class of the Two-Dimensional Euler Equations

The global existence and structures of solutions to multi-dimensional unsteady compressible Euler equations are interesting and important open problems. In this paper, we construct global classical solutions to the interaction of four orthogonal planar rarefaction waves with two axes of symmetry for the Euler equations in two space dimensions, in the case where the initial rarefaction waves are large. The bi-symmetric initial data is a basic type of four-wave two-dimensional Riemann problems. The solutions in this case are continuous, bounded and self-similar, and we characterize how large the rarefaction waves must be. We use the methods of hodograph transformation, characteristic decomposition, and phase space analysis. We resolve binary interactions of simple waves in the process.     

Construction of non-Gaussian self-similar random fields with hierarchical structure

In the present work we construct non-Gaussian self-similar random fields with hierarchical structure. The construction is based on non-Gaussian solutions of the main nonlinear equation of the hierarchical models theory. The existence of such solutions was proved originally by Sinai and the author and later by another method by Collet and Eckmann. Next we establish the uniqueness of a Gibbs state for the constructed self-similar field. Finally for a class of hierarchical models we prove the convergence of renormalization transformations of a random field at the critical point to the self-similar field.