Qualitative properties of pulsating fronts for reaction–advection–diffusion equations in periodic excitable media

In this paper, we study the pulsating fronts of reaction–advection-diffusion equations with two types of nonlinear term in periodic excitable media. Firstly, for the case with combustion nonlinearity, the unique front is proved to decay exponentially when it approaches the unstable limiting state. Secondly, for the degenerate monostable type nonlinearity, it is shown that the front with critical speed is unique, monotone and decays exponentially at negative end, while the fronts of noncritical speeds decay to zero non-exponentially.     

Factors affecting propagating fronts of addition polymerization: Velocity,front curvature,temperatue profile,conversion, and molecular weight distribution

Several properties of propagating fronts of addition polymerization were studied. A power function could be fit to the velocity dependence on initiator concentration, but not with the exponents predicted by current models or in agreement with other published work. Bubbles from the volatile by-products of initiator decomposition were found to affect the front velocity and curvature. The front velocity for triethylene glycol dimethacrylate polymerization was found to depend linearly on temperature over a moderate range. The conversion of methacrylic acid in fronts varied greatly with initiator type and concentration. Benzoyl peroxide produced much lower conversion than t-butyl peroxide, but fronts with tBPO propagated slower. A dual initiator system of BPO and tBPO produced rapidly propagating fronts with good conversion but the contribution of each initiator to the velocity was not additive. The possibility of chain branching was considered. The apparent molecular weight distributions were very broad, often trimodal, and found to depend on initiator type and concentration as well as the tube diameter. The temperature profiles were measured and found to be very sharp for BPO and broader for tBPO but both had front temperatures in excess of 200°C, indicating a high ceiling temperature. © 1995 John Wiley & Sons, Inc.     

An ignition-temperature model with two free interfaces in premixed flames†

In this paper we consider an ignition-temperature zero-order reaction model of thermo-diffusive combustion. This model describes the dynamics of thick flames, which have recently received considerable attention in the physical and engineering literature. The model admits a unique (up to translations) planar travelling wave solution. This travelling wave solution is quite different from those usually studied in combustion theory. The main qualitative feature of this travelling wave is that it has two interfaces: the ignition interface where the ignition temperature is attained and the trailing interface where the concentration of deficient reactants reaches zero. We give a new mathematical framework for studying the cellular instability of such travelling front solutions. Our approach allows the analysis of a free boundary problem to be converted into the analysis of a boundary value problem having a fully nonlinear system of parabolic equations. The latter is very suitable for both mathematical and numerical analysis. We prove the existence of a critical Lewis number such that the travelling wave solution is stable for values of Lewis number below the critical one and is unstable for Lewis numbers that exceed this critical value. Finally, we discuss the results of numerical simulations of a fully nonlinear system that describes the perturbation dynamics of planar fronts. These simulations reveal, in particular, some very interesting ‘two-cell’ steady patterns of curved combustion fronts.     

Algebraic Rate of Decay for the Excess Free Energy and Stability of Fronts for a Nonlocal Phase Kinetics Equation with a Conservation Law. I

This is the first of two papers devoted to the study of a nonlocal evolution equation that describes the evolution of the local magnetization in a continuum limit of an Ising spin system with Kawasaki dynamics and Kac potentials. We consider subcritical temperatures, for which there are two local equilibria, and begin the proof of a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface between regions of these two different local equilibria; i.e., the fronts. We shall show in the second paper that an initial perturbation v ₀ of a front that is sufficiently small in L ² norm, and sufficiently localized that x ² v ₀(x)² dx<, yields a solution that relaxes to another front, selected by a conservation law, in the L ¹ norm at an algebraic rate that we explicitly estimate. There we also obtain rates for the relaxation in the L ² norm and the rate of decrease of the excess free energy. Here we prove a number of estimates essential for this result. Moreover, the estimates proved here suffice to establish the main result in an important special case.on leave from     

Computing particle velocity and intensity in diffracting systems

In a previous paper the authors have shown how it is possible to compute the acoustic pressure due to an axi-symmetric radiator with different boundary conditions. The technique involves the conversion of a double integral to a single integral by means of a geometric transform, and may be applied with the Green's function appropriate to each boundary condition.The present work shows how the form of the single integrals makes it possible to derive and compute both the particle velocity and the wave intensity. It has become apparent that the particle velocity must be described as an elliptic vector. A new concept, the ‘diffraction front’, has been defined as a surface in which there is no net energy transfer, and this has been applied to the evaluation of the magnitude and direction of energy flow. Plots have been made of flow lines and diffraction fronts, and also three-dimensional representations of energy intensity have been made for those acoustic fields in which pressure has already been evaluated.     

Transition from pushed-to-pulled fronts in piecewise linear reaction–diffusion systems

The front dynamics in reaction–diffusion equations with a piecewise linear reaction term is studied. A transition from pushed-to-pulled fronts when they propagate into the unstable state is found using a variational principle. This transition occurs for a critical value of the discontinuity position in the reaction function. In particular, we study how the transition depends on the properties of the reaction term and on the delay time. Our results are in good agreement with the numerical solutions of the model.     

Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity

This paper is concerned with the existence of traveling wave fronts for delayed non-local diffusion systems without quasimonotonicity, which can not be answered by the known results. By using exponential order, upper-lower solutions and Schauder's fixed point theorem, we reduce the existence of monotone traveling wave fronts to the existence of upper-lower solutions without the requirement of monotonicity. To illustrate our results, we establish the existence of traveling wave fronts for two examples which are the delayed non-local diffusion version of the Nicholson's blowflies equation and the Belousov-Zhabotinskii model. These results imply that the traveling wave fronts of the delayed non-local diffusion systems without quasimonotonicity are persistent if the delay is small.     

Perturbation Analysis for Wetting Fronts in Richard's Equation

Perturbation methods are used to study the interaction of wetting fronts with impervious boundaries in layered soils. Solutions of Richards' equation for horizontal and vertical infiltration problems are considered. Asymptotically accurate solutions are constructed from outer solutions and boundary-layer corrections. Results are compared with numerical simulations to demonstrate a high level of accuracy.