Packet waves in a reaction-diffusion system

The finite-wavelength instability gives rise to a new type of wave in reaction-diffusion systems: packet waves, which propagate only within a wave packet, are found in experiments on the Belousov-Zhabotinsky reaction dispersed in water-in-oil AOT microemulsion (BZ-AOT) as well as in model simulations. Inwardly moving packet waves with negative curvature occur in experiments and in a model of the BZ-AOT system when the dispersion d omega(k)/dk is negative at the characteristic wave number k(0). This result sheds light on the origin of anti-spirals.     

Dash waves in a reaction-diffusion system

Patterns in reaction-diffusion systems generally consist of smooth traveling waves or of stationary, discontinuous Turing structures. Hybrid patterns that blend the properties of waves and Turing structures have not previously been observed. We report observation of dash waves, which consist of wave segments regularly separated by gaps, moving coherently in the Belousov-Zhabotinsky system dispersed in water-in-oil microemulsion. Dash waves emerge from the interaction between excitable and pseudo-Turing-unstable steady states. We are able to generate dash waves in simulations with simple models.     

Theory of spike spiral waves in a reaction-diffusion system

We discovered a type of spiral wave solutions in reaction-diffusion systems--spike spiral wave, which significantly differs from the spiral waves observed in the models of FitzHugh-Nagumo type. We present an asymptotic theory of these waves in the Gray-Scott model [Chem. Sci. Eng. 38, 29 (1983)]. We derive the kinematic relations describing the shape of this spiral, and find the dependence of its main parameters on the control parameters. The theory does not rely on the specific features of the Gray-Scott model and thus is expected to be applicable to a broad range of reaction-diffusion systems.     

Existence and stability of propagating fronts for an autocatalytic reaction-diffusion system

We study a one-dimensional reaction-diffusion system which describes an isothermal autocatalytic chemical reaction involving both a quadratic (A + B → 2B) and a cubic (A + 2B → 3B) autocatalysis. The parameters of this system are in the ratio D = D_B/D_A of the diffusion constants of the reactant A and the autocatalyst B, and the relative activity k of the cubic reaction. First, for all values of D > 0 and k ≥ 0, we prove the existence of a family of propagating fronts (or travelling waves) describing the advance of the reaction. In particular, in the quadratic case k = 0, we recover the results of Billingham and Needham [Phil. Trans. R. Soc. London A 334 (1991) 1–24]. Then, if D is close to 1 and k is sufficiently small, we prove using energy functionals that these propagating fronts are stable against small perturbations in exponentially weighted Sobolev spaces. This extends part of the results that are known for the scalar equation to which our system reduces when D = 1.     

Irregular wakes in reaction-diffusion waves

Reaction-diffusion equations have proved to be highly successful models for a wide range of biological and chemical systems, but chaotic solutions have been very rarely documented. We present a new mechanism for generating apparently chaotic spatiotemporal irregularity in such systems, by analysing in detail the bifurcation structure of a particular set of reaction-diffusion equations on an infinite one-dimensional domain, with particular initial conditions. We show that possible solutions include travelling fronts which leave behind either regular or irregular spatiotemporal oscillations. Using a combination of analytical and numerical analysis, we show that the irregular behaviour arises from the instability of oscillations induced by the passage of the front. Finally, we discuss the generality of this mechanism as a way in which spatiotemporal irregularities can arise naturally in reaction-diffusion systems.     

Statistics of waves propagating in a random medium

We present a review of part of the results obtained by the authors for the statistics of coherent radiation propagating in a random medium both in the framework of diagrammatic techniques and random matrix theory. Distribution functions for the total transmission coefficient and the angular transmission coefficient for the diffusive transport and the crossover between the diffusive and ballistic regimes are obtained.The authors acknowledge the financial support of the Israeli Academy of Sciences and of the Schottenstein Center.     

Eigenmodes for electromagnetic waves propagating in a toroidalcavity

A solution has been attempted by means of the Helmholtz equation for an electromagnetic wave propagating in an empty torus in a system of toroidal coordinates. The electromagnetic fields are expressed in terms of the Hertz vector to obtain a scalar Helmholtz equation. The latter has been solved by making use of an inverse aspect ratio expansion of the solution. Unlike most previous workers, the authors have obtained their solutions in terms of hypergeometric functions whose static limit is the toroidal harmonics. The cylindrical solutions in terms of Bessel functions can also be recovered by taking the appropriate large aspect ratio limit. The eigenmodes, with arbitrary toroidal and poloidal mode numbers, have been obtained by applying the boundary conditions on the metallic walls of infinite conductivity, and they cannot be distinguished as TE or TM modes. Eigenfrequencies for various toroidal and poloidal mode numbers are plotted against the inverse aspect ratio. First-order approximations to the fields in the toroidal cavity have also been derived     

Frequency-locking phenomena of propagating wave fronts in reaction-diffusion systems

We observe N:(N-1)(N>/=2) frequency-locking phenomena of propagating wave fronts when increasing the light intensity in a spatially extended system. The experiments were carried out using the light-sensitive form of the Belousov-Zhabotinsky reaction with Ru(bpy)(2+)3 as a catalyst. By constructing a mapping function, the characteristic devil's staircase can be reproduced when plotting wave period versus light intensity, in agreement with the experimental data.     

Inwards propagating waves in a limit cycle medium

The existence of a novel inwards propagating wave motion is demonstrated in a limit-cycle medium both for the FitzHugh-Nagumo and for modified Chernyak-Starobin-Cohen reaction-diffusion systems. The waves (pulses) are seen to be moving "backwards," that is, towards the point where the triggering pulse was initiated, instead of the regular propagation away from the origin. The feasibility of the phenomenon and some of its features are analyzed.     

Tunneling of evanescent waves into propagating waves

The tunneling of evanescent waves into propagating waves is related to the convolution of the high spatial frequencies of the source with those of the detectors. Such an approach is demonstrated by treating the evanescent waves which are diffracted from very narrow apertures in a plane screen (with dimensions much smaller than the wavelength) and are converted to propagating waves by tip detectors. The mechanism responsible for the conversion of evanescent waves into propagating waves is explained and a general formula for the conversion of evanescent waves into propagating waves is derived. PACS 42.25.Fx; 42.30.Kq; 42.25.Bs     

Unstable trigger waves induce various intricate dynamic regimes in a reaction-diffusion system of blood clotting

In this work we demonstrate that the unstable trigger waves, connecting stable and unstable spatially uniform steady states, can create intricate dynamic regimes in one-dimensional three-component reaction-diffusion model describing blood clotting. Among the most interesting regimes are the composite and replicating waves running at a constant velocity. The front part of the running composite wave remains constant, while its rear part oscillates in a complex manner. The rear part of the running replicating wave periodically gives rise to new daughter waves, which propagate in the direction opposite the parent wave. The domain of these intricate regimes in parameter space lies in the region of monostability near the region of bistability.     

Manipulation of self-aggregation patterns and waves in a reaction-diffusion system by optimal boundary control strategies

Reaction-diffusion systems are of considerable importance in many areas of physical sciences. For many reasons, an external manipulation of the dynamics of these processes is desirable. Here we show numerically how spatiotemporal behavior like pattern formation and wave propagation in a two component nonlinear reaction-diffusion model of bacterial chemotaxis can be externally controlled. We formulate the control goal as an objective functional and apply numerical optimization for the solution of the resulting control problem.     

圆管结构中的非线性周向导波*

采用将二阶微扰近似与模式展开分析相结合的方法,可从理论上有效求解圆管结构中的非线性周向导波问题。通过数值计算和有限元仿真发现,基频与二倍频周向导波模式的相速度匹配程度,可显著影响二倍频周向导波模式随传播周向角的积累增长程度。针对基频与二倍频周向导波模式的相速度和群速度均相匹配的情形,通过实验研究发现,周向导波确可存在强烈的非线性效应,且周向导波的二次谐波发生效应可对圆管早期损伤状态做出敏感的响应。文中给出的有关结果,可为进一步开展非线性周向导波的研究工作奠定理论和实验基础。     

Chemical waves as a result of instability in reaction-diffusion systems

Many-component reaction-diffusion systems are shown to be able to undergo the instability, leading to the spontaneous formation of waves. In the one-dimensional case the general equations, governing the dynamics in the neighbourhood of the wave-type instability point, are derived. The investigation of all steady state solutions of these equations shows that depending on the magnitude of only one essential parameter stable are either a uniform running wave and “leading center” regime or a uniform standing wave. All other solutions are unstable.     

Collective behavior of stabilized reaction-diffusion waves

Stabilized wave segments in the photosensitive Belousov-Zhabotinsky reaction are directionally controlled with intensity gradients in the applied illumination. The constant-velocity waves behave like self-propelled particles, and multiple waves interact via an applied interaction potential. Alignment arises from the intrinsic properties of the interacting waves, leading to processional and rotational behavior.