Pulse Dynamics in a Three-Component System: Existence Analysis

Abstract

In this article, we analyze the three-component reaction-diffusion system originally developed by Schenk et al. (PRL 78:3781-3784, 1997). The system consists of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor (or recovery variable). It has been used by several authors as a prototype three-component system that generates rich pulse dynamics and interactions, and this richness is the main motivation for the analysis we present. We demonstrate the existence of stationary one-pulse and two-pulse solutions, and travelling one-pulse solutions, on the real line, and we determine the parameter regimes in which they exist. Also, for one-pulse solutions, we analyze various bifurcations, including the saddle-node bifurcation in which they are created, as well as the bifurcation from a stationary to a travelling pulse, which we show can be either subcritical or supercritical. For two-pulse solutions, we show that the third component is essential, since the reduced bistable two-component system does not support them. We also analyze the saddle-node bifurcation in which two-pulse solutions are created. The analytical method used to construct all of these pulse solutions is geometric singular perturbation theory, which allows us to show that these solutions lie in the transverse intersections of invariant manifolds in the phase space of the associated six-dimensional travelling wave system. Finally, as we illustrate with numerical simulations, these solutions form the backbone of the rich pulse dynamics this system exhibits, including pulse replication, pulse annihilation, breathing pulses, and pulse scattering, among others.

Pinned fronts in heterogeneous media of jump type

“…Moreover, it also coincides with the FRS of the equivalent homogeneous problem. Therefore, the leading order results of the homogeneous case apply here, [5]. The FRS system is a conservative system with Hamiltonian…”

“…In the slow fields I ± s , the heteroclinic orbit Γ f has to be exponentially close to M ± 1,2 , and its evolution is determined by the slow (v, q, w, r) equations [5]. These are given by the slow reduced systems (SRS) of (2.2) 1 and (2.2) 2 , respectively.…”

“…The result of the theorem will follow by the Melnikov approach employed in [5], because this calculation will establish that…”

“…In particular, our results (Theorem 1.4 and 1.5) show that also the 2-component model exhibits pinned front and 2-front solutions. However, as will be clear later on, the dynamics of the 2-front solutions (and more general N -front solutions) are much richer in the 3-component model, see also [5,21]. Moreover, to analyze scattering and other more complex phenomena observed in [17], the analysis developed here for the full 3-component model will be required.…”

“…In [5,21,22], a detailed analysis of the existence [5], stability [21] and interaction [22] of localized structures for the homogeneous problem ((1.2) with γ(ξ) ≡ γ, constant) is given. Of the main results of these papers, three are needed for this paper:…”