# American Institute of Mathematical Sciences

May  2016, 21(3): 959-975. doi: 10.3934/dcdsb.2016.21.959

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

Received  August 2014 Revised  October 2015 Published  January 2015

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.
Citation: Shuangquan Xie, Theodore Kolokolnikov. Oscillations of many interfaces in the near-shadow regime of two-component reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 959-975. doi: 10.3934/dcdsb.2016.21.959
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