Nonlinear reaction-diffusion: a microscopic approach

A microscopic theory for reaction-difusion (R-D) processes is developed from Einstein’s master equation including a reactive term. The mean field formulation leads to a generalized R-D equation for the n-th order annihilation reaction A + A + A + ... + A → 0, and the steady state solutions exhibit long range power law behavior showing the relative dominance of sub-diffusion over reaction effects in constrained systems, or conversely short range concentration distribution with finite support describing situations where diffusion is slow and extinction is fast. We apply the theory to analyze experimental data for morphogen gradient formation in the wing disc of the Drosophila embryo.     

Stability analysis of fronts in a tristable reaction-diffusion system

A stability analysis is performed analytically for the tristable reaction-diffusion equation, in which a quintic reaction term is approximated by a piecewise linear function. We obtain growth rate equations for two basic types of propagating fronts, monotonous and nonmonotonous ones. Their solutions show that the monotonous front is stable whereas the nonmonotonous one is unstable. It is found that there are two values of the growth rate for the most dangerous modes (corresponding to the longest possible wavelengths), and , for the monotonous front, so that at the perturbation eigenfunction is positive whereas when it changes sign. It is also noted that the eigenvalue becomes negative in an inhomogeneous system with a particular (stabilizing) inhomogeneity. Counting arguments for the number of eigenmodes of the linear stability operator are presented.Received: 9 August 2004, Published online: 23 December 2004PACS: 05.45.-a Nonlinear dynamics and nonlinear dynamical systems - 47.20.Ma Interfacial instability - 47.54. + r Pattern selection; pattern formation     

Systematic analysis of the multivariate master equation for a reaction-diffusion system

The multivariate master equation for a reaction-diffusion system is analyzed using a singular perturbation approach. It is shown that in the vicinity of a bifurcation leading to two simultaneously stable steady states, the steady-state probability distribution reduces asymptotically to the exponential of the Landau-Ginzburg functional. On the other hand, for a system displaying quadratic nonlinearities and an absorbing state, critical behavior is ruled out.Supported in part by the Actions de Recherche Concertées of the Belgian government under convention no. 76/81 II 3.     

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.     

Nonlinear distribution function for a plasma obeying Vlasov-Maxwell system of equations

The nonlinear distribution function of Allis, generalised to include the transverse electromagnetic waves in a plasma, is used to set up the coupled wave equations for the longitudinal and the transverse modes. These are solved, keeping terms up to the cubic order of nonlinearity, by using the method of multiple scales. The equations of wave modulation are derived, which are solved to discuss the nature of the modulational instability and solitary wave propagation. It is found that the solutions so obtained satisfy conditions which are very similar to the well known Lighthill criterion for stability, appropriately modified due to the coupling of the two modes. The role of the average constant current due to any flow of the resonant and trapped electrons in determining the stability, is also discussed.     

反应扩散系统中时空斑图研究

时空斑图广泛地存在于反应扩散系统中,在延展的布鲁塞尔振子模型中,一维的时空斑图已经被研究过.本文中,我们对布鲁塞尔振子模型进行线性稳定性分析,模拟出两维的时空斑图,进一步阐明斑图形成的机制,形成斑图的机制是由于霍普夫失稳、短波失稳和图灵失稳以及它们之间的相互作用.当系统处于非平衡状态下,布鲁塞尔振子模型可以得到有序的时空斑图.?     

Resonant phase patterns in a reaction-diffusion system

Resonance regions similar to the Arnol'd tongues found in single oscillator frequency locking are observed in experiments using a spatially extended periodically forced Belousov-Zhabotinsky system. We identify six distinct 2:1 subharmonic resonant patterns and describe them in terms of the position-dependent phase and magnitude of the oscillations. Some experimentally observed features are also found in numerical studies of a forced Brusselator reaction-diffusion model.     

Phase transition in a three-states reaction-diffusion system

A one-dimensional reaction-diffusion model consisting of two species of particles and vacancies on a ring is introduced. The number of particles in one species is conserved while in the other species it can fluctuate because of creation and annihilation of particles. It has been shown that the model undergoes a continuous phase transition from a phase where the currents of different species of particles are equal to another phase in which they are different. The total density of particles and also their currents in each phase are calculated exactly.     

Symmetries of non-linear reaction-diffusion equations and their solutions

This paper is concerned with the symmetries of a certain class of non-linear reaction-diffusion equations. The symmetries are used for deriving solutions of these equations. Subsequently, we compare the solutions with those given by other authors.     

Amplitude equations for non-linear Rayleigh waves

We investigate asymptotic equations describing small amplitude surface elastic waves in the half-plane (Rayleigh waves). For hyperelastic materials such model equations are Hamiltonian systems, and are seen to lead to the formation of singularities in the surface elastic displacement. At the time of singularity formation the Fourier spectra of the solutions exhibit power law decay, and the observed exponents suggest the existence of both differentiable and non-differentiable singular profiles.     

Dissipative structures in a soluble non-linear reaction-diffusion system

The steady state solutions of a non-linear reaction diffusion system are evaluated exactly. This bifurcation diagram as well as their stability is discussed.     

Superlattice Turing structures in a photosensitive reaction-diffusion system

Families of complex superlattice structures, consisting of combinations of basic hexagonal or square patterns, are found in a photosensitive reaction-diffusion system. The structures are induced by simple illumination patterns whose wavelengths are appropriately related to that of the system's intrinsic Turing pattern. Computer simulations agree with the structures and their stability. The technique offers a general approach to generating superlattices for use in information storage and other applications.     

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.