Oscillations of many interfaces in the near-shadow regime of two-component reaction-diffusion systems

We consider the general class of two-component reaction-diffusion systems on a finite domain that admit interface solutions in one of the components, and we study the dynamics of n interfaces in one dimension. In the limit where the second component has large diffusion, we fully characterize the possible behaviour of n interfaces. We show that after the transients die out, the motion of n interfaces is described by the motion of a single interface on the domain that is 1/n the size of the original domain. Depending on parameter regime and initial conditions, one of the following three outcomes results: (1) some interfaces collide; (2) all n interfaces reach a symmetric steady state; (3) all n interfaces oscillate indefinitely. In the latter case, the oscillations are described by a simple harmonic motion with even-numbered interfaces oscillating in phase while odd-numbered interfaces are oscillating in anti-phase. This extends a recent work by [McKay, Kolokolnikov, Muir, DCDS B(17), 2012] from two to any number of interfaces.