Front Solutions for Bistable Differential-Difference Equations with Inhomogeneous Diffusion

This is the published version, also available here: http://dx.doi.org/10.1137/100807156.We consider a bistable differential-difference equation with inhomogeneous diffusion. Employing a piecewise linear nonlinearity, often referred to as McKean's caricature of the cubic, we construct front solutions which correspond, in the case of homogeneous diffusion, to monotone traveling front solutions or, in the case of propagation failure, to stationary front solutions. A general form for these fronts is given for essentially arbitrary inhomogeneous discrete diffusion, and conditions are given for the existence of solutions to the original discrete Nagumo equation. The specific case of one defect is considered in depth, giving a complete understanding of propagation failure and a grasp on changes in wave speed. Insight into the dynamic behavior of these front solutions as a function of the magnitude and relative position of the defects is obtained with the assistance of numerical results

Front Solutions for Bistable Differential-Difference Equations with Inhomogeneous Diffusion

We consider a bistable differential-difference equation with inhomogeneous diffusion. Employing a piecewise linear nonlinearity, often referred to as McKean\u27s caricature of the cubic, we construct front solutions which correspond, in the case of homogeneous diffusion, to monotone traveling front solutions or, in the case of propagation failure, to stationary front solutions. A general form for these fronts is given for essentially arbitrary inhomogeneous discrete diffusion, and conditions are given for the existence of solutions to the original discrete Nagumo equation. The specific case of one defect is considered in depth, giving a complete understanding of propagation failure and a grasp on changes in wave speed. Insight into the dynamic behavior of these front solutions as a function of the magnitude and relative position of the defects is obtained with the assistance of numerical results

Breakdown of the standard Perturbation Theory and Moving Boundary Approximation for "Pulled" Fronts

The derivation of a Moving Boundary Approximation or of the response of a coherent structure like a front, vortex or pulse to external forces and noise, is generally valid under two conditions: the existence of a separation of time scales of the dynamics on the inner and outer scale and the existence and convergence of solvability type integrals. We point out that these conditions are not satisfied for pulled fronts propagating into an unstable state: their relaxation on the inner scale is power law like and in conjunction with this, solvability integrals diverge. The physical origin of this is traced to the fact that the important dynamics of pulled fronts occurs in the leading edge of the front rather than in the nonlinear internal front region itself. As recent work on the relaxation and stochastic behavior of pulled fronts suggests, when such fronts are coupled to other fields or to noise, the dynamical behavior is often qualitatively different from the standard case in which fronts between two (meta)stable states or pushed fronts propagating into an unstable state are considered.Comment: pages Latex, submitted to a special issue of Phys. Rep. in dec. 9

Breathing Current Domains in Globally Coupled Electrochemical Systems: A Comparison with a Semiconductor Model

Spatio-temporal bifurcations and complex dynamics in globally coupled intrinsically bistable electrochemical systems with an S-shaped current-voltage characteristic under galvanostatic control are studied theoretically on a one-dimensional domain. The results are compared with the dynamics and the bifurcation scenarios occurring in a closely related model which describes pattern formation in semiconductors. Under galvanostatic control both systems are unstable with respect to the formation of stationary large amplitude current domains. The current domains as well as the homogeneous steady state exhibit oscillatory instabilities for slow dynamics of the potential drop across the double layer, or across the semiconductor device, respectively. The interplay of the different instabilities leads to complex spatio-temporal behavior. We find breathing current domains and chaotic spatio-temporal dynamics in the electrochemical system. Comparing these findings with the results obtained earlier for the semiconductor system, we outline bifurcation scenarios leading to complex dynamics in globally coupled bistable systems with subcritical spatial bifurcations.Comment: 13 pages, 11 figures, 70 references, RevTex4 accepted by PRE http://pre.aps.or