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.     

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.     

Pseudo-spectral methods and linear instabilities in reaction-diffusion fronts

We explore the application of a pseudo-spectral Fourier method to a set of reaction-diffusion equations and compare it with a second-order finite difference method. The prototype cubic autocatalytic reaction-diffusion model as discussed by Gray and Scott [Chem. Eng. Sci. 42, 307 (1987)] with a nonequilibrium constraint is adopted. In a spatial resolution study we find that the phase speeds of one-dimensional finite amplitude waves converge more rapidly for the spectral method than for the finite difference method. Furthermore, in two dimensions the symmetry preserving properties of the spectral method are shown to be superior to those of the finite difference method. In studies of plane/axisymmetric nonlinear waves a symmetry breaking linear instability is shown to occur and is one possible route for the formation of patterns from infinitesimal perturbations to finite amplitude waves in this set of reaction-diffusion equations. (c) 1996 American Institute of Physics.     

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.     

Kinematics of rigidly rotating spiral waves

Spiral waves rigidly rotating in excitable media are studied by use of a free-boundary approach. This study reveals the selection principle which determines the shape and the rotation frequency of spiral waves in an unbounded medium with a given excitability. It is shown that a rigidly rotating spiral in a medium with a strongly reduced refractoriness is supported within a range of the medium excitability restricted by two universal limits. At the low excitability limit the spiral core radius diverges, while at the high excitability limit it vanishes. The simulations performed for the medium excitability higher than the high excitability limit reveal nonstationary rotating waves, which considerably differ from well-studied meandering spiral waves. It is shown how the proposed free-boundary approach can be extended to the case of an arbitrary refractoriness. The predictions of the free-boundary approach are in good agreement with the results from numerical simulations of the underlying reaction-diffusion model and with asymptotics derived earlier for highly and weakly excitable media.     

Quasisoliton interaction of pursuit-evasion waves in a predator-prey system

We consider a system of partial differential equations describing two spatially distributed populations in a "predator-prey" interaction with each other. The spatial evolution is governed by three processes: positive taxis of predators up the gradient of prey (pursuit), negative taxis of prey down the gradient of predators (evasion), and diffusion resulting from random motion of both species. We demonstrate a new type of propagating wave in this system. The mechanism of propagation of these waves essentially depends on the taxis and is entirely different from waves in a reaction-diffusion system. Unlike typical reaction-diffusion waves, which annihilate on collision, these "taxis" waves can often penetrate through each other and reflect from impermeable boundaries.     

Nonlinear waves in nonequilibrium systems of the oscillatory type,part I

Nonlinear waves in mathematical models of nonequilibrium spatially uniform media with the oscillatory instability of the trivial state are considered. The models are based on the generalized Ginsburg-Landau equations. For the long-wave system, i.e. that described by two-component reaction-diffusion equations, we obtain the full stability conditions for monochromatic plane travelling waves. The basic part of the paper is devoted to the short-wave system which can be described by reaction-diffusion equations with not less than three components or by a two-component system with residual nonlocality. We construct the Ginsburg-Landau equation for this system, and we find its general quasistationary one-dimensional solution which is a travelling wave modulated by a travelling envelope wave. The stability of this solution is investigated with the especial emphasis on different important particular cases. The obtained results are compared with experimental observations of different waves on fronts of detonation and non-gaseous combustion (which also are characterized by the oscillatory short-wave instability of the trivial state), and the qualitative agreement between theoretical and experimental results is demonstrated.     

Model for spatial microtubule oscillations

Under particular in vitro conditions, oscillating spatial and temporal waves of assembled microtubules can be observed. A reaction-diffusion model is presented to reproduce these results. This model is based on a set of chemical reaction equations and extended to include spatial dependence and diffusion. The basic properties of the model are presented and the results are demonstrated to connect the observable waves with turbidimetric measurements. The results of the model are consistent with experimental findings.     

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.     

Propagation of fronts in activator-inhibitor systems with a cutoff

We consider a two-component system of reaction-diffusion equations with a small cutoff in the reaction term. A semi-analytical solution of fronts and how the front velocities vary with the parameters are given for the case when the system has a piecewise linear nonlinearity. We find the existence of a nonequilibrium Ising-Bloch bifurcation for the front speed when the cutoff is present. Numerical results of solutions to these equations are also presented and they allow us to consider the collision between fronts, and the existence of different types of traveling waves emerging from random initial conditions.     

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.     

Isotropic diffraction of a light beam by acoustic waves of fundamental frequency and harmonics

A set of equations describing acoustooptic diffraction of a light beam by acoustic waves of a fundamental frequency and its harmonics in an isotropic medium is obtained. The possibility of suppressing higher diffraction orders by adding the second or third harmonic to the fundamental monochromatic acoustic signal is theoretically justified. It is demonstrated that the maximum degree of suppression decreases with an increase in the light beam divergence. Results of simulation are presented for some particular cases of diffraction.     

Rotating spiral waves in excitable media: The analytical results

In this paper we calculate the rotation frequency of spiral waves as a function of the parameters of the excitable medium. We give the complete analytical solution in the special case of the Rinzel-Keller model and suggest an analytical procedure for general two-component relaxational reaction-diffusion models. This procedure is based on the Greenberg equation and stability analysis; it is applicable when the core of a spiral wave is large as compared with the characteristic diffusion length. Construction of spiral wave solutions for the waves inside ring channels, circular regions, or around holes is discussed. Multi-armed spiral waves are investigated.     

Computing the Maslov index of solitary waves, Part 1: Hamiltonian systems on a four-dimensional phase space

When solitary waves are characterized as homoclinic orbits of a finite-dimensional Hamiltonian system, they have an integer-valued topological invariant, the Maslov index. We develop a new robust numerical algorithm to compute the Maslov index, to understand its properties, and to study the implications for the stability of solitary waves. The algorithm reported here is developed in the exterior algebra representation, which leads to a fast algorithm with some novel properties. New results on the Maslov index for solitary wave solutions of reaction-diffusion equations, the fifth-order Korteweg–de Vries equation, and the long-wave–short-wave resonance equations are presented. Part 1 considers the case of a four-dimensional phase space, and Part 2 considers the case of a 2n-dimensional phase space with n>2.     

Regular wave propagation out of noise in chemical active media

A pacemaker, regularly emitting chemical waves, is created out of noise when an excitable photosensitive Belousov-Zhabotinsky medium, strictly unable to autonomously initiate autowaves, is forced with a spatiotemporal patterned random illumination. These experimental observations are also reproduced numerically by using a set of reaction-diffusion equations for an activator-inhibitor model, and further analytically interpreted in terms of genuine coupling effects arising from parametric fluctuations. Within the same framework we also address situations of noise-sustained propagation in subexcitable media.     

Evidence of negative-index refraction in nonlinear chemical waves

The negative index of refraction of nonlinear chemical waves has become a recent focus in nonlinear dynamics researches. Theoretical analysis and computer simulations have predicted that the negative index of refraction can occur on the interface between antiwaves and normal waves in a reaction-diffusion (RD) system. However, no experimental evidence has been found so far. In this Letter, we report our experimental design in searching for such a phenomenon in a chlorite-iodide-malonic acid (CIMA) reaction. Our experimental results demonstrate that competition between waves and antiwaves at their interface determines the fate of the wave interaction. The negative index of refraction was only observed when the oscillation frequency of a normal wave is significantly smaller than that of the antiwave. All experimental results were supported by simulations using the Lengyel-Epstein RD model which describes the CIMA reaction-diffusion system.     

Simulating biochemical networks at the particle level and in time and space: Green's function reaction dynamics

We present a technique, called Green's function reaction dynamics (GFRD), for particle-based simulations of reaction-diffusion systems. GFRD uses a maximum time step such that only single particles or pairs of particles have to be considered. For these particles, the Smoluchowski equations are solved analytically using Green's functions, which are used to set up an event-driven algorithm. We apply the technique to a model of gene expression. Under biologically relevant conditions, GFRD is up to 5 orders of magnitude faster than conventional particle-based schemes.     

Doppler effect of nonlinear waves and superspirals in oscillatory media

Nonlinear waves emitted from a moving source are studied. A meandering spiral in a reaction-diffusion medium provides an example in which waves originate from a source exhibiting a back-and-forth movement in a radial direction. The periodic motion of the source induces a Doppler effect that causes a modulation in wavelength and amplitude of the waves ("superspiral"). Using direct simulations as well as numerical nonlinear analysis within the complex Ginzburg-Landau equation, we show that waves subject to a convective Eckhaus instability can exhibit monotonic growth or decay as well as saturation of these modulations depending on the perturbation frequency. Our findings elucidate recent experimental observations concerning superspirals and their decay to spatiotemporal chaos.     

Effect of Bacterial Memory Dependent Growth by Using Fractional Derivatives Reaction-Diffusion Chemotactic Model

In this paper, numerical solutions of a reaction-diffusion chemotactic model of fractional orders for bacterial growth will be present. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. We compare the experimental result obtained with those obtained by simulation of the chemotactic model without fractional derivatives. The results show that the solution continuously depends on the time-fractional derivative. The resulting solutions spread faster than the classical solutions and may exhibit asymmetry, depending on the fractional derivative used. We present results of numerical simulations to illustrate the method, and investigate properties of numerical solutions. The Adomian’s decomposition method (ADM) is used to find the approximate solution of fractional ‘reaction-diffusion chemotactic model. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional order.