Drifting solitary waves in a reaction-diffusion medium with differential advection. Academic Article uri icon

abstract

  • A distinct propagation of solitary waves in the presence of autocatalysis, diffusion, and symmetry-breaking differential advection, is being studied. These pulses emerge at lower reaction rates of the autocatalytic activator, i.e., when the advective flow overcomes the fast excitation and induces a fluid type “drifting” behavior, making the phenomenon unique to reaction-diffusion-advection class systems. Using the spatial dynamics analysis of a canonical model, we present the properties and the organization of such drifting pulses. The insights underly a general understanding of localized transport in simple reaction-diffusion-advection models and thus provide a background to potential chemical and biological applications. Solitary waves are prominent generic solutions to reaction-diffusion RD systems and basic to many applied science disciplines 1. In one physical space dimension, these spatially localized propagating pulses are qualitatively described by a fast excitation leading front from a rest state followed by a slow recovery rear front to the same uniform state 1. Thus, in isotropic RD media a single symmetric suprathreshold localized perturbation results in simultaneously counterpropagating pulses or wave trains 2. However, in chemical and biological media transport can be facilitated by both diffusion and advection, and thus excitation properties of solitary waves can be subjected to linear and nonlinear convective instabilities 3. Several experiments in a spatially quasi-one-dimensional BelousovZhabotinsky chemical reaction have shown along with numerical simulations that excitable pulses propagate against the advective flow 4,5 and can propagate bidirectionally after splitting due to the “antirefractory” phenomenon, triggered by the imposed electrical field 4 .I n the latter case, the up- and down-stream traveling pulses retain the standard RD property of propagation by a fast excitation in the leading front. Consequently, the theoretical foundations of traveling and/or propagation failure of solitary waves in differential reaction-diffusion-advection RDA media, inherited the intuition of RD systems 6. In this Rapid Communication we analyze an RDA model and demonstrate a distinct solitary wave phenomenon that cannot emerge in RD systems: under certain conditions solitary waves may drift; i.e., the slow recovery becomes a leading front see Fig. 1. We reveal the regions and the properties of such drifting pulses and show that the phenomenon underlies a competition between a local kinetics of the activator and a differential advection. Our methods include a bifurcation theory of coexisting spatial solutions linear analysis and numerical continuations coupled to temporal stability; all the results agree well with direct numerical integrations. Applicability to chemical and biological media is also discussed. We start with a canonical RDA model that incorporates local kinetics of activator vx, t and inhibitor ux, t type,

publication date

  • January 1, 2010