Traveling periodic waves of chemical activity

It is shown that within the manifold of exact solutions a system of reaction-diffusion equations admits only travelling waves with planar symmetry. A derivation of the generic form of approximate (asymptotic) cylindrical and spiral travelling periodic wave solutions is given. If an exact solution homogeneous in space and periodic in time is admitted by the system of reaction-diffusion equations, then travelling periodic spiral waves are admissble as approximate solutions. This is the theoretical explanation for the travelling periodic waves of chemical activity observed in recent experiments.     

反应扩散模型在图灵斑图中的应用及数值模拟

反应扩散方程模型常被用于描述生物学中斑图的形成.从反应扩散模型出发,理论推导得到GiererMeinhardt模型的斑图形成机理,解释了非线性常微分方程系统的稳定常数平衡态在加入扩散项后会发生失稳并产生图灵斑图的过程.通过计算该模型,得到图灵斑图产生的参数条件.数值方法中采用一类有效的高精度数值格式,即在空间离散条件下采用Chebyshev谱配置方法,在时间离散条件下采用紧致隐积分因子方法.该方法结合了谱方法和紧致隐积分因子方法的优点,具有精度高、稳定性好、存储量小等优点.数值模拟表明,在其他条件一定的情况下,系统控制参数κ取不同值对于斑图的产生具有重要的影响,数值结果验证了理论结果.     

Nonlinear waves in a plasma cylinder

Nonlinear electron plasma waves in a cold plasma bounded by a cylindrical dielectric are investigated using the exact electron fluid equations. After the spatial wave structure is determined, a set of nonlinear equations governing the temporal behavior of the field quantities is solved. It is shown that finite amplitude waves and explosively unstable behavior can occur     

Elastic nonlinearity parameters of tetragonal crystals

The equations of motion for the propagation of finite amplitude elastic waves in crystals of tetragonal symmetry have been derived starting from the expression for the elastic strain energy. The equations have been solved for a finite amplitude sinusoidal wave propagating along the pure mode directions which are [100], [110] and [001] for the tetragonal group TI. The solutions corresponding to longitudinal wave propagation yield expressions for the amplitudes of the fundamental and generated second harmonic for these directions in terms of certain combinations of second and third order elastic constants of the medium. The results will aid the experimenter to determine these constants using ultrasonic harmonic generation technique.     

The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems

Many oscillatory biological systems show periodic travelling waves. These are often modelled using coupled reaction-diffusion equations. However, the effects of different movement rates (diffusion coefficients) of the interacting components on the predictions of these equations are largely unknown. Here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed, time period, wavelength, amplitude and stability of periodic wave solutions. We focus on two sets of kinetics that are commonly used in ecological applications: lambda-omega equations, which are the normal form of an oscillatory coupled reaction-diffusion system close to a supercritical Hopf bifurcation, and a standard predator-prey model. Our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one-parameter family of periodic travelling wave solutions. The position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients: in all cases, stability changes through an Eckhaus (‘sideband’) instability. These effects are always symmetrical in the two diffusion coefficients for the lambda-omega equations, but are asymmetric in the predator-prey equations, especially when the limit cycle of the kinetics is of large amplitude. In particular, there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator-prey scenario. Our results also show the existence of a one-parameter family of travelling waves, but not necessarily a Hopf bifurcation, for all values of the diffusion coefficients. Simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms, and our analysis of the travelling wave families assists in the understanding of these effects.     

反应扩散方程的差分解法——以行星冷却模型为例

反应扩散方程是一种非线性的抛物型方程,在多种物理模型中有广泛应用.本文考虑了一种类地行星冷却模型,通过对行星内部各层的热传导特性进行合理假设,构造出了一组反应扩散方程.根据实际物理情况,推导了该过程中特殊的边界条件及初始条件.然后基于有限差分法思想,利用计算机程序对该方程组进行数值求解,最终给出了行星内部温度分布随时间的变化情况.     

High-directional wave propagation in periodic loss modulated materials

Amplification/attenuation of light waves in artificial materials can become sensitive to the propagation direction by spatially modulating the gain/loss response of the medium on the wavelength scale. We give a numerical proof of the high anisotropy of the gain/loss in two dimensional periodic structures with square and rhombic lattice symmetry by solving the full set of Maxwell's equations using the finite difference time domain method. Anisotropy of amplification/attenuation leads to the narrowing of the angular spectrum of propagating radiation with wavevectors close to the edges of the first Brillouin Zone. The effect provides a novel and useful method to filter out high spatial harmonics from noisy beams.     

Symmetry in the problem on wave-flow modes of a thin viscous-fluid layer

It is shown that a suitable transformation in the transverse-coordinate shift makes the set of equations describing the flow of viscous fluid films in the long-wavelength approximation invariant with respect to the reflection transformation. When constructing the flow models with the Galerkin spectral method, this symmetry in the equation for the longitudinal velocity stimulates its representation with the help of a set of functions symmetric with respect to the transverse coordinate.     

圆孔非线性声阻抗三维时域仿真与特性分析

为研究有限振幅声波作用下圆孔的非线性声学特性,提出了基于三维时域计算流体动力学(CFD)仿真的圆孔非线性声阻抗提取方法,通过求解层流方程来模拟声信号在圆孔及上下游的传播,以及采用横向周期性边界条件来考虑高穿孔率时圆孔之间相互作用的影响。研究了不同幅值声波作用下孔径、厚度和穿孔率对声阻抗的影响规律,通过对质点振速幅值、频率和板厚等组成的无量纲参量进行非线性回归分析,得到了圆孔非线性声阻抗的拟合公式,并将其转换为可考虑多频声波影响的时域模型。最后结合声阻抗时域模型和有限差分方法计算了直通穿孔管消声器在小振幅和有限振幅声波作用下的传递损失,通过与实验测量结果的比较,验证了拟合公式的准确性和实用性。     

Rarefactive ion-acoustic electrostatic solitary structures in nonthermal plasmas

An investigation has been made of ion-acoustic solitary waves in an unmagnetized nonthermal plasma whose constituents are an inertial ion fluid and nonthermally distributed electrons. The properties of stationary solitary structures are briefly studied by the pseudo-potential approach, which is valid for arbitrary amplitude waves, and by the reductive perturbation method which is valid for small but finite amplitude limit. The time evolution of both compressive and rarefactive solitary waves, which are found to coexist in this nonthermal plasma model, is also examined by solving numerically the full set of fluid equations. The temporal behaviour of positive (compressive) solitary waves is found to be typical, i.e., the positive initial disturbance breaks up into a series of solitary waves with the largest in front. However, the behaviour of negative (rarefactive) solitary waves is quite different. These waves appear to be unstable and produce positive solitary waves at a later time. The relevancy of this investigation to observations in the magnetosphere of density depressions is briefly pointed out. Received 12 October 1999     

Nonequilateral hexagonal patterns

Rotational symmetry of pattern formation problems is the origin of a variety of patterns (rolls, squares, hexagons etc.) in convection and reaction-diffusion systems. Traditionally, only the patterns based on equilateral lattices in the Fourier space were considered. In the present paper, we develop an analysis of the patterns with slightly different lengths of the basic wave vectors. The analysis applies as well to systems with a broken rotational symmetry (convection in an inclined layer, etc.). We find, in the framework of the amplitude equations, existence and stability conditions for periodic nonequilateral patterns based on two and three wave vectors. In the latter case, special attention is paid to the case when the three amplitudes are coupled by the resonant interaction.     

Effects of axial magnetic field and nonextensivity on small amplitude dust acoustic solitary waves in dusty plasma

The nonlinear propagation of small amplitude dust‐acoustic (DA) solitary waves in magnetized dusty plasma consisting of negatively charged mobile dust fluid, and Boltzmann‐distributed electrons and ions with two distinct temperatures following a q‐nonextensive distribution are investigated. In this article, a number of nonlinear equations, namely, the Korteweg–de‐Vries (K‐dV) equations, have been derived by employing the reductive perturbation technique that is valid for a small but finite amplitude limit. The effects of nonextensivity of ions with two distinct temperatures and dust concentration on the amplitude and width of DA solitary waves are investigated theoretically. It is observed that both the nonextensive and low‐temperatures ions significantly modify the basic properties and polarities of DA solitary waves. It is shown that both positive and negative potential DA solitons occur in this case. The implications of these results to some astrophysical environments and space plasmas (e.g., stellar polytropes, peculiar velocity distributions of galaxies, and collisionless thermal plasma), and laboratory dusty plasma systems are briefly mentioned.     

Spectral broadening of lower hybrid waves due to ponderomotive effects

The nonlinear propagation of a lower hybrid (1.h.) disturbance in uniform and nonuniform plasmas is computed, consistently with the finite amplitude density modifications due to the ponderomotive forces originated by the radio frequency (r.f.) power itself. Strong spectral broadening is shown to occur due to the coupling with magnetized acoustic waves.     

Cylindrical electromagnetic waves in a nonlinear nondispersive medium: Exact solutions of the Maxwell equations

The excitation and propagation of cylindrical electromagnetic waves in a nonlinear nondispersive medium are analyzed. It is assumed that the medium lacks a center of symmetry and that the dependence of the electric displacement on the electric field can be approximated by an exponential function. For this case, a method for integrating the system of the Maxwell equations is proposed. Exact solutions to a set of nonlinear electromagnetic field equations are obtained by this method. It is shown that nonlinear effects described by these solutions can become apparent under experimental conditions.     

Diffraction of elastic waves by cracks: application to time-of-flight inspection

We obtain expressions for the angular variation of diffracted signal amplitude when infinite plane P or SV waves are incident on a semi-infinite crack. The importance of these calculations to non-destructive testing by the time-of-flight technique is discussed. We obtain the optimum angles to use for the case of a defect symmetrically placed with respect to transmitter and receiver. These angles are those for which the ratio of emitted signal amplitude to incident wave amplitude is a maximum.

We compare two methods of obtaining the exact solution for the diffraction of elastic waves at a semi-infinite crack. Both methods involve the solution of a set of integral equations, the equations being different in the two methods. In the far-field approximation the results from these methods are shown to be equivalent.     

Steady-state vectorial self-diffraction on a non-local photorefractive grating in a crystal of symmetry \overline{4}3m at symmetrical transmitting geometry

The steady-state two-wave interaction in a cubic crystal of the symmetry group 3m with the non-local photorefractive response in the absence of an external electric field is considered for the case of arbitrary interaction orientation with respect to the crystallographic coordinate system and for arbitrary intensities and polarization states of incident light waves. The self-diffraction problem is described on the basis of four coupled-wave equations in terms of the complex scalar amplitudes of components of the light waves with orthogonal linear polarization. The derived conservation laws are valid for the non-linear dependency of the photorefractive-grating amplitude on the modulation coefficient of the interference light pattern. It follows from these laws that the two non-unidirectional energy fluxes can form the total energy exchange between the two interacting light waves. A set of independent conservation laws allows us to decouple the coupled-wave equations and to obtain their analytical solution, at least, in the form of quadrature formulae. For example, such a solution is derived for the case of linearly polarized incident light waves and for the linearized dependency of the photorefractive-grating amplitude on the modulation coefficient. The explicit analytical expressions for the scalar amplitudes are obtained for the transversal electro-optic configuration of interaction. The possibility of polarization-state transformation of light waves without energy exchange between them is shown. Received: 30 July 2002 / Published online: 11 December 2002 RID="*" ID="*"Corresponding author. Fax: +7-3822/414321, E-mail: litvinov@ed.rk.tusur.ru     

Cookbook asymptotics for spiral and scroll waves in excitable media

Algebraic formulas predicting the frequencies and shapes of waves in a reaction-diffusion model of excitable media are presented in the form of four recipes. The formulas themselves are based on a detailed asymptotic analysis (published elsewhere) of the model equations at leading order and first order in the asymptotic parameter. The importance of the first order contribution is stressed throughout, beginning with a discussion of the Fife limit, Fife scaling, and Fife regime. Recipes are given for spiral waves and detailed comparisons are presented between the asymptotic predictions and the solutions of the full reaction-diffusion equations. Recipes for twisted scroll waves with straight filaments are given and again comparisons are shown. The connection between the asymptotic results and filament dynamics is discussed, and one of the previously unknown coefficients in the theory of filament dynamics is evaluated in terms of its asymptotic expansion. (c) 2002 American Institute of Physics.     

Theory of cavitons in complex plasmas

Nonlinear coupling between Langmuir waves with finite amplitude dispersive dust acoustic perturbations is considered. It is shown that the interaction is governed by a pair of coupled nonlinear differential equations. Numerical results reveal the formation of Langmuir envelope solitons composed of the dust density depression created by the ponderomotive force of bell-shaped Langmuir wave envelops. The associated ambipolar potential is positive. The present nonlinear theory should be able to account for the trapping of large amplitude Langmuir waves in finite amplitude dust density holes. This scenario may appear in Saturn's dense rings, and the Cassini spacecraft should be able to observe fully nonlinear cavitons, as presented herein. Furthermore, we propose that new electron-beam plasma experiments should be conducted to verify our theoretical prediction.     

Generation of difference-frequency radiation in GaAs in the field of a few-cycle laser pulse in the regime of weakly expressed chromatic dispersion

We derive a system of coupled modified Korteweg-de Vries equations in the approximation of unidirectional waves which describe the evolution of electric fields of mutually-orthogonal linearly polarized laser pulses with duration of several optical oscillations in a nonlinear crystal with quadratic nonlinearity in the regime of weakly expressed chromatic dispersion. An isotropic GaAs crystal is considered as a nonlinear crystal, and the direction of propagation of interacting pulses is assumed to coincide with the normal to the 〈110〉 plane of the crystal. The derived system of equations was solved numerically by the finite difference method. We obtained the dependence of the relative shift of the central wavelength of pumping pulse spectrum on the distance travelled. It is shown that the decrease in the crystal thickness down to values commensurate with the coherence length of difference-frequency radiation leads to an increase in conversion efficiency. In particular, it is shown that the efficiency of difference-frequency generation at 4.1 μm and 10 μm for the 30 fs duration of pumping at 1.98 μm and the 100 MV/m amplitude of the electric field is ?51 dB and ?14 dB, respectively.