Depinning Transitions in Discrete Reaction-Diffusion Equations

We consider spatially discrete bistable reaction-diffusion equations that admit wave front solutions. Depending on the parameters involved, such wave fronts appear to be pinned or to glide at a certain speed. We study the transition of traveling waves to steady solutions near threshold and give conditions for front pinning (propagation failure). The critical parameter values are characterized at the depinning transition and an approximation for the front speed just beyond threshold is given. 1. Introduction. Spatially discrete systems describe physical reality in many different fields: atoms adsorbed on a periodic substrate [13], motion of dislocations in crystals [32], propagation of cracks in a brittle material [35], microscopic theories of friction between solid bodies [18], propagation of nerve impulses along myelinated fibers [23, 24], pulse propagation through cardiac cells [24], calcium release waves in living cells [6], sliding of charge density waves [19], superconductor Josephson array junctions [39], or weakly coupled semiconductor superlattices [3, 9]. No one really knows why, but spatially discrete systems of equations often times have smooth solutions of the form u n (t) = u(n − ct), which are monotone functions approaching two different constants as (n − ct) → ±∞. Existence of such wave front solutions has been proved for particular discrete systems having dissipative dynamics [40]. In the case of discrete systems with conservative dynamics, a wave front solution was explicitly constructed by Flach et al. [16]. A general proof of wave front existence for discrete conservative systems with bistable sources is lacking. A distinctive feature of spatially discrete reaction-diffusion systems (not shared by continuous ones) is the phenomenon of wave front pinning: for values of a control parameter in a certain interval, wave fronts joining two different constant states fail to propagate [24]. When the control parameter surpasses a threshold, the wave front depins and starts moving [23, 19, 32, 9]. The existence of such thresholds is thought to be an intrinsically discrete fact, which is lost in continuum aproximations. The characterization of propagation failure and front depinning in discrete systems is thus an important problem, which is not yet well understood despite the numerous inroads In this paper, we study front depinning for infinite one-dimensional nonlinear spatially discrete reaction-diffusion (RD) systems. When confronted with a spatially discrete RD system, a possible strategy is to approximate it by a continuous RD system. For generic nonlinearities, the width of the pinning interval is exponentially